A functorial approach to monomorphism categories II: Indecomposables
Nan Gao
Julian Külshammer
Sondre Kvamme
Chrysostomos Psaroudakis

Nan Gao

Departement of Mathematics, Shanghai University

Shanghai 200444, PR China

Julian Külshammer

Department of Mathematics
Uppsala University
Box 480
75106 Uppsala,
Sweden<EMAIL_ADDRESS>
Chrysostomos Psaroudakis

Department of Mathematics
Aristotle University of Thessaloniki
Thessaloniki, 54124, Greece<EMAIL_ADDRESS>
Sondre Kvamme

Department of Mathematical Sciences, Norwegian University of Science and Technology

7491 Trondheim, Norway

We investigate the (separated) monomorphism category $\operatorname{mono}(Q,\Lambda)$ of a quiver over an Artin algebra $\Lambda$. We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from $\overline{\operatorname{mono}}(Q,\Lambda)$ to $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$, where  $\operatorname{mod}\Lambda$ is the category of finitely generated modules and $\overline{\operatorname{mod}}\, \Lambda$ and $\overline{\operatorname{mono}}(Q,\Lambda)$ denote the respective injectively stable categories. Furthermore, if $Q$ has at least one arrow, then we show that this is an equivalence if and only if $\Lambda$ is hereditary. In general, the epivalence induces a bijection between indecomposable objects in $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$ and non-injective indecomposable objects in $\operatorname{mono}(Q,\Lambda)$, and we show that the generalized Mimo-construction, an explicit minimal right approximation into $\operatorname{mono}{(Q,\Lambda)}$, gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of a radical-square-zero Nakayama algebra, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length $3$ with the same residue field.

The main tool to prove these results is the language of a free monad of an exact endofunctor on an abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras $H$ which were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and  Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the $\iota$quivers algebras which were introduced by Lu and Wang to realize $\iota$quantum groups via semi-derived Hall algebras.

§ INTRODUCTION

The fundamental theorem of finitely generated abelian groups is one of the oldest classification results in algebra. It was implicitly stated by Kronecker in <cit.>, and later more explicitly by Frobenius and Stickelberger in <cit.>. After its proof people started studying subgroups of finite abelian groups, with work by Miller [65, 66], Hilton [29], and Birkhoff [9]. In this case one isn't only interested in the abstract groups, but also in their embedding. The classification of all such embeddings can be reduced to determining all indecomposable embeddings into a finitely generated $\mathbb{Z}/(p^n)$-module, for $p$ a prime. The difficulty of this problem increases with $n$, with $n=5$ only first determined in [78] in 1999. The case $n=6$ is an open problem, called the Birkhoff problem after the paper [9]. For $n\geq 7$ the problem is controlled wild and there is no hope for a classification [74].

The study of submodules was revitalized by work of Ringel and Schmidmeier in a series of papers [74, 76, 75]. There they considered submodule categories of Artin algebras, with particular emphasis on the truncated polynomial ring $\Bbbk[x]/(x^n)$. These categories have connections to Hall algebras [84],  Littlewood–Richardson tableaux [46, 47, 83], metabelian groups [81], preprojective algebras [79], singularity categories and weighted projective lines [32, 41], and valuated groups [3, 77].

More general systems of submodules, called monomorphism categories, have also been studied in parallel, including their representation type [70, 85] and their Auslander–Reiten quivers [67, 89]. They have connections to cotorsion pairs [18, 31], Gorenstein homological algebra [14, 17, 61, 89, 90], parabolic vector bundles [68], weighted projective lines [86], and toplogical data analysis [8].

In this paper we make progress towards describing the indecomposable objects for any monomorphism category. As far as we know, our first main result (Theorem <ref>) is new even for classical submodule categories. To state it, we first explain the general setup. Let $\mathcal{B}$ be an abelian category and $Q$ a finite acyclic quiver with vertices $Q_0$ and arrows $Q_1$. The category of representation of $Q$ in $\mathcal{B}$, denoted $\operatorname{rep}(Q,\mathcal{B})$, is the category of functors $Q\to \mathcal{B}$, where $Q$ is considered as a category in the natural way. Explicitly, the objects are tuples $(B_\mathtt{i}, B_\alpha)_{\mathtt{i}\in Q_0, \alpha\in Q_1}$ where $B_{\mathtt{i}}$ is an object of $\mathcal{B}$ for each vertex $\mathtt{i}$ and $B_\alpha\colon B_{\mathtt{i}}\to B_{\mathtt{j}}$ is a morphism for each arrow $\alpha\colon\mathtt{i}\to \mathtt{j}$. The (separated) monomorphism category, denoted $\operatorname{mono}(Q,\mathcal{B})$, is a full subcategory of $\operatorname{rep}(Q,\mathcal{B})$. It consists of all representations for which

\[
B_{\mathtt{i},\arrowin}\colon \bigoplus_{\substack{\alpha\in Q_1\\t(\alpha)=\mathtt{i}}} B_{s(\alpha)}\xrightarrow{(B_\alpha)_{\alpha}} B_\mathtt{i}
\]

is a monomorphism for all vertices $\mathtt{i}$, where $s(\alpha)$ and $t(\alpha)$ denote the source and target of the arrow $\alpha$. If $Q=(\mathtt{1}\to \mathtt{2})$ then we recover the classical submodule categories studied by Birkhoff, Ringel and Schmidmeier. In general, $\operatorname{mono}(Q,\mathcal{B})$ is an exact category, and if $\mathcal{B}=\operatorname{mod}\Lambda$ is the category of finitely generated right $\Lambda$-modules for an Artin algebra $\Lambda$, then it has almost split sequences [61, 76].

To describe the indecomposables in $\operatorname{mono}(Q,\mathcal{B})$ we construct an epivalence, i.e. a full and dense functor which reflects isomorphisms. Here we follow the terminology of [39]. Such functors were first considered by Auslander in [6] as a way of saying that the representation theories of two additive categories are essentially the same. He called such functors representation equivalence. They are also sometimes called detecting, e.g. see [7].

To state our result, let $\overline{\mathcal{B}}$ and $\overline{\operatorname{mono}}(Q,\mathcal{B})$ be the quotients of $\mathcal{B}$ and $\operatorname{mono}(Q,\mathcal{B})$ by the ideal of morphisms factoring through injective objects, where we use the exact structure of $\operatorname{mono}(Q,\mathcal{B})$ to define its injectives. For a description of the injectives see Section <ref> or (<ref>).

Let $Q$ be a finite acyclic quiver and $\mathcal{B}$ an abelian category with enough injectives. Then the canonical functor

\[
\overline{\operatorname{mono}}(Q,\mathcal{B})\to \operatorname{rep}(Q,\overline{\mathcal{B}}).
\]

is an epivalence. Furthermore, if $Q$ has at least one arrow, then the functor is an equivalence if and only if $\mathcal{B}$ is hereditary.

This recovers the homology functor for perfect differential $kQ$-modules in <cit.> and (the dual of) the equivalence in <cit.> as special cases. In general, it gives a bijection between the indecomposable objects in $\overline{\operatorname{mono}}(Q,\mathcal{B})$ and in $\operatorname{rep}(Q,\overline{\mathcal{B}})$. If $\mathcal{B}$ is the module category of an Artin algebra, then these are further in bijection with the non-injective indecomposable objects in  $\operatorname{mono}(Q,\mathcal{B})$. From this we can deduce that a stable equivalence between two Artin algebras induces a bijection between the non-injective indecomposable objects in their corresponding monomorphism categories. We discuss consequences of this in more detail in Subsection <ref>.

Theorem <ref> is particularly useful for comparing local uniserial rings of finite Loewy length, such as $\mathbb{Z}/(p^n)$ and $\mathbbm{k}[x]/(x^n)$. On the one hand, there are several results on the representation theory of monomorphism categories over $\mathbbm{k}[x]/(x^n)$. For example,  the representation type of $\mono(\mathbb{A}_m,\operatorname{mod}\mathbbm{k}[x]/(x^n))$ for

\[
\mathbb{A}_m=\mathtt{1}\to \mathtt{2}\to \cdots \to \mathtt{m}
\]

was given in [85], and the classification of indecomposables and  the Auslander–Reiten quiver  in the representation-finite and tame cases when $m\leq 3$ were given in [67, 75]. On the other hand, there haven't been many new results for $\Lambda=\mathbb{Z}/(p^n)$, which might be surprising because of the similarity between $\mathbb{Z}/(p^n)$ and $\mathbbm{k}[x]/(x^n)$, both being local uniserial rings of Loewy length $n$. However, it can be explained by the lack of certain tools like covering theory. As a prominent example, the Auslander–Reiten quiver of $\operatorname{mono}(\mathbb{A}_2,\operatorname{mod}\mathbbm{k}[x]/(x^6))$ was determined in [75] using covering theory. The analogous question for $\operatorname{mono}(\mathbb{A}_2,\operatorname{mod}\mathbb{Z}/(p^6))$ is the Birkhoff problem, which is still open.

In general there is a hope that the representation theory of $\operatorname{mono}(Q,\operatorname{mod}\mathbbm{k}[x]/(x^n))$ and
$\operatorname{mono}(Q,\operatorname{mod}\mathbb{Z}(p^n))$ is similar. The following result goes a long way towards confirming this when $n\leq 3$. It is proved using Theorem <ref> and the fact that there is a stable equivalence

\[
\overline{\operatorname{mod}}\, \mathbb{F}_p[x]/(x^n)\cong \overline{\operatorname{mod}}\,\mathbb{Z}/(p^n)
\]

in this case. Here $\mathbb{F}_p$ denotes the finite field with $p$ elements.

Let $Q$ be a finite acyclic quiver and $n$ an integer less than or equal to $3$. Then, there exists a bijection which preserves partition vectors between indecomposable objects in $\mono(Q,\operatorname{mod}\mathbb{F}_p[x]/(x^n))$ and in $\mono(Q,\operatorname{mod}\mathbb{Z}/(p^n))$.

By partition vector we mean the following: For a representation $(M_\mathtt{i},M_\alpha)$ over $R=\mathbb{F}_p[x]/(x^n)$ or $R=\mathbb{Z}/(p^n)$ each $M_\mathtt{i}$ can be written as $M_{\mathtt{i}}=R/\mathfrak{m}^{n_1}\oplus R/\mathfrak{m}^{n_2} \oplus \cdots \oplus R/\mathfrak{m}^{n_k}$ where $\mathfrak{m}$ is the maximal ideal of $R$ and $n_1\geq n_2\geq \cdots \geq n_k$. The associated sequences of numbers $(n_1,n_2,\cdots, n_k)$ is the partition of $M_\mathtt{i}$. Doing this for each vertex $\mathtt{i}$ gives rise to the partition vector of the representation $(M_\mathtt{i},M_\alpha)$.

Another application of Theorem <ref> is a Gabriel-style classification of representation-finite monomorphism categories over $\radoperator^2$-zero Nakayama Artin algebras. Here $m$ is the number of simple $\Lambda$-modules and $t$ is the number of non-injective simple $\Lambda$-modules.

Let $Q$ be a finite connected acyclic quiver and let $\Lambda$ be a non-semisimple $\radoperator^2$-zero Nakayama Artin algebra. Then $\mono_Q(\Lambda)$ is of finite type if and only if $Q$ is Dynkin. In this case, the number of indecomposable objects is $m \cdot |Q_0|+t \cdot |\Phi^+|$, where $|\Phi^+|$ denotes the set of positive roots corresponding to the Dynkin type.

We don't assume our algebra to be linear over a field. In particular, the result applies to $\mathbb{Z}/(p^2)$.

Assume $\mathcal{B}$ is module category of an Artin algebra. Then Theorem <ref> provides a bijection.

\begin{align}
\renewcommand{\arraystretch}{1.8}
\begin{array}{ccc}
\renewcommand{\arraystretch}{1.1}
\begin{Bmatrix}
\text{Isomorphism classes of} \\
\text{indecomposable non-injective} \\
\text{objects in $\mono(Q,\mathcal{B})$}
\end{Bmatrix}
\xrightarrow{\cong}
\renewcommand{\arraystretch}{1.2}
\begin{Bmatrix}
\text{Isomorphism classes of} \\
\text{indecomposable objects} \\
\text{in $\rep (Q,\overline{\mathcal{B}})$}
\end{Bmatrix}
\end{array}\label{BijectionFromRepEquiv}
\renewcommand{\arraystretch}{1}
\end{align}

given by the functor $\operatorname{mono}(Q,\mathcal{B})\to \overline{\operatorname{mono}}(Q,\mathcal{B})\to \operatorname{rep}(Q,\overline{\mathcal{B}})$. In many cases, $\rep (Q,\overline{\mathcal{B}})$ is easier to study than $\operatorname{mono}(Q,\mathcal{B})$. For example, if $\mathcal{B}$ is the module category of a $\radoperator^2$-zero Nakayama Artin algebra, then $\overline{\mathcal{B}}$ is just the module category of a product of skew fields, and hence $\operatorname{rep}(Q,\overline{\mathcal{B}})$ can be computed using classical methods. This is how Theorem <ref> is shown.

To make best use of the bijection (<ref>), we would like to construct its inverse explicitly, so that we can obtain a description of the indecomposables in $\operatorname{mono}(Q,\mathcal{B})$ from the ones in $\operatorname{rep}(Q,\overline{\mathcal{B}})$. We explain the procedure to do this. Let $(B_\mathtt{i},B_\alpha)$ be an object in $\operatorname{rep}(Q,\overline{\mathcal{B}})$. For each vertex $\mathtt{i}$, choose an object $\widehat{B}_{\mathtt{i}}$ in $\mathcal{B}$ with no nonzero injective summands and which is isomorphic to $B_\mathtt{i}$ in $\overline{\mathcal{B}}$. For each arrow $\alpha\colon \mathtt{i}\to \mathtt{j}$, choose a lift $\widehat{B}_\alpha\colon \widehat{B}_{\mathtt{i}}\to \widehat{B}_\mathtt{j}$ of $B_\alpha$ to $\mathcal{B}$. This gives a (non-unique) representation $(\widehat{B}_\mathtt{i},\widehat{B}_\alpha)$ in $\operatorname{rep}(Q,\mathcal{B})$. Next, take the minimal right $\mono(Q,\mathcal{B})$-approximation of $(\widehat{B}_\mathtt{i},\widehat{B}_\alpha)$. This is denoted $\Mimo(\widehat{B}_\mathtt{i},\widehat{B}_\alpha)$ and called the Mimo-construction. It was introduced in [76] for $\mathbb{A}_2$ and generalized in [61] to finite acyclic quivers. For an explicit formula of it see Example <ref> or <cit.>.

Let $Q$ be a finite acyclic quiver and $\mathcal{B}=\operatorname{mod}\Lambda$ for an Artin algebra $\Lambda$. The association $(B_\mathtt{i},B_\alpha)\mapsto \Mimo (\widehat{B}_\mathtt{i},\widehat{B}_\alpha)$ above gives an inverse to (<ref>).

This theorem holds more generally if $\mathcal{B}$ has injective envelopes and is noetherian or artinian or locally noetherian.

To complete the picture, we give a description of the indecomposable injective objects in $\operatorname{mono}(Q,\mathcal{B})$. They are precisely given by
$f_!(J(\mathtt{i}))=(f_!(J(\mathtt{i}))_{\mathtt{j}},f_!(J(\mathtt{i}))_\alpha)$ where $J$ is indecomposable injective in $\mathcal{B}$ and

\begin{equation}\label{DescriptionInjectives}
f_!(J(\mathtt{i}))_\mathtt{k}=\bigoplus_{\substack{p\in Q_{\geq 0}\\ s(p)=\mathtt{i},t(p)=\mathtt{k}}}J \quad \text{and} \quad f_!(B(\mathtt{i}))_\alpha\colon \bigoplus_{\substack{p\in Q_{\geq 0}\\ s(p)=\mathtt{i},t(p)=\mathtt{k}}}J \to \bigoplus_{\substack{p\in Q_{\geq 0}\\ s(p)=\mathtt{i},t(p)=\mathtt{l}}}J
\end{equation}

for an arrow $\alpha\colon \mathtt{k}\to \mathtt{l}$, where $f_!(J(\mathtt{i}))_\alpha$ is induced by the identity map $J\xrightarrow{1} J$ between the components indexed by paths $p$ and $\alpha p$. We use this and Theorem <ref> to give an explicit description of the indecomposable objects in monomorphisms categories for linearly oriented $\mathbb{A}_n$-quivers, for a non-linearly oriented $\mathbb{A}_4$-quiver,  and for the Kronecker quiver, see Subsections <ref> and <ref>.

In most of the proofs we use the more abstract language of monads and Eilenberg–Moore categories, similar to [25]. This is to avoid the technical combinatorics arising from quiver representations, e.g. from the Mimo-construction and $f_!$ above. To see how they relate, consider the endofunctor

\[
X\colon \mathcal{C}\to \mathcal{C} \quad X(B_\mathtt{i})_{\mathtt{i}\in Q_0}=\left(\bigoplus_{\alpha\in Q_1,t(\alpha)=\mathtt{j}}B_{s(\alpha)}\right)_{\mathtt{j}\in Q_0}
\]

on $\mathcal{C}=\prod_{\mathtt{i}\in Q_0}\mathcal{B}$. The data of a representation of $Q$ in $\mathcal{B}$ is equivalent to an object $C\in \mathcal{C}$ and a morphism $X(C)\to C$. Furthermore, the representation lies in the monomorphism category if and only if $X(C)\to C$ is a monomorphism. Similarly, the data of a representation of $Q$ in $\overline{\mathcal{B}}$ is equivalent to an object $C\in \overline{\mathcal{C}}$ and a morphism $X(C)\to C$ in $\overline{\mathcal{C}}$. It follows from this that there are equivalences

\[
\operatorname{rep}(Q,\mathcal{B})\cong \mathcal{C}^{T(X)} \quad \text{and} \quad \Mono(X)\cong \operatorname{Mono}(Q,\mathcal{B}) \quad \text{and} \quad \operatorname{rep}(Q,\overline{\mathcal{B}})\cong \overline{\mathcal{C}}{}^{T(X)}
\]

where $\mathcal{C}^{T(X)}$ and $\overline{\mathcal{C}}{}^{T(X)}$ denote the Eilenberg–Moore categories of the free monad $T(X)$ on $\mathcal{C}$ and $\overline{\mathcal{C}}$, respectively, and $\Mono(X)$ is the full subcategory of $\mathcal{C}^{T(X)}$ where $X(C)\to C$ is a monomorphism. In this language several of the constructions become easier and more conceptual. For example, $f_!$ is the left adjoint of the forgetful functor $f^*\colon \mathcal{C}^{T(X)}\to \mathcal{C}$ from the Eilenberg–Moore category, and the Mimo-construction, given by the complicated formula above, is just obtained by taking a particular pushout, see Definitions <ref> and <ref>.  Our results hold for any exact, locally nilpotent endofunctor $X$ which preserves injectives on an abelian category. The constructions and proofs use the Eilenberg–Moore category of its free monad as illustrated above.

Since we are working in a more general setting, our result also applies to generalizations of quiver representations, such as representations of modulations, see Example <ref>. They are known under several different names in the literature (and with varying hypothesis), such as representations of pro-species of algebras in [48], representations of phyla in [25], representations over diagrams of abelian categories in [15], and twisted representations in [24]. Monomorphism categories of modulations have connections to Gorenstein homological algebra [48, 15], and to cotorsion pairs and model structures  [15]. An important class of examples are given by prospecies over selfinjective rings. More explicitly, given a quiver $Q$, one associates to each vertex $\mathtt{i}$ a selfinjective algebra $\Lambda_\mathtt{i}$, and to each arrow $\alpha\colon \mathtt{i}\to \mathtt{j}$ a $\Lambda_\mathtt{i}$-$\Lambda_\mathtt{j}$-bimodule $M_\alpha$ which is projective as left $\Lambda_\mathtt{i}$-module and as right $\Lambda_\mathtt{j}$-module. It turns out that the monomorphism category of the corresponding modulation is equal to the category of Gorenstein projectives modules over the tensor algebra $T(M)$ of $M=\bigoplus_{\alpha\in Q_1}M_\alpha$ by $\Lambda=\prod_{\mathtt{i}\in Q_0}\Lambda_{\mathtt{i}}$, and its stable category is equivalent to the singularity category of $T(M)$, see [48].

One of the most important classes of prospecies over selfinjective rings arises from the GLS-algebras $H=H(C,D,\Omega)$  introduced in [26]. Here $C=(c_{\mathtt{i},\mathtt{j}})$ is a symmetrizable Cartan matrix with symmetrizer $D=\operatorname{diag}(d_\mathtt{i})$ and orientation $\Omega$. We can apply our results to study the indecomposable non-projective Gorenstein projective $H$-modules, or equivalently the indecomposable objects in the singularity category of $H$. An algebra is called Cohen–Macaulay finite if it has finitely many finitely generated indecomposable Gorenstein-projective modules up to isomorphism.

Let $C=(c_{\mathtt{i},\mathtt{j}})_{\mathtt{i},\mathtt{j}\in I}$ be a symmetrizable Cartan matrix with symmetrizer $D=\operatorname{diag}(d_\mathtt{i}\mid \mathtt{i}\in I)$ and orientation $\Omega$. Assume $d_\mathtt{i}\leq 2$ for all $\mathtt{i}\in I$. Let $I'\subseteq I$ be the subset of all elements $\mathtt{i}$ for which $d_\mathtt{i}=2$, and let $C|_{I'\times I'}=(c_{\mathtt{i},\mathtt{j}})_{\mathtt{i},\mathtt{j}\in I'}$ be the corresponding submatrix of $C$. Then $H=H(C,D,\Omega)$ is Cohen-Macaulay finite if and only if  $C|_{I'\times I'}$ is Dynkin as a symmetric Cartan matrix. Furthermore, in this case there is a bijection between the positive roots of $C|_{I'\times I'}$ and the isomorphism classes of indecomposable objects in the singularity category of $H$.

More generally, we show that there is a bijection between isomorphism classes of indecomposable objects in the singularity category of $H$, and finite-dimensional indecomposable representations over the quiver determined by the Cartan matrix $C|_{I'\times I'}$ with orientation $\Omega|_{I'\times I'}$. For this no finiteness assumptions are necessary. It is a consequence of a more general result on modulations over $\radoperator^2$-zero cyclic Nakayama algebras, see Theorem <ref>. The bijection is obtained from an analogue of the epivalence in Theorem <ref> for modulations, and we get an explicit description of the indecomposable object in the singularity category associated to a given representation from an analogue of Theorem <ref>. The more general result also applies to modules over $\iota$quiver algebras studied in [59, 57, 58], see Subsection <ref>. There the category of Gorenstein projective modules plays an important role, in particular since it is used to realize $\iota$quantum groups via Hall algebras [59, 57], and since for Dynkin quivers it is equivalent to the category of projectives over the regular Nakajima–Keller–Scherotzke categories [58].

The structure of the paper is as follows. Section <ref> contains the necessary background on monads and exact categories, respectively. In Section <ref> we define the monomorphism category of the free monad on an endofunctor. The section is divided into four parts: In Subsection <ref>  we define the free monad of an endofunctor, and provide examples of it. In Subsection <ref> we restrict to the abelian case.  Subsection <ref> introduces the top functor and recalls its basic properties. Finally, Subsection <ref> introduces the key player of the paper, the monomorphism category. Section <ref> deals with injective objects and the existence of injective envelopes for the monomorphism category. In Section <ref> we prove Theorem <ref>. It starts with a discussion of contravariantly finiteness of the monomorphism category in Subsection <ref>, proceeds with the proof of the existence of the epivalence in Subsection <ref>, and finishes by discussing the hereditary case in Subsection <ref>. Section <ref> introduces the $\Mimo$ construction in our language. Subsection <ref> deals with the general case while Subsection <ref> makes the setup explicit in the case of modulations. Section <ref> contains the proof of Theorem <ref>, with a short discussion on maximal injective summands in the beginning. Section <ref> discusses applications to representations of quivers over Artin algebras. Subsection <ref> is on stable equivalences and the induced bijections between indecomposables in monomorphism categories. In Subsection <ref> we prove Theorem <ref> and discuss connections to other results in the literature. In Subsection <ref> we prove Theorem <ref>, and explicitly compute indecomposables for monomorphism categories over $\radoperator^2$-zero Nakayama algebras, using Theorem <ref>. In Subsection <ref> we compute the indecomposables in the monomorphism category of the Kronecker quiver over $\mathbbm{k}[x]/(x^2)$. In Section <ref> we apply our results to representations of modulations over $\radoperator^2$-zero cyclic Nakayama algebras to obtain Theorem <ref>. We then consider $\iota$quivers and modules over the GLS algebras in Subsection <ref>, and finish by proving Theorem <ref> from the introduction.

§ PRELIMINARIES

§.§ Notation

We ignore set-theoretic issues in this paper.
All categories are assume to be additive and idempotent-complete, and all functors are assumed to be additive. For functors $F\colon \mathcal{C}\to \tilde{\mathcal{C}}$ and $G\colon \tilde{\mathcal{C}}\to \mathcal{C}$ we write $F\dashv G$
to denote that $F$ is left adjoint to $G$. The unit and counit of the adjunction are denoted by $\eta\colon \Id \to GF$ and $\varepsilon\colon FG\to \Id$, respectively. For a ring $\Lambda$ we let $\Mod \Lambda$, respectively $\operatorname{mod}\Lambda$, denote the category of right $\Lambda$-modules, respectively finitely presented right $\Lambda$-modules. Throughout the paper $\mathbbm{k}$ is a commutative ring.

§.§ Monads and modules over monads

In this subsection we recall the definition of a monad and its Eilenberg–Moore category, depending on the context also called the category of algebras or the category of modules over the monad. Our main examples arise from modulations on quivers, see Example <ref>.

Let $\mathcal{C}$ be a category. A monad on $\mathcal{C}$ is a tuple $(T,\eta,\mu)$ where $T\colon \mathcal{C}\to \mathcal{C}$ is a functor and $\eta\colon \Id\to T$ and $\mu\colon T^2\to T$ are natural transformations such that the diagrams

\[\begin{tikzcd}
\end{tikzcd}
\quad \text{ and }
\begin{tikzcd}
\end{tikzcd}\]

commute.  By standard abuse of notation we sometimes denote a monad $(T,\eta,\mu)$ simply by $T$.

An important source of instances of monads are adjunctions: If $\eta$ and $\varepsilon$ denote the unit and counit of an adjunction $L\dashv R$, then the tuple $(R\circ L,\eta,R(\varepsilon_L))$
defines a monad. We refer to <cit.> for more details on this.

Conversely, monads give rise to adjunctions via the Eilenberg–Moore category, which we recall next. This is sometimes called the category of algebras or the category of modules over the monad in the literature.

Let $(T,\eta,\mu)$ be a monad on $\mathcal{C}$.

* A $T$-module is a pair $(M,h)$ where $M$ is an object in $\mathcal{C}$ and $h\colon T(M)\to M$ is a morphism  satisfying $h\circ \eta_M=1_M$ and $h\circ \mu_M = h\circ T(h)$.

* The Eilenberg–Moore category $\mathcal{C}^{T}$ of $T$ is the category whose objects are $T$-modules, and where a morphism $\map\colon (M,h)\to (M',h')$ between $T$-modules is given by a morphism $\map\colon M\to M'$ in $\mathcal{C}$ satisfying $\map\circ h=h'\circ T(\map)$.

* $f^*\colon \mathcal{C}^{T}\to \mathcal{C}$ denotes the forgetful functor given by $f^*(M,h)=M$ and $f^*(\map)=\map$.

* $f_!\colon \mathcal{C}\to \mathcal{C}^{T}$ denotes the functor given by $f_!(M)=(T(M),\mu_M)$ and $f_!(\map)=T(\map)$.

Let $T$ be a monad on $\mathcal{C}$. Then $f^*\colon \mathcal{C}^T\to \mathcal{C}$ is right adjoint to $f_!\colon \mathcal{C}\to \mathcal{C}^T$.

This follows from <cit.>.

We fix $\eta$ and $\varepsilon$ to be the unit and counit of the adjunction $f_!\dashv f^*$.
We finish by giving a sufficient criterion for the Eilenberg–Moore category to be abelian.

Let $\mathcal{C}$ be an abelian category, and let $(T,\eta,\mu)$ be a monad on $\mathcal{C}$. Assume that $T$ is a right exact functor. The following hold:

* The Eilenberg–Moore category $\mathcal{C}^T$ is abelian.

* A sequence $(M,h)\to (M',h')\to (M'',h'')$ in $\mathcal{C}^{T}$ is exact if and only if the sequence $M\to M'\to M''$ in $\mathcal{C}$ is exact.

§.§ Exact categories

Here we recall some basic properties of exact categories, in particular results on injective envelopes.

An exact category is an additive category $\mathcal{E}$ endowed with a class of kernel-cokernel pairs, called conflations, satisfying certain properties, see <cit.> or [13] for more details. If  $E_1\xrightarrow{i}E_2\xrightarrow{p}E_3$ is a conflation, then $i$ is called an inflation and $p$ is called a deflation.
Let $\mathcal{E}'$ be a full subcategory of an exact category $\mathcal{E}$. We say that $\mathcal{E}'$ is extension-closed if for any conflation $E_1\to E_2\to E_3$ with $E_1$ and $E_3$ in $\mathcal{E}'$, the middle term $E_2$ must be in $\mathcal{E}'$. In this case $\mathcal{E}'$ inherits an exact structure whose conflations are the conflations in $\mathcal{E}$ where all the terms are in $\mathcal{E}'$ see <cit.>.

An object $I$ in an exact category $\mathcal{E}$ is called $\emphbf{injective}$ if for any inflation $E\to E'$ the morphism $\operatorname{Hom}_{\mathcal{E}}(E',I)\to \operatorname{Hom}_{\mathcal{E}}(E,I)$ is surjective. The exact category $\mathcal{E}$ is said to have enough injectives if for any object $E$ in $\mathcal{E}$ there exists an inflation $E\to I$ with $I$ injective. In this case we let $\overline{\mathcal{E}}$ denote the quotient of $\mathcal{E}$ by the ideal of morphisms factoring through an injective object.

Let $g\colon E_1\to E_2$ be a morphism in $\mathcal{E}$. We say that $g$ is left minimal if any morphism $g'\colon E_2\to E_2$ satisfying $g'\circ g=g$ is an isomorphism. An injective envelope of an object $E$ in $\mathcal{E}$ is a left minimal inflation $i\colon E\to I$ where $I$ is injective. Note that an injective envelope of an object is unique up to isomorphism. The radical of $\mathcal{E}$ is the ideal defined by

\[
\operatorname{Rad}_{\mathcal{E}}(E,E')\coloneqq\{g\in \operatorname{Hom}_{\mathcal{E}}(E,E')\mid 1_{E'}-g\circ g'\text{ is invertible for all } g'\in \operatorname{Hom}_{\mathcal{E}}(E',E)\}
\]

for all $E,E'\in \mathcal{E}$. Recall that the radical is symmetric, and so we have the equality

\[
\operatorname{Rad}_{\mathcal{E}}(E,E')\coloneqq\{g\in \operatorname{Hom}_{\mathcal{E}}(E,E')\mid 1_E-g'\circ g\text{ is invertible for all } g'\in \operatorname{Hom}_{\mathcal{E}}(E',E)\}.
\]

We refer to <cit.> for more information on injective envelopes, and to <cit.> and [37] for more information on the radical. We only need the following results relating them. The proof of the second claim is based on the argument given in <cit.>.

Let $\mathcal{E}$ be a exact category, and let $E\in \mathcal{E}$. The following hold:

*  If $i\colon E\to I$ is an injective envelope, then $p\colon I\rightarrow \operatorname{coker}i$ is in the radical of $\mathcal{E}$.

*  Assume $\mathcal{E}$ is abelian with the natural exact structure. Furthermore, assume $\mathcal{E}$ has injective envelopes. Let $g\colon J\to E$ be a morphism in $\mathcal{E}$. If $J$ is injective and $E$ has no nonzero injective summands, then the inclusion $\operatorname{ker}g\to J$ is an injective envelope. In particular, $g$ is in the radical of $\mathcal{E}$.

To prove statement (<ref>) let $k\colon \operatorname{coker}i\to I$ be an arbitrary morphism. Then the endomorphism $k'=1_I-k\circ p$ satisfies $k'\circ i=i$. Since $i$ is an injective envelope, we must have that $k'$ is an isomorphism. Since $k$ was arbitrary, it follows that $p$ is in the radical of $\mathcal{C}$.

To prove (<ref>) let $u\colon \operatorname{ker}g\to E(\operatorname{ker}g)$ denote the injective envelope of $\operatorname{ker}g$. Then the monomorphism $i\colon\operatorname{ker}g\to J$ lifts to a split monomorphism  $E(\operatorname{ker}g)\to J$. Consider the commutative diagram where the lower row is a split exact sequence

\[
\begin{tikzcd}
\end{tikzcd}
\]

Since the left square commutes, we get an induced morphism $\operatorname{coker}i\to J'$. Since $J'$ is injective, we can lift this to a morphism $E\to J'$ such that the rightmost square in the diagram commutes. Since $J\to J'$ is a split epimorphism, the morphism $E\to J'$ is a split epimorphism. Therefore $J'$ must be a summand of both $J$ and $E$. But being a summand of $J$ implies that $J'$ is injective, and $E$ has no nonzero injective summands. Therefore $J'=0$. Hence $i$ is an injective envelope. Since $g$ factors through the cokernel of $i$, it must be in the radical by part (<ref>).

We finish by proving a uniqueness result on the decomposition of an object by a maximal injective summand. It holds for abelian categories with injective envelopes by the previous lemma. It also holds for the monomorphism category by Lemma <ref>.

Let $\mathcal{E}$ be an exact category, and let $E_1$ and $E_2$ be objects in $\mathcal{E}$. Assume that any morphism from an injective object to $E_1$ or $E_2$ is in the radical of $\mathcal{E}$. Let

\[
E_1\oplus I_1\xrightarrow{\cong} E_2\oplus I_2
\]

be an isomorphism where $I_1$ and $I_2$ are injective. Then the restrictions $E_1\to E_2$ and $I_1\to I_2$ are isomorphisms.

\[
\phi=\begin{pmatrix}
\phi_1&\phi_2\\
\phi_3&\phi_4
\end{pmatrix}\colon E_1\oplus I_1\to E_2\oplus I_2
\quad \text{and} \quad \psi=\begin{pmatrix}
\psi_1&\psi_2\\
\psi_3&\psi_4
\end{pmatrix}\colon E_2\oplus I_2\to E_1\oplus I_1
\]

denote the isomorphism and its inverse. Then we have that $\psi_3\circ \phi_2 + \psi_4\circ \phi_4=1_{I_1}$. By assumption, $\phi_2$ is in the radical of $\mathcal{E}$ . Hence, by definition of the radical the composite $\psi_4\circ \phi_4=1_{I_1}-\psi_3\circ \phi_2$ is an isomorphism. By a similar argument the composite $\phi_4\circ \psi_4$ is also an isomorphism. Hence $\phi_4\colon I_1\to I_2$ must itself be an isomorphism. Now consider the commutative diagram with split exact rows

\[
\begin{tikzcd}
0\arrow{r}&I_1\arrow{r}\arrow{d}{\phi_4}&E_1\oplus I_1\arrow{d}{\phi}\arrow{r}{}&E_1\arrow{r}\arrow{d}{\phi_1}&0\\
0\arrow{r}&I_2\arrow{r}&E_2\oplus I_2\arrow{r}{}&E_2\arrow{r}&0.
\end{tikzcd}
\]

Since the two leftmost vertical maps are isomorphisms, the map $\phi_1\colon E_1\to E_2$ must be an isomorphism. This proves the claim.

§ THE MONOMORPHISM CATEGORY OF THE FREE MONAD

Fix a $\mathbbm{k}$-linear additive category $\mathcal{C}$.  Throughout the section $X\colon \mathcal{C}\to \mathcal{C}$ is assumed to be a $\Bbbk$-linear functor which is locally nilpotent, i.e. for any object $M\in \mathcal{C}$ there exists an $n\geq 0$ such that $X^n(M)=0$. From Subsection <ref> we assume $\mathcal{C}$ is abelian and $X$ is exact, and from Section <ref> onwards we also assume that $X$ preserves injective objects. The assumptions on $X$ are introduced  to capture the essential properties of monads arising from representations of finite acyclic quivers in additive and abelian categories, see Example <ref>. This is done using the free monad on $X$, which we define. In addition to this we define the monomorphism category and the analogue of the top functor.

The assumption we make on $X$ differs from the ones in [25]. Several of the results in [25] rely on the existence of a relative Nakayama functor on the Eilenberg–Moore category of the free monad on $X$, and this is not necessary to assume in this paper. On the other hand, many of the proofs in [25] still go through. Another technical condition we use in [25] is a relative version of Nakayama's lemma, which says that if $M$ is non-zero, then there are no epimorphisms $X(M)\twoheadrightarrow M$, see <cit.>. Under the assumption that $X$ is locally nilpotent, this is automatically satisfied when $X$ preserves epimorphisms, and in particular if $X$ is an exact functor on an abelian category.

§.§ The free monad

The definition of the free monad mimics the construction of the path algebra of a quiver, and more generally the tensor algebra of a bimodule over an algebra.

The free monad on $X$ is the monad $(T(X),\eta,\mu)$ where $T(X)\colon \mathcal{C}\to \mathcal{C}$ is given by

\[
T(X)(M)=\coprod_{i\geq 0}X^i(M)
\]

and where $\eta\colon \Id\to T(X)$ is the canonical inclusion and $\mu\colon T(X)\circ T(X)\to T(X)$ is given componentwise by the canonical identification $X^iX^j(M)\xrightarrow{\cong}X^{i+j}(M)$.

Since $X$ is locally nilpotent, the coproduct $\coprod_{i\geq 0}X^i(M)$ is finite for each object $M\in \mathcal{C}$. We therefore identify it with the direct sum $\bigoplus_{i\geq 0}X^i(M)$.

In [25] it is assumed that $X$ preserves countable coproducts when defining the free monad. However this is only used to conclude that the canonical map $X(\coprod_{i\geq 0}X^i(M))\to \coprod_{i\geq 1}X^i(M)$ is an isomorphism, which follows here from the fact that since $X$ is locally nilpotent, the coproduct is finite and is therefore preserved by the additive functor $X$.

The Eilenberg–Moore category of a free monad $T(X)$ has an alternative simpler description. Let $(X\Downarrow \Id_{\mathcal{C}})$ be the category whose objects are pairs $(M,h_1)$ where $M\in \mathcal{C}$ and $h_1\colon X(M)\to M$ is a morphism, and where a morphism $\map\colon (M,h_1)\to (M',h_1')$ in $(X\Downarrow \Id_{\mathcal{C}})$ is a morphism $\map\colon M\to M'$ in $\mathcal{C}$ satisfying $\map\circ h_1= h_1'\circ X(\map)$.  Note that we have a functor

\begin{align*}
\mathcal{C}^{T(X)}\to (X\Downarrow \Id_{\mathcal{C}}) \quad (\bigoplus_{i\geq 0}X^i(M)\xrightarrow{h}M)\mapsto (M,h_1)
\end{align*}

where $h_1\colon X(M)\to M$ is the restriction of $h$ to $X(M)$.

The functor $\mathcal{C}^{T(X)}\to (X\Downarrow \Id_{\mathcal{C}})$ above is an isomorphism of categories.

This follows from the proof of Lemma 5.18 in [25].

We will identify the categories $\mathcal{C}^{T(X)}$ and $(X\Downarrow \Id_{\mathcal{C}})$. We use sans serif typestyle $\module{M},\module{N},\dots$ to denote objects in $\mathcal{C}^{T(X)}$, so that the same letter without the sans serif typestyle denotes the underlying object in $\mathcal{C}$, i.e. $M=f^*(\module{M})$ and $N=f^*(\module{N})$. The induced morphism $X(M)\to M$ is then denoted by $h_{\module{M}}$ and called the structure map of $\module{M}$.

In Section <ref> we saw that the forgetful functor $f^*\colon \mathcal{C}^{T(X)}\to \mathcal{C}$ has a left adjoint $f_!\colon \mathcal{C}\to \mathcal{C}^{T(X)}$.  It is given by $f_!(M)=(\bigoplus_{i\geq 0}X^i(M),\iota_M)$ where the structure map $\iota_M$ is the canonical inclusion

\begin{equation}\label{StructureMapRelProj}
\iota_M\colon \bigoplus_{i\geq 1}X^i(M)\to \bigoplus_{i\geq 0}X^i(M)
\end{equation}

Any summand of an object of the form $f_!(M)$ is called relative projective. If $\mathcal{C}=\operatorname{Mod} \Lambda$ for a semisimple Artin algebra $\Lambda$, then the relative projectives coincide with the projectives in $\mathcal{C}^{T(X)}$, cf. Proposition <ref>. The following lemma implies that the relative projectives behave as projectives for objectwise split epimorphisms in $\mathcal{C}^{T(X)}$.

Let $N$ be an object in $\mathcal{C}$ and let $g\colon \module{M}\to \module{M}'$ be a morphism in $\mathcal{C}^{T(X)}$. Assume $f^*(g)$ is a split epimorphism. Then any morphism $f_!(N)\to \module{M}'$ factors through $g$.

By the adjunction $f_!\dashv f^*$, the map

\[
\operatorname{Hom}_{\mathcal{C}^{T(X)}}(f_!(N),g)\colon \operatorname{Hom}_{\mathcal{C}^{T(X)}}(f_!(N),\module{M})\to \operatorname{Hom}_{\mathcal{C}^{T(X)}}(f_!(N),\module{M}')
\]

is isomorphic to the map

\[
\operatorname{Hom}_{\mathcal{C}}(N,f^*(g))\colon \operatorname{Hom}_{\mathcal{C}}(N,f^*(\module{M}))\to \operatorname{Hom}_{\mathcal{C}}(N,f^*(\module{M}')).
\]

The latter must be an epimorphism since $f^*(g)$ is a split epimorphism. This proves the claim.

A sequence $\module{M}_1\to \module{M_2}\to \module{M}_3$ in $\mathcal{C}^{T(X)}$ is called objectwise split exact if the sequence

\[
f^*(\module{M}_1)\to f^*(\module{M_2})\to f^*(\module{M}_3)
\]

is split exact in $\mathcal{C}$. We show that such a sequence is an exact sequence in $\mathcal{C}^{T(X)}$, i.e. a kernel-cokernel pair. Note that this holds even though $\mathcal{C}^{T(X)}$ is not assumed to be abelian. We also show that the free monad $T(X)$ has relative global dimension one, i.e. any object in $\mathcal{C}^{T(X)}$ has a objectwise split resolution of length one by relative projective objects. Here the structure morphism of $f_!(M)$ is considered as a morphism $\iota_M\colon f_!X(M)\to f_!(M)$ in $\mathcal{C}^{T(X)}$.

The following hold:

*  Any objectwise split exact sequence is an exact sequence.

*  For each $\module{M}\in \mathcal{C}^{T(X)}$ the sequence

\[
0\to f_!X(M)\xrightarrow{\iota_M-f_!(h_\module{M})}f_!(M)\xrightarrow{\varepsilon_{\module{M}}}\module{M}\to 0
\]

is objectwise split exact.

Let $\module{M}_1\to \module{M}_2\to \module{M}_3$ be a objectwise split exact sequence. By <cit.> the map $\module{M}_1\to \module{M_2}$ is a kernel of $\module{M}_2\to \module{M}_3$ since $f^*(\module{M}_1)\to f^*(\module{M}_2)$ is a kernel of $f^*(\module{M}_2)\to f^*(\module{M}_3)$. Since $T(X)$ preserves split exact sequences, <cit.> implies that $\module{M}_2\to \module{M}_3$ is a cokernel of $\module{M}_1\to \module{M}_2$ since  $f^*(\module{M}_1)\to f^*(\module{M}_2)\to f^*(\module{M}_3)$ is split exact. This proves (<ref>).

The fact that

\[
0\to f^*f_!X(M)\xrightarrow{f^*(\iota_M-f_!(h_\module{M}))}f^*f_!(M)\xrightarrow{f^*(\varepsilon_{\module{M}})}f^*(\module{M})\to 0
\]

is a split exact sequence can be shown in a similar way as in the proof of Lemma 6.17 in [25]. Note that the infinite sums in that proof are finite in our case, since $X$ is locally nilpotent.

Next we show that the relative projectives are always isomorphic to objects of the form $f_!(M)$. They can also be characterized by their structure morphism being a split monomorphism.

Let $\module{M}\in \mathcal{C}^{T(X)}$. The following are equivalent:

*  $\module{M}$ is relative projective.

*  $h_{\module{M}}\colon X(M)\to M$ is a split monomorphism in $\mathcal{C}$.

*  $\module{M}$ is isomorphic to an object of the form $f_!(N')$ for $N'\in \mathcal{C}$.

Clearly the class of objects $\module{M}\in \mathcal{C}^{T(X)}$ for which $h_{\module{M}}$ is a split monomorphism is closed under direct summands. Since $\iota_N$ is a split monomorphism for all $N\in \mathcal{C}$, this shows (<ref>)$\Rightarrow$(<ref>).

For (<ref>)$\Rightarrow$(<ref>), consider the split exact sequence

\[
0\to X(M)\xrightarrow{h_{\module{M}}}M\to N'\to 0
\]

in $\mathcal{C}$ and choose a section $i\colon N'\to M$. Then for each $m\geq 0$ the morphism $X^m(i)\colon X^n(N')\to X^n(M)$ is a section to the split exact sequence

\[
0\to X^{m+1}(M)\xrightarrow{X^m(h_{\module{M}})}X^m(M)\to X^m(N')\to 0.
\]

Using the $X^m(i)$'s, we get isomorphisms

\[
M\cong N'\oplus X(M)\cong N'\oplus X(N')\oplus X^2(M)\cong \cdots \cong N'\oplus X(N')\oplus \cdots \oplus X^n(N')
\]

where $n$ is some integer with $X^{n+1}(M)=0$. Since these isomorphisms commute with the structure morphisms of $\module{M}$ and $f_!(N')$, we get an isomorphism $\module{M}\cong f_!(N')$.  The remaining direction (<ref>)$\Rightarrow$(<ref>) is obvious.

An object $M\in\mathcal{C}$ is indecomposable if and only if $f_!(M)$ is indecomposable.

By Proposition <ref> any summand of $f_!(M)$ is of the form $f_!(N)$. Taking the cokernel of their structure morphisms, we see that this is equivalent to $N$ being a summand of $M$. Hence $f_!(M)$ is indecomposable if and only if $M$ is indecomposable.

Let $Q=(Q_0,Q_1)$ be a quiver, where $Q_0$ and $Q_1$ denote the set of vertices and arrows of $Q$, respectively. A $\mathbbm{k}$-modulation $\mathfrak{B}$ of $Q$ is an assignment of a $\mathbbm{k}$-linear additive category $\mathcal{B}_\mathtt{i}$ to each vertex $\mathtt{i}\in Q_0$ and a $\mathbbm{k}$-linear functor $F_\alpha\colon \mathcal{B}_\mathtt{i}\to \mathcal{B}_\mathtt{j}$ to each arrow $\alpha\colon \mathtt{i}\to \mathtt{j}$. Associated to a modulation $\mathfrak{B}$ we have the category $\rep \mathfrak{B}$ of $\mathfrak{B}$-representations. Explicitly, its objects are collections $(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}$ where $B_\mathtt{i}$ is an object of $\mathcal{B}_\mathtt{i}$ and $B_\alpha\colon F_\alpha(B_\mathtt{i})\to B_\mathtt{j}$ is a morphism in $\mathcal{B}_\mathtt{j}$. A morphism of $\mathfrak{B}$-representations $(B_\mathtt{i},B_\alpha)\to (B'_\mathtt{i},B'_\alpha)$ is a collection of morphism $(\varphi_\mathtt{i}\colon B_\mathtt{i}\to B'_\mathtt{i})_{\mathtt{i}\in Q_0}$ such that the following diagram commutes for every $\alpha\in Q_1$:

\[
\begin{tikzcd}
F_\alpha(B_\mathtt{i})\arrow{r}{B_\alpha}\arrow{d}{F_\alpha(\varphi_\mathtt{i})} &B_\mathtt{j}\arrow{d}{\varphi_\mathtt{j}}\\
F_\alpha(B'_\mathtt{i})\arrow{r}{B'_\alpha} &B'_\mathtt{j}
\end{tikzcd}
\]

Note that the category of representations can be identified with the sections of the Grothendieck construction of a certain functor obtained from the $\mathbbm{k}$-modulation, see Remark 4.6 in [25].

Assume $Q$ is finite and acyclic and set $\mathcal{C}=\prod_{\mathtt{i}\in Q_0}\mathcal{B}_\mathtt{i}$. Define the functor

\begin{align}\label{DefXModulation}
& X\colon \mathcal{C}\to \mathcal{C}, \quad  \quad (B_\mathtt{i})_{\mathtt{i}\in Q_0}\mapsto (\bigoplus_{\substack{\alpha\in Q_1\\t(\alpha)=\mathtt{i}}}F_{\alpha}(B_{s(\alpha)}))_{\mathtt{i}\in Q_0}.
\end{align}

By the assumptions on $Q$ it follows that $X$ is nilpotent, i.e. $X^n=0$ for some $n\gg 0$. The category $\rep \mathfrak{B}$ of $\mathfrak{B}$-representations can be identified with Eilenberg–Moore category $\mathcal{C}^{T(X)}$ of the free monad $T(X)$ on $\mathcal{C}$. We  describe the Eilenberg–Moore adjunction $f_!\dashv f^*$ explicitly, following <cit.>. The forgetful functor is given by

\[
f^*\colon \rep \mathfrak{B}\to \mathcal{C}, \quad \quad f^*(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}=(B_\mathtt{i})_{\mathtt{i}\in Q_0}.
\]

For $f_!$, we need some notation. Let $Q_{\geq 0}$ denote the collection of all paths in $Q$, and for $p\in Q_{\geq 0}$ let $s(p)$ and $t(p)$ denote its source and target, respectively. If $p=\alpha_n\alpha_{n-1}\cdots \alpha_1$ set

\begin{align*}
F_p&\coloneqq F_{\alpha_n}\circ F_{\alpha_{n-1}}\circ \dots\circ F_{\alpha_1}\colon \mathcal{B}_{s(p)}\to \mathcal{B}_{t(p)}
\end{align*}

The functor $f_!$ applied to $B=(B_\mathtt{i})_{\mathtt{i}\in Q_0}\in \mathcal{C}$ is then given by

\begin{align}\label{Formula:f_!}
f_!(B)&=\left(\bigoplus_{\substack{p\in Q_{\geq 0}\\t(p)=\mathtt{i}}}F_p(B_{s(p)}), f_!(B)_\alpha\right)_{\substack{\mathtt{i}\in Q_0\\\alpha\in Q_1}}
\end{align}

\begin{align*}
f_!(B)_\alpha&\colon \bigoplus_{\substack{p\in Q_{\geq 0}\\t(p)=s(\alpha)}} F_\alpha F_p(B_{s(p)})\to \bigoplus_{\substack{q\in Q_{\geq 0}
\\t(q)=t(\alpha)}}F_q(B_{s(q)})
\end{align*}

is induced by the identity map $F_\alpha F_p(B_{s(p)})\xrightarrow{1}F_q(B_{s(q)})$ for $q=\alpha p$. For more details on this construction see <cit.>.

Example <ref> can be extended to a modulation $\mathfrak{B}$ on any quiver $Q$ (not necessarily finite or acylic). To ensure that the functor $X$ given by (<ref>) is locally nilpotent, we need to restrict to the subcategory $\mathcal{C}$ consisting of objects $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}$ of $\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ satisfying:

(A) There exists an integer $n\geq 0$ (depending on $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}$) such that $B_{s(p)}= 0$ for any path $p$ of length greater than $n$.

Clearly, $X$ restricts to a locally nilpotent endofunctor on $\mathcal{C}$. Since $\mathcal{C}=\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ if $Q$ is finite and acyclic, this generalizes Example <ref>. Note that $\mathcal{C}$ is nonzero if and only if $Q$ has a sink vertex.

As a special case of the previous examples we can consider the modulation $\mathfrak{B}$ given by $\mathcal{B}_\mathtt{i}=\mathcal{B}$ for all $\mathtt{i}\in Q_0$ and $F_\alpha=\operatorname{Id}$ for all $\alpha\in Q_1$. In this case, the category of representation of $\mathfrak{B}$ is just the category $\rep (Q,\mathcal{B})$ of representations of $Q$ in $\mathcal{B}$, whose objects are collections $(B_\mathtt{i},B_{\alpha})_{\mathtt{i}\in Q_0,\alpha\in Q_1}$ of objects $B_\mathtt{i}$ and morphisms $B_{\alpha}\colon B_{s(\alpha)}\to B_{t(\alpha)}$ in $\mathcal{B}$. Note that $\rep (Q,\mathcal{B})$ is equivalent to the category $\mathcal{B}^{\mathbbm{k} Q}$ of $\mathbbm{k}$-linear functors $\mathbbm{k} Q\to \mathcal{B}$, where $Q$ is considered as a category with objects in $Q_0$ and with morphisms given by $Q_{\geq 0}$, where $\mathbbm{k} Q$ denotes the $\mathbbm{k}$-linearization of the category $Q$.

§.§ The free monad on an abelian category.

In this subsection $\mathcal{C}$ is assumed to be abelian and $X\colon \mathcal{C}\to \mathcal{C}$ is assumed to be exact. We have the following basic results for $T(X)$ and its Eilenberg–Moore category.

The following hold:

*  The functor $T(X)\colon \mathcal{C}\to \mathcal{C}$ is exact

*  The category $\mathcal{C}^{T(X)}$ is abelian.

*  The functors $f_!\colon \mathcal{C}\to \mathcal{C}^{T(X)}$ and $f^*\colon \mathcal{C}^{T(X)}\to \mathcal{C}$ are exact.

*  If $X$ preserves injectives, then $T(X)$ preserves injectives.

Since $X$ is exact and taking coproducts is a right exact functor, $T(X)$ must be right exact. Since $X$ is locally nilpotent, we have an isomorphism $T(X)\cong \prod_{i\geq 0}X^i$. Since taking products is a left exact functor, $T(X)$ must also be left exact. Hence, $T(X)$ is exact. The fact that $\mathcal{C}^{T(X)}$ is abelian and $f^*$ is exact follows from Proposition <ref>. Now the exactness of $f_!$ follows from the exactness of $T(X)$ and the description of exact sequences in $\mathcal{C}^{T(X)}$ from Proposition <ref>. Finally, if $I\in \mathcal{C}$ is injective and $X$ preserves injectives, then $X^i(I)$ is injective for all $i\geq 0$. Since injective objects are closed under products, $T(X)(I)\cong \prod_{i\geq 0}X^i(I)$ must be injective, so $T(X)$ preserves injective objects.

If $X$ preserves injective objects, then $X$ descends to an endofunctor on the stable injective category $\overline{\mathcal{C}}$. By Lemma <ref> (<ref>) the functor $T(X)$ also preserves injective objects, and therefore descends to a monad on $ \overline{\mathcal{C}}$, denoted in the same way.

Assume $\mathcal{C}$ has enough injectives and $X$ preserves injectives. Then $T(X)$, considered as a monad on $\overline{\mathcal{C}}$, is equal to the free monad of $X$ on $\overline{\mathcal{C}}$. In particular, we have an isomorphism
\overline{\mathcal{C}}{}^{T(X)}\xrightarrow{\cong} (X\Downarrow \Id_{\overline{\mathcal{C}}}).

Since $X$ is locally nilpotent,  the sum $\bigoplus_{i\geq 0}X^i$ is finite and therefore preserved by the functor $\mathcal{C}\to \overline{\mathcal{C}}$. Hence, $T(X)$ is the free monad of $X$ on $\overline{\mathcal{C}}$, which proves the claim.

Let $Q$ be a finite and acyclic quiver and $\mathfrak{B}$ a modulation on $Q$ as in Example <ref>, and assume the categories $\mathcal{B}_\mathtt{i}$ are abelian. Then $X$ is exact if and only if if the functors $F_\alpha\colon \mathcal{B}_i\to \mathcal{B}_j$ are exact, and $X$ preserves injectives if and only if the functors $F_\alpha$ preserve injectives. Assume these two conditions hold and the categories $\mathcal{B}_\mathtt{i}$ have enough injectives for all vertices $\mathtt{i}$. Then we have another modulation $\overline{\mathfrak{B}}$ on $Q$, given by the injectively stable category $\overline{\mathcal{B}_\mathtt{i}}$ at vertex $\mathtt{i}$, and by the functor $G_\alpha\colon \overline{\mathcal{B}_\mathtt{i}}\to \overline{\mathcal{B}_\mathtt{j}}$ induced from $F_\alpha$ at an arrow $\alpha\colon \mathtt{i}\to \mathtt{j}$. Setting $\mathcal{C}=\prod_{\mathtt{i}\in Q_0}\mathcal{B}_\mathtt{i}$, we get equivalences $\overline{\mathcal{C}}\cong\prod_{\mathtt{i}\in Q_0}\overline{\mathcal{B}_\mathtt{i}}$ and $\operatorname{rep}\overline{\mathfrak{B}}\cong \overline{\mathcal{C}}{}^{T(X)}$. If $\mathcal{B}_\mathtt{i}=\mathcal{B}$ for all $\mathtt{i}\in Q_0$ and $F_\alpha=\operatorname{Id}$ for all $\alpha\in Q_1$ as in Example <ref>, then  this gives $\operatorname{rep}(Q,\overline{\mathcal{B}})\cong \overline{\mathcal{C}}{}^{T(X)}$.

Let $Q$ be any quiver, not necessarily finite or acyclic. Assume the categories $\mathcal{B}_\mathtt{i}$ are abelian with enough injectives. Then the abelian category $\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ has enough injectives, and they are given componentwise, i.e. $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}$ is injective if and only if $B_\mathtt{i}$ is injective for all $\mathtt{i}\in Q_0$. Assume the functors $F_\alpha$ are exact and preserve injectives.  To ensure that the functor $X$ given by (<ref>)
is exact and preserves injectives, we need to restrict to objects $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}$ of $\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ satisfying:

(B) For each vertex $\mathtt{j}\in Q_0$ one of the following conditions hold:

* There are only finitely many arrows $\alpha$ with $t(\alpha)=\mathtt{j}$ and $B_{s(\alpha)}\neq 0$, or

* $\mathcal{B}_{\mathtt{j}}$ is a locally noetherian Grothendieck category.

Indeed, $X$ is exact and preserves injectives in the first case since finite direct sums are exact and preserve injectivity. It holds in the second case since infinite coproducts are exact in Grothendieck categories, and since infinite coproducts of injectives are injective in locally noetherian categories.

Now let $\mathcal{C}$ be the subcategory $\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ consisting of all objects satisfying (A) in Example <ref> and (B). Since $\mathcal{C}$ is closed under subobjects, extensions, and quotients, it is a Serre subcategory of $\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$. In particular, it is abelian. It also has enough injectives, since for any $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}$ in $\mathcal{C}$ we can find a monomorphism $(B_{\mathtt{i}})_{\mathtt{i}\in Q_0}\to (J_{\mathtt{i}})_{\mathtt{i}\in Q_0}$ with $J_\mathtt{i}$ injective for all $\mathtt{i}\in Q_0$ and $J_\mathtt{i}\neq 0$ if and only if $B_\mathtt{i}\neq 0$. Furthermore, $X$ restricts to an exact, locally nilpotent endofunctor on $\mathcal{C}$ which preserves injective objects. Hence, it satisfies the standing assumptions in this section. Again, $\mathcal{C}$ is nonzero if and only if $Q$ has a sink vertex, and $\mathcal{C}=\prod_{\mathtt{i} \in Q_0}\mathcal{B}_\mathtt{i}$ if $Q$ is finite and acyclic.

Let $\mathfrak{B}$ be a modulation on a quiver $Q$ such that $\mathcal{B}_{s(\alpha)}=\operatorname{Mod}R$ and $\mathcal{B}_{t(\alpha)}=\operatorname{Mod}S$ for two rings $R$ and $S$ and an arrow $\alpha$. We investigate common situations in which the functor $F_\alpha\colon \mathcal{B}_{s(\alpha)}\to \mathcal{B}_{t(\alpha)}$ is exact and preserves injectives, analogous to <cit.>.

Assume $F_{\alpha}=\operatorname{Hom}_R(M,-)$ for an $S$-$R$-bimodule $M$. We claim that $F_\alpha$ is exact and preserves injectives if and only if $M$ is projective as a right $R$-module and flat as a left $S$-module. Indeed, exactness of $F_\alpha$ is clearly equivalent to $M$ being projective as a right $R$-module. Now $F_\alpha(I)=\operatorname{Hom}_R(M,I)$ is an injective $S$-module if and only if

\[
\operatorname{Hom}_S(-,\operatorname{Hom}_R(M,I))\cong \operatorname{Hom}_R(-\otimes_S M,I)
\]

is an exact functor. This holds for every injective $R$-module if and only if $-\otimes_SM$ is exact, i.e. $M$ is flat as a left $S$-module.

Now assume $F_{\alpha}=-\otimes_RN$ for an $R$-$S$-bimodule $N$, and that $N$ is finitely presented as a left $R$-module. We claim that $F_\alpha$ is exact and preserves injectives if and only if $N$ is projective as a left $R$-module and $\operatorname{Hom}_{R^{\operatorname{op}}}(N,R)$ is flat as a left $S$-module. Indeed, $F_\alpha$ being exact is equivalent to $N$ being flat as an $R$-module, and since $N$ is finitely presented this is again equivalent to $N$ being projective as an $R$-module. If $N$ is finitely presented and projective, then

\[
F_\alpha=-\otimes_R N\cong\operatorname{Hom}_R(\operatorname{Hom}_{R^{\operatorname{op}}}(N,R),-).
\]

Hence $F_\alpha$ preserves injectives if and only if $\operatorname{Hom}_{R^{\operatorname{op}}}(N,R)$ is flat as a left $S$-module by the argument above.

As a special case of Example <ref>, let $\mathfrak{B}$ be a modulation on $Q=(\mathtt{1}\to \mathtt{2})$ given by a tensor product $-\otimes_R N\colon \operatorname{Mod}R\to \operatorname{Mod}S$ by an $R$-$S$-bimodule $N$. Then the category of $\mathfrak{B}$-representations is equivalent to $\operatorname{Mod}\Lambda$, where $\Lambda$ is the triangular matrix ring

\[
\Lambda\coloneqq\begin{pmatrix}R&0\\ N&S
\end{pmatrix}
\]

Such rings have for example been studied in [21] and <cit.>.  Their monomorphism categories (defined below, see Example <ref>) occur in [60, 88, 91] when describing Gorenstein projective $\Lambda$-modules.

§.§ The top functor

We consider the cokernel functor $\Kopf_X$ and its right adjoint $S$:

\begin{align*}
\Kopf_{X}\colon\mathcal{C}^{T(X)}\to \mathcal{C} \quad \module{M}\mapsto \coker h_\module{M}  \\
S\colon \mathcal{C}\to \mathcal{C}^{T(X)} \quad M\mapsto (X(M)\xrightarrow{0}M).
\end{align*}

Applying $\Kopf_{X}$ is analogous to taking the top of a module, while $S(M)$ can be thought of as the analogue of the semisimple module concentrated at $M$.  Here we collect their most important properties. Lemma <ref> (<ref>) can be considered as an analogue of Nakayama's Lemma.

The following hold:

*  We have an adjunction $\Kopf_X\dashv S$.

*  $\Kopf_Xf_!(M)\cong M$ naturally in $M\in \mathcal{C}$.

*  $\Kopf_X(\iota_M)=0$ for all $M\in \mathcal{C}$.

*  If $\Kopf_X(\module{M})=0$, then $\module{M}=0$.

Part (<ref>) follows from <cit.>. Part (<ref>) and (<ref>) are obvious.  Part (<ref>) follows since there are no epimorphisms $X(M)\twoheadrightarrow M$ if $M\neq 0$, see Remark <ref>.

We also need a result on the existence of the left derived functors of $\Kopf_X$.

The left derived functors $L_j\Kopf_X$ exist for all $j>0$. Furthermore, they vanish on relative projective objects.

By <cit.> it suffices to show that the functors $f_!f^*$ and $\Kopf_X\circ f_!f^*$ are exact. But this follows immediately from the fact that $f_!$ and $f^*$ are exact and the isomorphism $\Kopf_X\circ f_!\cong \operatorname{Id}_{\mathcal{C}}$ in Lemma <ref> (<ref>).

We finish with the following description of the left derived functors of $\Kopf_X$.

For $\module{M}\in \mathcal{C}^{T(X)}$ we have

\[
L_1\Kopf_X(\module{M})=\ker h_\module{M} \quad \text{and} \quad L_i\Kopf_X(\module{M})=0 \quad i\geq 2.
\]

This follows by the same proof as in <cit.>.

§.§ The monomorphism category

The monomorphism category $\Mono(X)$ is the full subcategory of $\mathcal{C}^{T(X)}$ consisting of objects $\module{M}$ where the structure map $h_\module{M}$ is a monomorphism.

Note that the objects of the monomorphism category can equivalently be characterized using the first left derived functor of $\Kopf_X$.

Let $\module{M}\in \mathcal{C}^{T(X)}$. Then $\module{M}\in \Mono(X)$ if and only if $L_1\Kopf_X(\module{M})=0$

This follows immediately from the description of $L_1\Kopf_X$ in Lemma <ref>.

Recall that a subcategory of an abelian category is called $\emphbf{resolving}$ if it is generating and closed under extensions, kernels of epimorphisms, and direct summands.

The category $\Mono(X)$ is resolving and closed under subobjects in $\mathcal{C}^{T(X)}$. In particular, it is an exact category whose conflations are epimorphisms and whose inflations are monomorphisms with cokernel in $\Mono(X)$.

For any object $\module{M}\in \mathcal{C}^{T(X)}$ the counit $\varepsilon_{\module{M}}\colon f_!f^*(\module{M})\twoheadrightarrow \module{M}$ is an epimorphism. Since any object in the image of $f_!$ is contained in $\Mono(X)$, this shows that $\Mono(X)$ is generating. The fact that $\Mono(X)$ is closed under extensions and subobjects follows from the fact that monomorphisms are closed under extensions and subobjects. Since kernels of epimorphisms and direct summands are special cases of subobjects, this shows that $\Mono(X)$ is resolving.

Finally, since $\Mono(X)$ is closed under extensions, it inherits an exact structure from $\mathcal{C}^{T(X)}$ where the inflations and the deflations are the monomorphism and the epimorphisms whose cokernel and kernel lies in $\Mono(X)$, respectively. Since $\Mono(X)$ is closed under kernels of epimorphisms, the deflations coincide with the epimorphisms in $\Mono(X)$.

Next we show how $\Kopf_X$ detects inflations and isomorphisms in $\Mono(X)$.

Let $g\colon \module{M}\to \module{N}$ be a morphism in $\Mono(X)$. The following hold:

*  $g$ is an inflation in $\Mono(X)$ if and only if $\Kopf_X(g)$ is a monomorphism.

*  $g$ is an isomorphism if and only if $\Kopf_X(g)$ is an isomorphism.

To prove part (<ref>) assume that $g$ is a monomorphism with cokernel in $\Mono(X)$. Applying $\Kopf_X$ to the exact sequence

\[
0\to \module{M}\xrightarrow{g} \module{N}\to \operatorname{coker}g\to 0
\]

and using that $L_1\Kopf_X(\operatorname{coker}g)=0$, we get that $\Kopf_X(g)$ is a monomorphism.

Conversely, assume $\Kopf_X(g)$ is a monomorphism, and consider the exact sequence

\[
0\to \ker g\to \module{M}\to \operatorname{Im}g\to 0.
\]

Since $\operatorname{Im}g$ is a subobject of $\module{N}$, it is contained in $\Mono(X)$. Therefore $L_1\Kopf_X(\operatorname{Im}g)=0$, so applying $\Kopf_X$ to the inclusion $\ker g\to \module{M}$ gives a monomorphism $\Kopf_X(\ker g)\to \Kopf_X(\module{M})$. Furthermore, since the composite $\Kopf_X(\ker g)\to \Kopf_X(\module{M})\xrightarrow{\Kopf_X(g)} \Kopf_X(\module{N})$ is $0$ and $\Kopf_X(g)$ is a monomorphism, it follows that $\Kopf_X(\ker g)=0$. Hence $\ker g=0$ by Lemma <ref> (<ref>), so $g$ must be a monomorphism. Finally, applying $\Kopf_X$ to the short exact sequence

\[
0\to \module{M}\xrightarrow{g} \module{N}\to \coker g\to 0
\]

and using that $\Kopf_X(g)$ is a monomorphism and $L_1\Kopf_X(\module{N})=0$, we get that $L_1\Kopf_X(\coker g)=0$. Hence $\coker g\in \Mono(X)$ by Lemma <ref>. This proves (<ref>).

To prove part (<ref>) observe that if $g$ is an isomorphism, then $\Kopf_X(g)$ must be an isomorphism. Conversely, assume that $\Kopf_X(g)$ is an isomorphism. Then $g$ must be a monomorphism by part (<ref>). Hence we have a right exact sequence

\[
\Kopf_X(\module{M})\xrightarrow{\Kopf_X(g)} \Kopf_X(\module{N})\to \Kopf_X(\coker g)\to 0
\]

Since $\Kopf_X(g)$ is an isomorphism, we must have that $\Kopf_X(\coker g)=0$. Therefore $\coker g=0$ by Lemma <ref> (<ref>), so $g$ must be an isomorphism.

As a consequence of this we get a criterion for a morphism to be in the radical.

Let $g\colon \module{M}\to \module{N}$ be a morphism in $\Mono(X)$, and assume $\Kopf_X(g)$ is in the radical of $\mathcal{C}$. Then $g$ is in the radical of $\Mono(X)$.

This follows immediately from the definition of the radical in Subsection <ref> and the fact that $\Kopf_X$ reflects isomorphisms by Lemma <ref> (<ref>).

Let $Q$ be a finite and acyclic quiver and $\mathfrak{B}$ a modulation on $Q$ by abelian categories $\mathcal{B}_\mathtt{i}$ and exact functors $F_\alpha$ as in Example <ref>. Let $(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}$ be a $\mathfrak{B}$-representation. For each vertex $\mathtt{i}\in Q_0$ consider the map
\[B_{\mathtt{i},\operatorname{in}}\colon \bigoplus_{\substack{\alpha\in Q_1\\t(\alpha)=\mathtt{i}}}F_{\alpha}(B_{s(\alpha)})\xrightarrow{(B_\alpha)} B_\mathtt{i}.\]
Then $\Kopf_X$ is given by

\[
\Kopf_X(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}=(\coker B_{\mathtt{i},\operatorname{in}})_{\mathtt{i}\in Q_0}.
\]

Furthermore, by Lemma <ref> it follows that

\[
L_1\Kopf_X(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}=(\ker B_{\mathtt{i},\operatorname{in}})_{\mathtt{i}\in Q_0}.
\]

Hence, $(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}\in \Mono (X)$ if and only if $B_{\mathtt{i},\operatorname{in}}$ is a monomorphism for all $\mathtt{i}\in Q_0$.

§ INJECTIVE OBJECTS

Recall that $\Mono(X)$ is an exact category by Proposition <ref>. In this section we investigate the injective objects in $\Mono(X)$ under the exact structure. We show that up to isomorphism they are the objects of the form $f_!(I)$ , where $I$ is injective in $\mathcal{C}$, see Proposition <ref> and Corollary <ref>. This improves on Lemma 6.5 in [25] for the free monad, since that result only implies that injective objects are summands of objects of the form $f_!(I)$. Furthermore the proof in [25] relies on the existence of a relative Nakayama functor, and this assumption is removed from the proofs below. We finish this section by characterizing when $\Mono(X)$ has enough injectives and when it has injective envelopes. Throughout $\mathcal{C}$ is a $\mathbbm{k}$-linear abelian category, and $X\colon \mathcal{C}\to \mathcal{C}$ is an exact functor which is locally nilpotent and preserves injective objects.

We first show that the functor $\Kopf_X$ induces a surjective map on morphisms spaces when the codomain is of the form $f_!(I)$ with $I$ injective.

Let $g\colon \Kopf_X \module{M}\to I$ be a morphism in $\mathcal{C}$ with $I$ injective and $\module{M}\in \Mono(X)$. Then there exists a morphism $k\colon \module{M}\to f_!(I)$ in $\mathcal{C}^{T(X)}$ with $\Kopf_X(k)=g$.

Let $k_0$ denote the composite $M\xrightarrow{} \operatorname{coker}h_{\module{M}}\xrightarrow{g} I$. Applying $X$, we get a morphism $X(k_0)\colon X(M)\to X(I)$, and since $X(I)$ is injective and $h_\module{M}\colon X(M)\to M$ is a monomorphism, we can find a morphism $k_1\colon M\to X(I)$ satisfying $k_1\circ h_\module{M}=X(k_0)$. Repeating this procedure, we get morphisms $k_i\colon M\to X^i(I)$ satisfying $k_i\circ h_{\module{M}}=X(k_{i-1})$ for each $i\geq 1$. These induce a morphism $M\to \bigoplus_{i\geq 0}X^i(I)$, and since $k_0\circ h_{\module{M}}=0$ this lifts to a morphism $k\colon \module{M}\to f_!(I)$ in $\mathcal{C}^{T(X)}$. By construction we have that $\Kopf_X(k)=g$, so we are done.

We can now show that $f_!(I)$ is injective in $\Mono(X)$ when $I$ is injective in $\mathcal{C}$.

Let $I\in \mathcal{C}$ be injective. Then

\[
\Ext^i_{\mathcal{C}^{T(X)}}(\module{M},f_!(I))=0 \quad \text{for all $i>0$.}
\]

We prove the result using Yoneda-$\Ext$. Let $\xi\in \Ext^i_{\mathcal{C}^{T(X)}}(\module{M},f_!(I))$. We want to find a representative of $\xi$ whose leftmost map is a split monomorphism. To this end, note that since $\Mono(X)$ is
resolving, it satisfies the dual of condition (C2) in <cit.>. Therefore, the dual of <cit.> implies that the induced functor

\[
D^-(\Mono(X))\to D^-(\mathcal{C}^{T(X)})
\]

between the derived categories is fully faithful, where we consider $\Mono(X)$ as an exact category. Hence, we can find a representative of $\xi$ of the form

\[
0\to f_!(I)\to \module{N}_1\to \cdots \to \module{N}_i\to \module{M}\to 0
\]

where all the terms are in $\Mono(X)$. Since $\Mono(X)$ is closed under subobjects, also all the intermediate kernels are in $\Mono(X)$. Let $i\colon f_!(I)\to \module{N}_1$ denote the leftmost morphism. By Lemma <ref> the morphism $\Kopf_X (i)\colon I\to \Kopf_X\module{N}_1$ is monic. Since $I$ is injective, $\Kopf_X (i)$ is a split monomorphism, so we can choose a left inverse $g\colon \Kopf_X\module{N}_1\to I$ of it. By Lemma <ref> we can find a morphism $k\colon \module{N}_1\to f_!(I)$ satisfying $\Kopf_X(k)=g$. By construction, if we apply $\Kopf_X$ to the composite $k\circ i\colon f_!(I)\to f_!(I)$ we get the identity morphism on $I$. Hence, by Lemma <ref> (<ref>) the morphism $k\circ i$ is an isomorphism, so $i$ must be a split monomorphism. This proves the claim.

To show the converse of Proposition <ref> we use  the following result.

Let $\module{M}\in \Mono(X)$, and assume $M$ is injective in $\mathcal{C}$. Then $\module{M}\cong f_!(J)$ for some injective object $J$ in $\mathcal{C}$.

Since $\module{M}\in \Mono(X)$, the map $h_\module{M}\colon X(M)\to M$ is a monomorphism. Since $M$ is injective, $X(M)$ must be injective, so $h_\module{M}$ is a split monomorphism. By Proposition <ref> the claim follows.

Let $\module{M}$ be an injective object in $\Mono(X)$ considered as an exact category. Then $\module{M}\cong f_!(J)$ for some injective object $J$ in $\mathcal{C}$.

It follows from the assumptions that $\operatorname{Ext}^1_{\mathcal{C}^{T(X)}}(f_!(N),\module{M})=0$ for any $N\in \mathcal{C}$. Now the adjunction $f_!\dashv f^*$ induces an isomorphism

\[
\operatorname{Ext}^1_{\mathcal{C}^{T(X)}}(f_!(N),\module{M})\cong \operatorname{Ext}^1_{\mathcal{C}}(N,f^*(\module{M}))
\]

see <cit.>. Therefore $\operatorname{Ext}^1_{\mathcal{C}}(N,f^*(\module{M}))=0$ for all $N\in \mathcal{C}$. This implies that $f^*(\module{M})$ is injective in $\mathcal{C}$. Hence by Proposition <ref> we get that $\module{M}\cong f_!(J)$ for some injective object $J\in \mathcal{C}$.

We use our results to investigate when $\Mono(X)$ has enough injectives.

Assume $\mathcal{C}$ has enough injectives. Then $\Mono(X)$ has enough injectives.

Let $\module{M}\in \Mono(X)$ be arbitrary, and choose a monomorphism $i\colon \Kopf_X(\module{M})\rightarrow I$ in $\mathcal{C}$ with $I$ injective. Note that $f_!(I)$ is injective by Proposition <ref>. By Lemma <ref> we can find a morphism $j\colon \module{M}\to f_!(I)$ satisfying $\Kopf_X(j)=i$. Now by Lemma <ref> (<ref>) we have that $j$ is an inflation in $\Mono(X)$. This proves the claim.

We finish by showing the existence of injective envelopes in $\Mono(X)$.

The following hold:

*  Let $g\colon \module{M}\to f_!(I)$ be a morphism in $\Mono(X)$ with $I\in \mathcal{C}$ injective. Then $g$ is an injective envelope in $\Mono(X)$ if and only if $\Kopf_X(g)\colon \Kopf_X(M)\to I$ is an injective envelope in $\mathcal{C}$.

*  If $\mathcal{C}$ has injective envelopes, then $\Mono(X)$ has injective envelopes.

To prove part (<ref>) we first assume $\Kopf_X(g)$ is an injective envelope. Then $g$ must be a monomorphism with cokernel in $\Mono(X)$ by Lemma <ref>. It remains to show that $g$ is left minimal. Assume $k\colon f_!(I)\to f_!(I)$ is a morphism satisfying $k\circ g=g$. Applying $\Kopf_X$ we get that $\Kopf_X(k)\circ \Kopf_X(g)=\Kopf_X(g)$. Since $\Kopf_X(g)$ is an injective envelope, it is left minimal, so $\Kopf_X(k)$ must be an isomorphism. Hence $k$ must be an isomorphism by Lemma <ref> (<ref>), so $g$ is left minimal.

Conversely, assume $g$ is an injective envelope. Since $g$ is an inflation, $\Kopf_X(g)\colon \Kopf_X(\module{M})\to I$ must be a monomorphism. Hence, it only remains to show that $\Kopf_X(g)$ is left minimal, so let $k\colon I\to I$ be a morphism satisfying $k\circ \Kopf_X(g)=\Kopf_X(g)$. Consider  $k'\coloneqq g-f_!(k)\circ g\colon \module{M}\to f_!(I)$. Since $\Kopf_X(k')=0$, there exists a morphism $r\colon \module{M}\to f_!X(I)$ such that the left triangle in

\[
\begin{tikzcd}
\end{tikzcd}
\]

is commutative. Since $f_!X(I)$ is injective in $\Mono(X)$ and $g\colon \module{M}\to f_!(I)$ is a monomorphism, the morphism $r$ extends to a morphism $s\colon f_!(I)\to f_!X(I)$ via $g$. Then clearly

\[
g=(\iota \circ s+f_!(k))\circ g
\]

and hence $\iota \circ s+f_!(k)$ is an isomorphism since $g$ is an injective envelope. Finally, since $\Kopf_X(\iota)=0$, it follows that $\Kopf_X(\iota\circ s+f_!(k))=k$, which must therefore also be an isomorphism. This shows that $\Kopf_X(g)$ is left minimal.

To prove part (<ref>) let $\module{M}\in \Mono(X)$ be arbitrary, and let $i\colon \Kopf_X\module{M}\to I$ be an injective envelope in $\mathcal{C}$. By Lemma <ref> we can find a morphism $j\colon \module{M}\to f_!(I)$ satisfying $\Kopf_X(j)=i$. Then $j$ must be an injective envelope by the first part of this lemma.

§ THE EPIVALENCE

Throughout this section we let $\mathcal{C}$ be a $\mathbbm{k}$-linear abelian category with enough injectives and $X\colon \mathcal{C}\to \mathcal{C}$ an exact functor which is locally nilpotent and preserves injectives. Our goal is to show that the canonical functor

\[
\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}
\]

is an epivalence, and an actual equivalence if $\mathcal{C}$ is hereditary. The proof that the functor is dense is using a particular construction of right $\Mono(X)$-approximations, which we investigate first.

§.§ Contravariantly finiteness

Recall from Lemma <ref> that we have an exact sequence

\[
0\to f_!X(M)\xrightarrow{\iota_M-f_!(h_\module{M})} f_!(M)\xrightarrow{\varepsilon_{\module{M}}} \module{M}\to 0
\]

for all $\module{M}\in \mathcal{C}^{T(X)}$. Applying $\Kopf_{X}$ to this gives an exact sequence

\[
0\to L_1\Kopf_{X} \module{M}\to X(M)\xrightarrow{h_\module{M}} M\to \Kopf_{X} \module{M}\to 0
\]

since $L_1\Kopf_{X} f_!(M)=0$.

Fix a monomorphism $j\colon L_1\Kopf_{X}\module{M}\hookrightarrow J$ into an injective module $J$, and fix a lift $e\colon X(M)\to J$ of $j$. Define the object $\Mo\module{M}\in \mathcal{C}^{T(X)}$ and the morphism $p_{\module{M}}\colon \Mo\module{M}\to \module{M}$ to be such that the lower sequence in the following diagram is a pushout of the upper sequence:

\begin{equation}\label{Defining Mo}
\begin{tikzcd}[column sep=1.5cm]
0\arrow{r} &f_!X(M)\arrow{r}{\iota_M-f_!(h_\module{M})}\arrow{d}{f_!(e)}&f_!(M)\arrow{d}{r}\arrow{r}{\varepsilon_{\module{M}}}&\module{M}\arrow[equals]{d}\arrow{r}&0\\
0\arrow{r} &f_!(J)\arrow{r}{s}&\Mo\module{M}\arrow{r}{p_{\module{M}}}&\module{M}\arrow{r}&0
\end{tikzcd}
\end{equation}

Note that $\Mo\module{M}$ is not well-defined up to isomorphism, since it depends on the choice of $J$, the monomorphism $j$, and the lift $e$.  In Proposition <ref> we show that it is well-defined in a quotient category of $\mathcal{C}^{T(X)}$.

We first show that $\Mo \module{M}$ gives a right $\Mono (X)$-approximation.

The morphism $p_{\module{M}}\colon \Mo\module{M}\to \module{M}$ is a right $\Mono(X)$-approximation for $\module{M}\in \mathcal{C}^{T(X)}$.

We first show that $\Mo \module{M}$ is contained in $\Mono(X)$. To this end, we apply $\Kopf_{X}$ to the diagram (<ref>). This yields the diagram

\[
\begin{tikzcd}
L_1\Kopf_{X} \module{M}\arrow{r}\arrow[equals]{d}&X(M)\arrow{d}{e}\arrow{r}{h_\module{M}} &M\arrow{d}\arrow{r} &\Kopf_{X}\module{M}\arrow[equals]{d}\\
\end{tikzcd}
\]

with exact rows. By commutativity of the leftmost square the map $L_1\Kopf_{X}\module{M}\to J$ is equal to $j$, whence is a monomorphism. Furthermore, the lower row can be extended to an exact sequence

\[
0\to L_1\Kopf_{X}\Mo\module{M}\to L_1\Kopf_{X}\module{M}\xrightarrow{j} J
\]

since $L_1\Kopf_X(f_!(J))=0$. Hence, it follows that $L_1\Kopf_{X}\Mo\module{M}=0$. Thus, $\Mo\module{M}\in \Mono(X)$.

To see that $p_\module{M}$ is a right approximation, apply $\Hom_{\mathcal{C}^{T(X)}}(\module{N},-)$ with $\module{N}\in \Mono(X)$ to the exact sequence

\[
0\to f_!(J)\to \Mo\module{M}\xrightarrow{p_{\module{M}}} \module{M}\to 0.
\]

This gives an epimorphism

\[
\Hom_{\mathcal{C}^{T(X)}}(\module{N},p_{\module{M}})\colon \Hom_{\mathcal{C}^{T(X)}}(\module{N},\Mo\module{M})\xrightarrow{} \Hom_{\mathcal{C}^{T(X)}}(\module{N},\module{M}).
\]

since $\Ext^1_{\mathcal{C}^{T(X)}}(\module{N},f_!(J))=0$ by Proposition <ref>. This proves the claim.

Next we show that $\Mo\module{M}$ satisfies a universal property. Here $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$ denotes the quotient of the category $\mathcal{C}^{T(X)}$ by the ideal of morphisms factoring through an object of the form $f_!(J)$ where $J\in \inj \mathcal{C}$.

Let $g\colon\module{N}\to \module{M}$ be a morphism in $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$ with $\module{N}\in\Mono(X)$. Then there exists a unique morphism

\[
g'\colon \module{N}\to \Mo\module{M}
\]

in $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$ such that the equality $p_{\module{M}}\circ g' = g$ holds.

The existence of $g'$ follows from $p_{\module{M}}$ being a right $\Mono(X)$-approximation. We show uniqueness: Assume $g'\colon \module{N}\to \Mo\module{M}$ and $g''\colon \module{N}\to \Mo\module{M}$ are two morphisms in $\mathcal{C}^{T(X)}$ such that $p_{\module{M}}\circ g'$ and $p_{\module{M}}\circ g''$ are equal as morphisms in $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$.  Then there exists $J'\in \inj \mathcal{C}$ and morphisms $u\colon \module{N}\to f_!(J')$ and $v\colon f_!(J')\to \module{M}$ such that

\[
v\circ u = p_{\module{M}}\circ (g'-g'')
\]

in $\mathcal{C}^{T(X)}$. Furthermore, since $p_{\module{M}}$ is a right $\Mono(X)$-approximation and $f_!(J')\in \Mono(X)$, there exists a morphism $w\colon f_!(J')\to \Mo\module{M}$ satisfying

\[
v=p_{\module{M}}\circ w.
\]

Hence, $p_{\module{M}}\circ (g'-g''-w\circ u)=0$ which implies that $g'-g''-w\circ u$ factors through $\ker p_{\module{M}}=f_!(J)$. Therefore, $g'-g''=(g'-g''-w\circ u)+w\circ u$ factors through $f_!(J)\oplus f_!(J')\cong f_!(J\oplus J')$. This shows that $g'$ and $g''$ are equal in $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$.

It follows from Proposition <ref> that $\Mo\module{M}$ and $p_{\module{M}}$ are unique up to isomorphism in $\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$, independently of the choice of $j\colon L_1\Kopf_{X}\module{M}\to J$ and the lift $e\colon X(M)\to J$. In fact, the universal property of $\Mo\module{M}$ implies that the assignment $\module{M}\mapsto \Mo\module{M}$ can be made into a functor, which the following result shows. Here we write

\[
\overline{\Mono}(X)=\tfrac{\Mono(X)}{f_!(\inj \mathcal{C})}
\]

since by Proposition <ref> and Proposition <ref> the subcategory of injectives in $\Mono(X)$ is $f_!(\inj \mathcal{C})$.

The assignment $\module{M}\mapsto \Mo\module{M}$ induces a functor

\[
\Mo\colon \tfrac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})} \to \overline{\Mono}(X)
\]

which is right adjoint to the inclusion functor $i \colon \overline{\Mono}(X)\to \frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}$. Furthermore, the counit of the adjunction $i\dashv \Mo$ at $\module{M}$ is $p_{\module{M}}$.

Composing with $p_{\module{M}}$ gives an isomorphism

\[
\Hom_{\overline{\Mono}(X)}(\module{N},\Mo\module{M})\xrightarrow{\cong}\Hom_{\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}}(\module{N},\module{M})
\]

for $\module{N}\in \Mono(X)$ by Proposition <ref>. Since $\Mo\module{M}\in \Mono(X)$, it follows from Yoneda's lemma that the assignment $\module{M}\mapsto \Mo\module{M}$ defines a functor
$\Mo\colon \frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})} \to \overline{\Mono}(X)$ which makes the isomorphism

\[
\Hom_{\frac{\mathcal{C}^{T(X)}}{f_!(\inj \mathcal{C})}}(i(-),\module{M})\cong \Hom_{\overline{\Mono}(X)}(-,\Mo\module{M})
\]

natural in $\module{M}$. If $\module{N}=\Mo\module{M}$, then the image of the identity $1_{\Mo\module{M}}$ is $p_{\module{M}}$, which is therefore the counit at $\module{M}$. This proves the claim.

§.§ The general case

In this subsection, we show that the inclusion of the monomorphism category into the Eilenberg–Moore category of the stable category induces an epivalence.

A functor is called an epivalence if it is full, dense and reflects isomorphisms.

The remainder of the subsection is concerned with proving the following result.

The canonical functor $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is an epivalence.

It was observed in <cit.> that epivalences preserve and reflect several important representation-theoretic concepts. In particular, Theorem <ref> has the following consequences:

(i) An object in $\overline{\Mono}(X)$ is indecomposable if and only if its image in $\overline{\mathcal{C}}{}^{T(X)}$ is indecomposable.

(ii) There is a bijection between isomorphism classes of objects in $\overline{\Mono}(X)$ and isomorphism classes of objects in $\overline{\mathcal{C}}{}^{T(X)}$, which restricts to a bijection between the indecomposables.

(iii) $\overline{\Mono}(X)$ is Krull–Remak–Schmidt if and only if $\overline{\mathcal{C}}{}^{T(X)}$ is

Under mild assumptions on $\mathcal{C}$ there is a bijection between the indecomposable non-injective objects in $\Mono(X)$ and the indecomposable objects in $\overline{\Mono}(X)$. Therefore, given an indecomposable object in $\overline{\mathcal{C}}{}^{T(X)}$, we have an associated unique (up to isomorphism) indecomposable non-injective object in $\Mono(X)$. In Section <ref> we show that the Mimo-construction, a refinement of $\Mo$ as defined in the previous subsection,  gives an explicit way to describe it. In particular, Theorem <ref> reduces the study of indecomposable objects in $\Mono(X)$ to the study of indecomposable objects in $\overline{\mathcal{C}}{}^{T(X)}$, which is often much simpler.

The canonical functor $\Mono(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is dense.

An object $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$ is given by a morphism $h_\module{M}\colon X(M)\to M$ in $\overline{\mathcal{C}}$ by Lemma <ref>. Choose a lift $h'\colon X(M)\to M$ to $\mathcal{C}$ of $h_{\module{M}}$, and let $\module{M}'$ be the  object in $\mathcal{C}^{T(X)}$ corresponding to it. By Theorem <ref> the object $\Mo \module{M}'$ is in $\Mono(X)$, and by construction it must be isomorphic to $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$. This proves the claim.

Next we show that the functor $\Mono(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is full.

Let $\module{M}\in \Mono(X)$ and $\module{N}\in \mathcal{C}^{T(X)}$. Then any morphism $\module{M}\to \module{N}$ in $\overline{\mathcal{C}}{}^{T(X)}$ can be lifted to a morphism in $\mathcal{C}^{T(X)}$. In particular, the functor $\Mono(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is full.

Let $\overline{g}\colon \module{M}\to \module{N}$ be a morphism in $\overline{\mathcal{C}}{}^{T(X)}$, and let $g'\colon M\to N$ be an arbitrary lift of $f^*(\overline{g})$ to $\mathcal{C}$. Since $f^*(\overline{g})\circ h_{\module{M}}$ and $h_{\module{N}}\circ X(f^*(\overline{g}))$ are equal in $\overline{\mathcal{C}}$, the difference

\[
g'\circ h_{\module{M}}-h_{\module{N}}\circ X(g')\colon X(M)\to N
\]

is equal to a composite $X(M)\xrightarrow{u'} J\xrightarrow{v'}N$ where $J$ is injective. Furthermore, since $h_\module{M}$ is a monomorphism, the map $u'$ can be lifted to a morphism $u_0\colon M\to J$. Similarly, since $X(J)$ is injective, the map $X(u_0)\colon X(M)\to X(J)$ can be lifted to a morphism $u_1\colon M\to X(J)$. Repeating this argument, we get maps $u_i\colon M\to X^i(J)$ for each $i\geq 0$ such that $u_i\circ h_\module{M}=X(u_{i-1})$ for all $i>0$. Now let $I\coloneqq\bigoplus_{i\geq 0}X^i(J)$, and note that $I$ is injective since $X$ is locally nilpotent and therefore $\bigoplus_{i\geq 0}X^i(J)$ is a finite sum. Let $v\colon I\to N$ be the unique map given on component $X^i(J)$ as the composite

\[
X^i(J)\xrightarrow{X^i(v')} X^i(N)\xrightarrow{X^{i-1}(h_\module{N})}X^{i-1}(N)\xrightarrow{X^{i-2}(h_\module{N})}\cdots \xrightarrow{h_\module{N}}N.
\]

Furthermore,  let $u\colon M\to I$ be the unique map given on component $X^i(J)$ as $u_i\colon M\to X^i(J)$. If we let $g=g'-v\circ u$, then a short computation shows that $g\circ h_{\module{M}}=h_\module{N}\circ X(g)$, so $g$ is a morphism $\module{M}\to \module{N}$ in $\mathcal{C}^{T(X)}$. Since $g$ is equal to $\overline{g}$ in $\overline{\mathcal{C}}{}^{T(X)}$ and $\overline{g}$ was arbitrary, this proves the claim.

The canonical functor $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ reflects isomorphisms.

Let $g\colon \module{M}\rightarrow\module{N}$ be a morphism in $\mathcal{C}^{T(X)}$ which becomes an isomorphism in $\overline{\mathcal{C}}{}^{T(X)}$. Consider the commutative diagram with exact rows

\[
\begin{tikzcd}[column sep=2cm]
0 \arrow{r} & f_!X(M)\arrow{d}{f_!Xf^*(g)}\arrow{r}{\iota_M-f_!(h_\module{M})}&f_!(M)\arrow{d}{f_!f^*(g)}\arrow{r}{\varepsilon_{\module{M}}}& \module{M} \arrow{d}{g}\arrow{r} & 0 \\
0 \arrow{r}& f_!X(N)\arrow{r}{\iota_{N}-f_!(h_\module{N})}&f_!(N) \arrow{r}{\varepsilon_{\module{N}}} & \module{N} \arrow{r} & 0.
\end{tikzcd}
\]

Since $f^*(g)\colon M\to N$ is an isomorphism in $\overline{\mathcal{C}}$, the maps $f_!f^*(g)$ and $f_!Xf^*(g)$ are isomorphisms in $\overline{\Mono}(X)$. Now $\module{M}$ is isomorphic to the cone of $\iota_M-f_!(h_\module{M})$ and $\module{N}$ is isomorphic to the cone of $\iota_N-f_!(h_\module{N})$ when we consider $\overline{\Mono}(X)$ as a right triangulated category as in [49], see also [10] and [1]. Since the two leftmost vertical maps $f_!Xf^*(g)$ and $f_!f^*(g)$ are isomorphisms in $\overline{\Mono}(X)$, the map $g$ between the cones must also be an isomorphism in $\overline{\Mono}(X)$, see for example the proof of <cit.>. This proves the claim.

This follows from Lemma <ref>, Lemma <ref> and Lemma <ref>.

Let $\mathbbm{k}$ be a field and $Q$ a finite acyclic quiver. Consider  $\operatorname{rep}(Q,\operatorname{mod}\mathbbm{k}[x]/(x^2))$ as in Example <ref>. Then the category $\Mono(X)$ is equivalent to the category of perfect differential $kQ$-modules considered in [80]. Moreover, since $\overline{\operatorname{mod}}\, \mathbbm{k}[x](x^2)\cong \operatorname{mod}\mathbbm{k}$, it follows that

\[
\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \mathbbm{k}[x](x^2))\cong \operatorname{rep}(Q,\operatorname{mod}\mathbbm{k}).
\]

Hence we have an epivalence $\overline{\Mono(X)}\to \operatorname{rep}(Q,\operatorname{mod}\mathbbm{k})$. The composite

\[
\Mono(X)\to \overline{\Mono(X)}\to \operatorname{rep}(Q,\operatorname{mod}\mathbbm{k})
\]

can be identified with the homology functor in <cit.>.

§.§ The hereditary case

Recall that an object $M\in \mathcal{C}$ is called a cosyzygy (of an object $N$) if there exists an exact sequence $0\to N\to I\to M\to 0$ in $\mathcal{C}$ with $I$ injective. The category $\mathcal{C}$ is called hereditary if any cosyzygy in $\mathcal{C}$ is injective. Our goal is to show that the functor in Theorem <ref> becomes an actual equivalence when $\mathcal{C}$ is hereditary. We start by proving that relative projective objects are closed under cosyzygies in $\Mono(X)$.

Assume we have an exact sequence

\[
0\to f_!(M)\xrightarrow{g} f_!(J)\to \module{N}\to 0
\]

in $\mathcal{C}^{T(X)}$ with $\module{N}\in \Mono(X)$ and $J\in \mathcal{C}$ injective. Then $\module{N}\cong f_!(N')$ for some $N'\in \mathcal{C}$.

Choose an exact sequence

\[
0\to M\xrightarrow{i}I\xrightarrow{q} M'\to 0
\]

in $\mathcal{C}$ with $I$ injective. Applying $f_!(-)$, we get an exact sequence

\[
0\to f_!(M)\xrightarrow{f_!(i)}f_!(I)\xrightarrow{f_!(q)} f_!(M')\to 0
\]

in $\mathcal{C}^{T(X)}$. This implies that both $\module{N}$ and $f_!(M')$ are cosyzygies of $\module{M}$ in $\Mono(X)$. But the cosyzygy of an object is well-defined in $\overline{\Mono}(X)$, see [28]. Hence, there exist injective objects $J_1,J_2\in \mathcal{C}$ and an isomorphism

\[
\module{N}\oplus f_!(J_1)\cong f_!(M')\oplus f_!(J_2).
\]

In particular, $\module{N}$ is a direct summand of $f_!(M'\oplus J_2)$, and therefore by Proposition <ref> we have an isomorphism $\module{N}\cong f_!(N')$ for some $N'\in \mathcal{C}$.

Next we show that if a morphism in $\Mono(X)$ factors componentwise through an injective, then it must factor through a relative projective.

Let $g\colon \module{M}\to \module{N}$ be a morphism in $\Mono(X)$, and assume $f^*(g)\colon M\to N$ factors through an injective object in $\mathcal{C}$. Then $g$ factors through an object of the form $f_!(K)$ where $K$ is a cosyzygy of $X(M)$.

Let $r\colon M\to J$ be a monomorphism with $J$ injective. Since $f^*(g)$ factors through an injective object, it must also factor through $r$, so we can write $f^*(g)=s\circ r$ for some map $s\colon J\to N$. Let $\module{K}'$ be the pushout of $\varepsilon_{\module{M}}\colon f_!(M)\to \module{M}$ along $f_!(r)\colon f_!(M)\to f_!(J)$. We then get the following commutative diagram

\[
\begin{tikzcd}
f_!(M)\arrow{d}{\varepsilon_{\module{M}}}\arrow{r}{f_!(r)}&f_!(J)\arrow{d}\arrow{r}{f_!(s)}& f_!(N) \arrow{d}{\varepsilon_{\module{N}}} \\
\module{M}\arrow{r}{}&\module{K}' \arrow{r}{} & \module{N}.
\end{tikzcd}
\]

where the left hand square is the pushout square and the bottom right morphism is uniquely defined such that the right hand square commutative and the composite $\module{M}\to \module{K}'\to \module{N}$ is equal to $g$. Note that the left hand square is also a pullback square since $f_!(r)$ is a monomorphism. Therefore, since $\varepsilon_{\module{M}}$ is an epimorphism with kernel $f_!X(M)$, the same must hold for $f_!(J)\to \module{K}'$, so we get an exact sequence

\[
0\to f_!X(M)\to f_!(J)\to \module{K}'\to 0.
\]

By Proposition <ref> it follows that $\module{K}'\cong f_!(K)$ for some object $K\in \mathcal{C}$. Applying $\Kopf_X$ to the exact sequence, we get the exact sequence

\[
0\to X(M)\to J\to K\to 0.
\]

This shows that $K$ is a cosyzygy of $X(M)$, which proves the claim.

We can now prove the main result of this subsection.

The following are equivalent:

*  Any object in the image of $X$ has injective dimension at most $1$.

*  $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is an equivalence.

In particular, this holds if $\mathcal{C}$ is hereditary.

We already know that the functor in (<ref>) is full and dense, so the statement is equivalent to the functor being faithful. By Proposition <ref> any morphism $g$ in $\Mono(X)$ which is $0$ in $\overline{\mathcal{C}}{}^{T(X)}$ must factor through an object of the form $f_!(K)$ where $K$ is a cosyzygy in $\mathcal{C}$ of an object of the form $X(M)$. If (<ref>) holds, then $K$ must be injective, so $g$ must be $0$ in $\overline{\Mono}(X)$.

Next assume (<ref>) holds, and let $M\in \mathcal{C}$ be arbitrary. Choose an exact sequence

\[
0\to X(M)\to J\to K\to 0
\]

with $J$ injective. Our goal is to show that $K$ is injective. Applying $f_!$ and taking the pushout along $\iota_M$ gives a commutative diagram

\[
\begin{tikzcd}
\end{tikzcd}
\]

with exact rows. Note that $\module{E}\in \Mono(X)$ since it is an extension of two objects in $\Mono(X)$. Since $f^*(\iota_M)$ is a split monomorphism, $f^*(g)$ must be a split monomorphism and $f^*(p)$ must factor through $f^*f_!(J)$. Hence, $p$ is $0$ in $\overline{\mathcal{C}}{}^{T(X)}$. It follows from  assumption (<ref>) that $p$ factors through an object of the form $f_!(I)$ with $I$ injective in $\mathcal{C}$. Hence $\Kopf_X(p)\colon \Kopf_X\module{E}\to K$ factors through $I$. We claim that $\Kopf_X(p)$ is also a split epimorphism. Indeed, applying $\Kopf_X$ to the diagram above gives a commutative diagram with exact rows

\[
\begin{tikzcd}
\end{tikzcd}
\]

Since $\Kopf_X(\iota_M)=0$ and the left hand square is a pushout, the map $M\to \Kopf_X\module{E}$ is a split monomorphism. Therefore $\Kopf_X(p)$ must be a split epimorphism. Since $\Kopf_X(p)$ factors through $I$, it follows that the induced map $I\to K$ is also a split epimorphism. Hence $K$ is a summand of an injective object, and must therefore be injective.

Consider the category $\operatorname{rep}(Q,\mathcal{B})$ of representations of a finite acyclic quiver $Q$ in $\mathcal{B}$ as in Example <ref>, and assume $\mathcal{B}$ has enough injectives. Let $\Mono(Q,\mathcal{B})$ be the monomorphism category, and consider the functor

\[
\overline{\Mono}(Q,\mathcal{B})\to\operatorname{rep}(Q,\overline{\mathcal{B}})
\]

in Theorem <ref>.  If $Q$ has at least one arrow, then the image of $X$ contains at least one copy of $\mathcal{B}$. Then it follows from Theorem <ref> that the functor above is an equivalence if and only if $\mathcal{B}$ is hereditary. In contrast, if $Q$ has no arrows, than $X$ is the zero functor, so (<ref>) is automatically satisfied and in (<ref>), both categories coincide with $\prod_{\mathtt{i}\in Q_0} \overline{\mathcal{C}}$.

We are particularly interested in situations where $\overline{\mathcal{C}}$ is abelian. This implies that $\overline{\mathcal{C}}{}^{T(X)}$ is abelian, and hence $\overline{\Mono}(X)$ is abelian by Theorem <ref>. Using the Mimo-construction in Theorem <ref>, it follows that describing the indecomposable non-injective objects in $\Mono(X)$ is equivalent to describing the indecomposable objects in an abelian category, namely $\overline{\mathcal{C}}{}^{T(X)}$.

Assume $\mathcal{C}$ is the category of left $\Lambda$-modules for an Artin algebra $\Lambda$. Then $\mathcal{C}$ is hereditary if and only if $\Lambda$ is left hereditary. It was shown in <cit.> that for such rings the stable category of $\mathcal{C}$ modulo projectives, denoted $\underline{\mathcal{C}}$, is abelian if and only if the injective envelope of $\Lambda$ is projective. Since $\overline{\mathcal{C}}\cong \underline{\mathcal{C}}$ by <cit.>, this is also equivalent to $\overline{\mathcal{C}}$ being abelian. Now the injective envelope of $\Lambda$ being projective holds precisely if $\Lambda$ is isomorphic to a finite direct products of complete blocked triangular matrix algebras over division rings, see <cit.>.

Consider the category $\operatorname{rep}(Q,\mathcal{B})$ of representations of a finite acyclic quiver $Q$ in $\mathcal{B}$ as in Example <ref>. Assume furthermore that $\mathcal{B}=\operatorname{Mod}\mathbbm{k}\mathbb{A}_n$ where $\mathbbm{k}\mathbb{A}_n$ is the path algebra of a linearly oriented $\mathbb{A}_n$-quiver over a field $\mathbbm{k}$. Since $\overline{\operatorname{Mod}}\mathbbm{k}\mathbb{A}_n\cong \operatorname{Mod}\mathbbm{k}\mathbb{A}_{n-1}$ we get that

\[
\mathcal{C}=\prod_{\mathtt{i}\in Q_0}\operatorname{Mod}\mathbbm{k}\mathbb{A}_n \quad \text{and} \quad \overline{\mathcal{C}}\cong \prod_{\mathtt{i}\in Q_0}\operatorname{Mod}\mathbbm{k}\mathbb{A}_{n-1}.
\]

Hence by Theorem <ref>  we get an equivalence

\[
\overline{\Mono}(Q,\operatorname{Mod}\mathbbm{k}\mathbb{A}_{n})\cong \rep (Q,\operatorname{Mod}\mathbbm{k}\mathbb{A}_{n-1}).
\]

By letting $Q$ be the linearly oriented $\mathbb{A}_m$-quiver, we recover (the dual of) <cit.>. Note that

\[
\prod_{\mathtt{i}\in Q_0}\operatorname{Mod}\mathbbm{k}\mathbb{A}_n\cong \operatorname{Mod}\Lambda
\]

where $\Lambda=\prod_{\mathtt{i}\in Q_0}\mathbbm{k}\mathbb{A}_n$ satisfies the condition in Remark <ref>.

There are other examples of hereditary categories $\mathcal{C}$ such that $\overline{\mathcal{C}}$ is abelian. Indeed, let $\mathcal{C}=\operatorname{art}\Lambda$ be the category of artinian modules over a Dedekind domain $\Lambda$. This is abelian, closed under injective envelopes <cit.>, and hereditary. We claim that any artinian module must be a finite sum of indecomposable injective modules and modules of finite length. Indeed, since any indecomposable finite length module over $\Lambda$ is of the form $\Lambda/\mathfrak{m}^n$ where $\mathfrak{m}$ is a maximal ideal, the finite length modules form a uniserial category in the sense of [44]. By <cit.> any artinian module is a filtered colimit of finite length modules, so the claim follows from <cit.>.

Consider the stable category $\overline{\mathcal{C}}$. Since any injective module over a Dedekind domain has infinite length <cit.>, the objects in $\overline{\mathcal{C}}$ are up to isomorphism precisely the modules of finite length. Furthermore, since $\Lambda$ is hereditary there are no nonzero morphisms from injective modules to modules of finite length, so $\overline{\mathcal{C}}$ must be equivalent to the category $\operatorname{fl}\Lambda$ of finite length modules over $\Lambda$, which is abelian.

§ THE MIMO-CONSTRUCTION

We fix a $\mathbbm{k}$-linear abelian category $\mathcal{C}$ with enough injectives, and an exact functor $X\colon \mathcal{C}\to \mathcal{C}$ which is locally nilpotent and preserves injectives. The goal of this section is to extend the Ringel–Schmidmeier's $\Mimo$-construction [76] to our setting. In particular, we show that it is a minimal right $\Mono(X)$–approximation.

§.§ Definition and properties

We start with the definition of the Mimo-construction.

Let $\module{M}$ be an object in $\mathcal{C}^{T(X)}$, and consider the construction $\Mo \module{M}$ in Definition <ref>. If $J$ is an injective envelope of $L_1\Kopf_{X}\module{M}$, then we write $\Mimo \module{M}\coloneqq \Mo \module{M}$ and call this the Mimo-construction of $\module{M}$.

Note that $\Mimo\module{M}$ might not exist for all $\module{M}$, unless $\mathcal{C}$ has injective envelopes. We show that if $\Mimo\module{M}$ exists, then it is a minimal right approximation and therefore unique up to isomorphism.

Let $\module{M}\in \mathcal{C}^{T(X)}$, and assume $\Mimo \module{M}$ exists. Then the canonical morphism $p_{\module{M}}\colon \Mimo\module{M}\to \module{M}$ is a minimal right $\Mono(X)$-approximation.

By Theorem <ref> we only need to show $p_\module{M}$ is right minimal. Let $\varphi\colon \Mimo\module{M}\to \Mimo\module{M}$ be a morphism satisfying $p_{\module{M}}\circ \varphi = p_{\module{M}}$. Consider the following commutative diagram with exact rows

\[
\begin{tikzcd}
\end{tikzcd}
\]

where $\psi$ is induced from the commutativity of the right hand square. Applying $\Kopf_{X}$ yields the commutative diagram:

\[
\begin{tikzcd}
0\arrow{r} &L_1\Kopf_{X}\module{M}\arrow[equals]{d}\arrow{r}{j}&J\arrow{d}{\Kopf_{X}(\psi)}\arrow{r}&\Kopf_{X}(\Mimo\module{M})\arrow{d}{\Kopf_{X}(\varphi)}\arrow{r}&\Kopf_{X}\module{M}\arrow{r}\arrow[equals]{d}&0\\
0\arrow{r} &L_1\Kopf_{X}\module{M}\arrow{r}{j}&J\arrow{r}&\Kopf_{X}(\Mimo\module{M})\arrow{r}&\Kopf_{X}\module{M}\arrow{r}&0 .
\end{tikzcd}
\]

As $j$ is a minimal left approximation, it follows that $\Kopf_{X}(\psi)$ is an isomorphism. Therefore $\psi$ is an isomorphism by Lemma <ref> part (<ref>). The $5$-Lemma then implies that $\varphi$ is an isomorphism.

Next we show that the isomorphism class of $\Mimo \module{M}$ only depends on the isomorphism class of $\module{M}$ in  $\overline{\mathcal{C}}{}^{T(X)}$. This plays an important role in Section <ref>. First we prove an analogue for $\Mo \module{M}$.

Let $\module{M},\module{N}\in \mathcal{C}^{T(X)}$. If $\module{M}\cong \module{N}$ in $\overline{\mathcal{C}}{}^{T(X)}$, then $\Mo\module{M}\cong \Mo\module{N}$ in $\overline{\Mono}(X)$.

Consider the composite of the isomorphisms $\Mo \module{M}\xrightarrow{\cong} \module{M}\xrightarrow{\cong} \module{N}\xrightarrow{\cong} \Mo\module{N}$ in $\overline{\mathcal{C}}{}^{T(X)}$. By Lemma <ref> the functor $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ is full, and hence this composite can be lifted to a morphism $\Mo \module{M}\to \Mo\module{N}$ in $\overline{\Mono}(X)$. By Lemma <ref> the functor $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ reflects isomorphisms, so the lift has to be an isomorphism in $\overline{\Mono}(X)$. This proves the claim.

We are not claiming that $\Mo$ is a functor on $\overline{\mathcal{C}}{}^{T(X)}$ in Lemma <ref>. As one can see from the proof, the choice of the isomorphism $\Mo\module{M}\cong \Mo\module{N}$ is not unique.

To obtain a similar result for $\Mimo$, we first need to prove a result on the non-existence of nonzero injective summands.

Assume $\mathcal{C}$ has injective envelopes. Let $\module{M}\in \mathcal{C}^{T(X)}$, and assume $M$ has no nonzero injective summands. Let $g\colon f_!(I)\to \Mimo\module{M}$ be a morphism in $\Mono(X)$. If $I$ is injective, then $g$ must be in the radical of $\Mono(X)$. In particular, $\Mimo\module{M}$ has no nonzero summands in $\operatorname{add} f_!(\inj \mathcal{C})$.

By definition, $g$ is in the radical if for any morphism $g'\colon \Mimo \module{M}\to f_!(I)$ the difference $1_{f_!(I)}-g'\circ g$ is an isomorphism. For this, it suffices to show that $g'\circ g$ is in the radical, since then by definition $1_{f_!(I)}-1_{f_!(I)}\circ (g'\circ g)=1_{f_!(I)}-g'\circ g$ is an isomorphism. Let $r\colon f_!(M)\to \Mimo\module{M}$ and $s\colon f_!(J)\to \Mimo\module{M}$ be as in diagram (<ref>) (where $\Mo\module{M}$ is replaced by $\Mimo\module{M}$). Consider the diagram

\[
\begin{tikzcd}
&f_!(M)\oplus f_!(J)\arrow{u}{(r,s)}\arrow{ru}[swap]{(g'\circ r,g'\circ s)}&
\end{tikzcd}
\]

We want to show the existence a morphism $k$ making the left triangle commutative. To this end, note that the triangle identities for the adjunction $f_!\dashv f^*$ imply that $f^*(\varepsilon_\module{M})$ is a split epimorphism. Thus, the sequence

\[
0\to f^*f_!X(M)\xrightarrow{f^*(\iota_M)-f^*f_!(h_\module{M})} f^*f_!(M)\xrightarrow{f^*(\varepsilon_{\module{M}})} M\to 0
\]

is split exact. Hence, $f^*(\iota_M)-f^*f_!(h_\module{M})$ is a split monomorphism. Therefore the short exact sequence (obtained by the definition of $\Mimo\module{M}$ as a pushout)

\begin{equation*}
\resizebox{\textwidth}{!}{$ 0\to f^*f_!X(M)\xrightarrow{\begin{pmatrix} f^*f_!(e) \\ f^*(\iota_M)-f^*f_!(h_\module{M}) \end{pmatrix}} f^*f_!(J)\oplus f^*f_!(M)\xrightarrow{\begin{pmatrix} f^*(s) & f^*(r) \end{pmatrix}} f^*(\Mimo\module{M})\to 0$ }
\end{equation*}

is split exact. It follows that $(f^*(s), f^*(r))$ is a split epimorphism. By Lemma <ref> this implies the existence of the morphism $k$ above.

Next we claim that the morphism $(\Kopf_X(g'\circ r),\Kopf_X(g'\circ s))$ is in the radical. Indeed, the morphism $\Kopf_X(g'\circ r)\colon M\to I$ is in the radical by Lemma <ref> (<ref>) since $M$ has no nonzero injective summands. Also, the composite

\[
L_1\Kopf_{X}\module{M}\xrightarrow{j}J\xrightarrow{\Kopf_{X}(g'\circ s)}I
\]

must be $0$, by construction of $\Mimo$. Hence $\Kopf_{X}(g'\circ s)$ factors through the cokernel of $j$, which is a radical morphism by Lemma <ref> (<ref>), since $j$ is an injective envelope. Hence $\Kopf_{X}(g'\circ s)$ must itself be a radical morphism. This shows that $(\Kopf_X(g'\circ r),\Kopf_X(g'\circ s))$ is in the radical. By Lemma <ref> the morphism  $(g'\circ r,g'\circ s)$ is in the radical of $\Mono(X)$. Since $g'\circ g$ factors through $(g'\circ r,g'\circ s)$, this proves the claim.

We can now prove a similar result to Lemma <ref> for the Mimo-construction. This is an analogue of <cit.>.

Assume $\mathcal{C}$ has injective envelopes. Let $\module{M}$ and $\module{N}$ be objects in $\mathcal{C}^{T(X)}$. Assume that $M$ and $N$ have no nonzero injective summands, and that $\module{M}\cong \module{N}$
in $\overline{\mathcal{C}}{}^{T(X)}$. Then $\Mimo\module{M}\cong \Mimo\module{N}$ in $\Mono(X)$.

By Lemma <ref> it follows that $\Mimo\module{M}\cong \Mimo\module{N}$ in $\overline{\Mono}(X)$. Hence, there exist injective objects $I_1,I_2\in \mathcal{C}$ such that $ \Mimo\module{M}\oplus f_!(I_1)$ and $\Mimo\module{N}\oplus f_!(I_2)$ are isomorphic. The claim now follows from Lemma <ref> and Lemma <ref>.

§.§ Mimo for modulations

To describe $\Mimo \module{M}$ for a modulation, we first give a description of it as an object in $(X\Downarrow \operatorname{Id}_{\mathcal{C}})$ and of $r,s$ and $p_{\module{M}}$ in (<ref>) as morphisms in $\mathcal{C}$. Recall that $f^*f_!(J)=\bigoplus_{i\geq 0}X^i(J)$ and $f^*f_!(M)=\bigoplus_{i\geq 0}X^i(M)$.

Fix the notation as in Definition <ref>. Then there exists an isomorphism $f^*(\Mo \module{M})\cong M\oplus \bigoplus_{i\geq 0}X^i(J)$ such that:

*  $f^*(s)=\begin{pmatrix}0\\1\end{pmatrix}\colon \bigoplus_{i\geq 0}X^i(J)\to M\oplus \bigoplus_{i\geq 0}X^i(J)$.

*  $f^*(r)\colon  \bigoplus_{i\geq 0}X^i(M)\to M\oplus \bigoplus_{i\geq 0}X^i(J)$ is induced by the maps

* $X^k(M)\xrightarrow{h_\module{M}\circ \cdots \circ X^{k-1}(h_{\module{M}})}M$ for all $k\geq 0$. (where $k=0$ gives the identity map).

* $X^{j+k+1}(M)\xrightarrow{X^{j}(e)\circ X^{j+1}(h_{\module{M}})\circ \cdots \circ X^{j+k}(h_{\module{M}})} X^j(J)$ for all $j,k\geq 0$.

*  $f^*(p_{\module{M}})=\begin{pmatrix}1&0\end{pmatrix}\colon  M\oplus \bigoplus_{i\geq 0}X^i(J)\to M$.

*   $h_{\Mo\module{M}}\colon X(M)\oplus \bigoplus_{i\geq 0}X^{i+1}(J)\to M\oplus \bigoplus_{i\geq 0}X^i(J)$ is induced by the maps

* $X^{i+1}(J)\xrightarrow{1} X^{i+1}(J)$ for all $i\geq 0$.

* $X(M)\xrightarrow{h_{\module{M}}}M$.

* $X(M)\xrightarrow{e}J$.

Assume we have an object $\module{M}'\in (X\Downarrow \operatorname{Id}_{\mathcal{C}})$ defined by $f^*(\module{M}')=M\oplus \bigoplus_{i\geq 0}X^i(J)$ and by a map $h_{\module{M}'}\colon Xf^*(\module{M}')\to f^*(\module{M'})$ as in (<ref>). Recall that $h_{f_!(M)}\colon \bigoplus_{i\geq 0}X^{i+1}(M)\to \bigoplus_{i\geq 0}X^{i}(M)$ and $h_{f_!(J)}\colon \bigoplus_{i\geq 0}X^{i+1}(J)\to \bigoplus_{i\geq 0}X^{i}(J)$ are induced by the maps $X^{i+1}(M)\xrightarrow{1}X^{i+1}(M)$ and $X^{i+1}(J)\xrightarrow{1}X^{i+1}(J)$ for $i\geq 0$, respectively. A straightforward computation then gives that

\begin{align*}
h_{\module{M}'}\circ X(r')=r'\circ h_{f_!(M)} \quad \text{and} \quad  h_{\module{M}'}\circ X(s')=s'\circ h_{f_!(J)}
\end{align*}

where $r'$ and $s'$ are defined using the formulas in (<ref>) and (<ref>), respectively. Hence there exist well-defined  morphisms $r\colon f_!(M)\to \module{M}'$ and $s\colon f_!(J)\to \module{M}'$ in $\mathcal{C}^{T(X)}\cong (X\Downarrow \operatorname{Id}_{\mathcal{C}})$ such that $f^*(r)=r'$ and $f^*(s)=s'$. Now observe that the sequence

\[
0\to f_!X(M)\xrightarrow{\begin{pmatrix}\iota_M-f_!(h_{\module{M}})\\f_!(e)\end{pmatrix}}f_!(M)\oplus f_!(J)\xrightarrow{\begin{pmatrix}r&-s\end{pmatrix}} \module{M}'\to 0
\]

is exact, since it is exact when applying $f^*$. This implies that the left square in

\begin{equation*}
\begin{tikzcd}[column sep=2cm]
0\arrow{r} &f_!X(M)\arrow{r}{\iota_M-f_!(h_\module{M})}\arrow{d}{f_!(e)}&f_!(M)\arrow{d}{r}\arrow{r}{\varepsilon_{\module{M}}}&\module{M}\arrow[equals]{d}\arrow{r}&0\\
0\arrow{r} &f_!(J)\arrow{r}{s}&\module{M'}\arrow[dashed]{r}{p_{\module{M}}}&\module{M}\arrow{r}&0
\end{tikzcd}
\end{equation*}

is a pushout square. But since $\Mo \module{M}$ is also defined by this pushout square, we get an isomorphism $\module{M'}\cong \Mo \module{M}$. Finally, since $f^*(p_{\module{M}})$ as defined in (<ref>) is the cokernel of $f^*(s)$, it must lift to a map $p_{\module{M}}\colon \module{M'}\to \module{M}$ such that the lower sequences in the diagram is exact. Since $f^*(p_{\module{M}})\circ f^*(r)=f^*(\varepsilon_{\module{M}})$, the right square must be commutative. This proves the claim.

Let $Q$ be a finite and acyclic quiver and $\mathfrak{B}$ a modulation by abelian categories with enough injectives and exact functors preserving injective objects, see Example <ref>. Let $(B_\mathtt{i},B_\alpha)_{\mathtt{i}\in Q_0,\alpha\in Q_1}$ be a $\mathfrak{B}$-representation. We want to compute $\Mo (B_\mathtt{i},B_\alpha)$ as in Definition <ref>. To this end, for each $\mathtt{k}\in Q_0$ choose an injective object $J_\mathtt{k}$ in $\mathcal{B}_{\mathtt{k}}$ and a map $e_{\mathtt{k}}\colon \bigoplus_{\substack{\alpha\in Q_1, t(\alpha)=\mathtt{k}}} F_{\alpha}(B_{s(\alpha)})\to J_{\mathtt{k}}$ whose restriction to $L_1\Kopf_X(B_\mathtt{i},B_\alpha)_{\mathtt{k}}=\ker B_{\mathtt{k},\operatorname{in}}$ is a monomorphism, see Example <ref>. From Lemma <ref> it follows that

\[
\Mo (B_\mathtt{i},B_\alpha)_{\mathtt{k}}= B_{\mathtt{k}}\oplus \bigoplus_{p\in Q_{\geq 0},t(p)=\mathtt{k}}F_p(J_{s(p)})
\]

By Lemma <ref> (<ref>) the morphism $\Mo (B_\mathtt{i},B_\alpha)_{\beta}\colon \Mo (B_\mathtt{i},B_\alpha)_{\mathtt{j}}\to \Mo (B_\mathtt{i},B_\alpha)_{\mathtt{k}}$ associated to an arrow $\beta\colon \mathtt{j}\to \mathtt{k}$ is the  map

\[
F_{\beta}(B_{\mathtt{j}})\oplus \bigoplus_{p\in Q_{\geq 0},t(p)=\mathtt{j}}F_\beta F_p(J_{s(p)})\to B_{\mathtt{k}}\oplus \bigoplus_{q\in Q_{\geq 0},t(q)=\mathtt{k}}F_q(J_{s(q)})
\]

which is induced by the identity $F_\beta F_p(J_{s(p)})\xrightarrow{1}F_q(J_{s(q)})$ for $q=\beta p$, the map $B_\beta\colon F_{\beta}(B_\mathtt{j})\to B_\mathtt{k}$, and the composite $F_{\beta}(B_\mathtt{j})\xrightarrow{} \bigoplus_{\substack{\alpha\in Q_1, t(\alpha)=\mathtt{k}}} F_{\alpha}(B_{s(\alpha)})\xrightarrow{e_{\mathtt{k}}} J_{\mathtt{k}}$ where the first map is the inclusion. If the restriction of $e_\mathtt{k}$ to $\ker B_{\mathtt{k},\operatorname{in}}$ is an injective envelope for all $\mathtt{k}\in Q_0$, then $\Mo(B_\mathtt{i},B_\alpha)=\Mimo (B_\mathtt{i},B_\alpha)$ and we get a formula for the Mimo-construction.

Consider the category of representations $\operatorname{rep}(Q,\mathcal{B})$ as in Example <ref>. In this case the Mimo-construction $\operatorname{Mimo}(B_\mathtt{i},B_\alpha)=(B'_\mathtt{i},B'_\alpha)$ of an object $(B_\mathtt{i},B_\alpha)$ is given as follows: Choose an injective envelope $j_\mathtt{i}\colon K_\mathtt{i}\to J_\mathtt{i}$ for each $\mathtt{i}\in Q_0$, where $K_\mathtt{i}$ is the kernel of the morphism

\[
B_{\mathtt{i},\arrowin}\colon \bigoplus_{\substack{\alpha\in Q_1\\t(\alpha)=\mathtt{i}}} B_{s(\alpha)}\xrightarrow{(B_\alpha)_\alpha} B_\mathtt{i}.
\]

Let $e_{\mathtt{i}}\colon \bigoplus_{\alpha\in Q_1, t(\alpha)=\mathtt{i}} B_{s(\alpha)}\to J_\mathtt{i}$ be a lift of $j_i$ via the inclusion $K_\mathtt{i}\to \bigoplus_{\alpha\in Q_1, t(\alpha)=\mathtt{i}} B_{s(\alpha)}$. Then

\[
B'_{\mathtt{i}}= B_{\mathtt{i}}\oplus \bigoplus_{p\in Q_{\geq 0},t(p)=\mathtt{i}}J_{s(p)}
\]

where $Q_{\geq 0}$ is the set of paths in $Q$, and $s(p)$ and $t(p)$ denotes the source and target of $p$, respectively. For an arrow $\beta\colon \mathtt{i}\to \mathtt{k}$, the morphism

\[
B'_\beta\colon B_{\mathtt{i}}\oplus \bigoplus_{p\in Q_{\geq 0},t(p)=\mathtt{i}}J_{s(p)}\to B_{\mathtt{k}}\oplus \bigoplus_{q\in Q_{\geq 0},t(q)=\mathtt{k}}J_{s(q)}
\]

is induced by the identity $J_{s(p)}\xrightarrow{1}J_{s(q)}$ for $q=\beta p$, the structure map $B_\beta\colon B_\mathtt{i}\to B_\mathtt{k}$, and the composite $B_\mathtt{i}\xrightarrow{} \bigoplus_{\substack{\alpha\in Q_1, t(\alpha)=\mathtt{k}}} B_{s(\alpha)}\xrightarrow{e_{\mathtt{k}}} J_{\mathtt{k}}$ where the first map is the canonical inclusion.

This formula has already been obtained in <cit.>. In <cit.> they show that it gives a right $\Mono(X)$-approximation. In Theorem <ref> we prove the same results. Note that the proofs are shorter and more transparent in our language.

§ A CHARACTERIZATION OF THE INDECOMPOSABLE OBJECTS IN $\MONO(X)$

By Theorem <ref> there is a bijection between isomorphism classes of indecomposable objects in $\overline{\Mono}(X)$ and $\overline{\mathcal{C}}{}^{T(X)}$, induced  by the epivalence

\[
\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}.
\]

Under some mild additional assumptions this is also in bijection with isomorphism classes of non-injective indecomposable objects in $\Mono(X)$. The goal in this section is to provide an explicit formula for this latter bijection. More precisely, we show that the Mimo-construction extends to objects in $\overline{\mathcal{C}}{}^{T(X)}$, and that it gives a bijection between indecomposable objects in $\overline{\mathcal{C}}{}^{T(X)}$ and non-injective indecomposable objects in $\Mono(X)$. We illustrate the usefulness of this result on examples in Section <ref>.  For the result to hold $\mathcal{C}$ must admit maximal injective summands, see Definition <ref>. We therefore start by showing that any noetherian, artinian, or locally noetherian category admits maximal injective summands.

§.§ Maximal injective summand

Let $\mathcal{C}$ be an abelian category with injective envelopes. We say that $\mathcal{C}$ admits maximal injective summands if for any $M\in \mathcal{C}$ there exists an isomorphism

\[
M\cong M'\oplus I
\]

where $I$ is injective and $M'$ has no nonzero injective summands.

Note that $M'$ and $I$ are unique up to isomorphism by Lemma <ref> (<ref>) and Lemma <ref>.

Recall that an object $M\in \mathcal{C}$ is artinian if any decreasing sequence

\[
\cdots\subseteq M_1\subseteq M_0 \subseteq M
\]

of subobjects of $M$ stabilizes, and noetherian if any increasing sequence

\[
M_0\subseteq M_1\subseteq M_2\subseteq \cdots \subseteq M
\]

of subobjects of $M$ stabilizes.  The category $\mathcal{C}$ is artinian if each object in $\mathcal{C}$ is artinian, and noetherian if each object in $\mathcal{C}$ is noetherian. Finally, $\mathcal{C}$ is called locally noetherian if it is a Grothendieck category with a generating set of noetherian objects. Note that any locally noetherian category has injective envelopes, see <cit.>.

Let $\mathcal{C}$ be an abelian category with injective envelopes. Then $\mathcal{C}$ admits maximal injective summands if one of the following conditions hold:

*  Any injective object in $\mathcal{C}$ can be written as a finite direct sum of indecomposables.

*  $\mathcal{C}$ is artinian.

*  $\mathcal{C}$ is noetherian.

*  $\mathcal{C}$ is locally noetherian.

Assume property (<ref>). Since $\mathcal{C}$ has injective envelopes, the endomorphism ring of any indecomposable injective object in $\mathcal{C}$ has local endomorphism ring by the proof of <cit.>. Hence, the injective objects in $\mathcal{C}$ form a Krull–Remak–Schmidt category, so any injective object can be written uniquely as a sum of indecomposable objects up to permutation and isomorphism.

Now let $M\in \mathcal{C}$ be arbitrary, and let $M\to E(M)$ be its injective envelope. Since any injective summand of $M$ must be an injective summand of $E(M)$, it follows that the number of indecomposable summands of $M$ must be bounded by the number of indecomposable summands of $E(M)$, which is finite by hypothesis. So we can choose an injective summand $I$ of $M$ with the maximal amount of indecomposable summands. Writing $M\cong I\oplus M'$ we see that $M'$ has no injective summands, since otherwise there would exist an injective summand of $M$ with more indecomposable summands than $I$. This shows that $\mathcal{C}$ admits maximal injective summands.

Next we show that $\mathcal{C}$ being noetherian or artinian implies condition (<ref>). Indeed, assume an injective object $I$ cannot be written as a finite direct sum of indecomposable injectives. Then there exist nonzero injective objects $I_n,I_n'$ with $I=I_0'\oplus I_0$ and $I_n=I'_{n+1}\oplus I_{n+1}$ for $n\geq 0$. We then have a strictly decreasing and a strictly increasing sequence of subobjects of $I$

\begin{align*}
\cdots \subset I_2\subset I_1\subset I_0\subset I \quad \text{and} \quad 	I_0'\subset I_0'\oplus I_1'\subset I_0'\oplus I_1'\oplus I_2'\subset \cdots \subset I.
\end{align*}

Hence, $\mathcal{C}$ can't be noetherian or artinian.

Finally, assume $\mathcal{C}$ is a locally noetherian category. Then injective objects are closed under filtered colimits, see <cit.>.  Therefore, Zorn's lemma implies that $M$ has a maximal injective subobject $I$. Furthermore, the inclusion $I\to M$ must be split, since $I$ is injective. Hence, we have an isomorphism $M\cong M'\oplus I$ for some object $M'$. If $M'$ has a nonzero injective summand $J$, then $I\oplus J$ is an injective subobject of $M$ which strictly contains $I$. This contradicts the maximality of $I$. Hence $M'$ has no nonzero injective summands.

If $\mathcal{C}$ is the category of quasi-coherent sheaves over a noetherian scheme, or the category of all modules over a noetherian ring, then $\mathcal{C}$ is locally noetherian. The existence of maximal injective summands in the latter case was first shown in [63].

Following [35] a ring $\Lambda$ is called right co-noetherian if injective envelopes of simple right $\Lambda$-modules are artinian. By <cit.> this is equivalent to injective envelopes of artinian right modules being artinian. Hence, the category $\mathcal{C}=\operatorname{art}\Lambda$ of artinian right $\Lambda$-modules is an artinian abelian category with injective envelopes, and therefore admits maximal injective summands. Examples of co-noetherian rings are commutative noetherian rings <cit.>, Quasi-Frobenius rings <cit.>, Noetherian PI rings <cit.>, module finite algebras over commutative noetherian rings <cit.>, finite normalizing extensions of a right co-noetherian ring <cit.>, and the first Weyl algebra of a commutative ring finitely generated as an algebra over the integers <cit.>. A commutative ring is co-noetherian if and only if its localizations at any maximal ideal is noetherian <cit.>.

There exist abelian categories with injective envelopes which do not satisfy Definition <ref>. Indeed, let $\mathcal{C}$ be any Grothendieck category which is not locally noetherian, e.g. the category of all modules over a non-noetherian ring. Since $\mathcal{C}$ is Grothendieck, it has injective envelopes, see <cit.>. Also, since $\mathcal{C}$ is not locally noetherian, there exists a set $\mathcal{J}$ of injective objects such that the sum $M=\bigoplus_{J\in \mathcal{J}}J$ is not injective, see <cit.>. We claim that $M$ has no maximal injective summand. Assume otherwise, i.e. that $M\cong M'\oplus I$ where $J$ is injective and $M'$ has no nonzero injective summands. Let $\mathcal{J}'\subset \mathcal{J}$ be a finite subset, and let $I'=\bigoplus_{J\in \mathcal{J'}}J$ be the corresponding injective object. Choose a left inverse $M\to I'$ to the inclusion $I'\to M$. Via the isomorphism $M\cong M'\oplus I$ we get morphisms

\[
\begin{pmatrix}g_1\\g_2\end{pmatrix}\colon I'\to M'\oplus I \quad \text{and} \quad \begin{pmatrix}g'_1&g'_2\end{pmatrix}\colon M'\oplus I\to I'
\]

such that $g'_1\circ g_1+g'_2\circ g_2=1_{I'}$. By Lemma <ref> (<ref>) the morphism $g_1$ is in the radical of $\mathcal{C}$, so $g_2'\circ g_2=1_{I'}-g_1'\circ g_1$ must be an isomorphism. In particular, $g_2\colon I'\to I$ is a monomorphism. Now consider the morphism $\bigoplus_{J\in \mathcal{J}}J=M\cong M'\oplus I\to I$. We have shown that this is a monomorphism when restricted to the direct sum of any finite subset of $\mathcal{J}$. Since $\bigoplus_{J\in \mathcal{J}}J$ is the filtered colimit of such finite sums, and filtered colimits in Grothendieck categories are exact, the morphism $\bigoplus_{J\in \mathcal{J}}J\to I$ must itself be a monomorphism. Since it is clearly an epimorphism, it must be an isomorphism. But this implies that $M=\bigoplus_{J\in \mathcal{J}}J$ is injective, which is a contradiction.

§.§ Construction and main result

Let $\mathcal{C}$ be an abelian category with injective envelopes and maximal injective summands, and let $X\colon \mathcal{C}\to \mathcal{C}$ be an exact functor which is locally nilpotent and preserves injective objects. Our goal is to define the Mimo-construction directly on objects in the Eilenberg–Moore category of the stable category $\overline{\mathcal{C}}$. To do this, we need the following lemma.

Let $\module{M}\in \overline{\mathcal{C}}{}^{T(X)}$. Then there exists an object $\widehat{\module{M}}\in \mathcal{C}^{T(X)}$ which is isomorphic to $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$ and for which $f^*(\widehat{\module{M}})=\widehat{M}$ has no injective summands.

By Lemma <ref> the data of an object $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$ is the same as the data of a morphism $h_\module{M}\colon X(M)\to M$ in $\overline{\mathcal{C}}$. Choose a lift $h'\colon X(M)\to M$ to $\mathcal{C}$ of $h_{\module{M}}$, and write $M\cong\widehat{M}\oplus J$ where $J$ is injective and $\widehat{M}$ has no injective summands. Now $h'$ gives a morphism $X(\widehat{M})\oplus X(J)\to \widehat{M}\oplus J$. Let $h''\colon X(\widehat{M})\to \widehat{M}$ be the restriction of $h'$, and let $\widehat{\module{M}}=(\widehat{M},h'')$ be the corresponding object in $\mathcal{C}^{T(X)}$. Clearly $\widehat{\module{M}}$ is isomorphic to $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$ and $\widehat{M}$ has no injective summands by construction.

We can now extend the Mimo-construction to objects in $\overline{\mathcal{C}}{}^{T(X)}$. Here the superscript $\cong$ indicates that we are considering isomorphism classes of objects.

For any $\module{M}\in \overline{\mathcal{C}}{}^{T(X)}$, choose an object $\widehat{\module{M}}$ in $\mathcal{C}^{T(X)}$ which is isomorphic to $\module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$ and such that $f^*(\widehat{\module{M}})=\widehat{M}$ has no nonzero injective summands in $\mathcal{C}$. Then the association $\module{M}\mapsto \Mimo \widehat{\module{M}}$ induces a well-defined map

\begin{align}\label{MimoMapOnStable}
\{\text{objects in }\overline{\mathcal{C}}{}^{T(X)}\}^{\cong} \xrightarrow{\Mimo} \{\text{objects in } \Mono(X)\}^{\cong}.
\end{align}

Furthermore, it is independent of choice of $\widehat{\module{M}}$.

It is well-defined and independent of choice by Corollary <ref> and Lemma <ref>.

We want to show that the map in Proposition <ref> restricts to an isomorphism between indecomposable objects in $\overline{\mathcal{C}}{}^{T(X)}$ and non-injective indecomposable objects in $\Mono(X)$. We first show that it can be used to give an inverse to the bijection

\begin{align}\label{Bijection:RepEquivIndec}
\{\text{objects in } \overline{\Mono}(X)\}^{\cong}\to \{\text{objects in }\overline{\mathcal{C}}{}^{T(X)}\}^{\cong}
\end{align}

coming from the epivalence $\overline{\Mono}(X)\to \overline{\mathcal{C}}{}^{T(X)}$ in Theorem <ref>.

Composing (<ref>) with the functor $\Mono(X)\to \overline{\Mono}(X)$ gives an inverse to the bijection (<ref>). In particular, (<ref>) is injective and preserves indecomposables.

Let $\module{M}\in \overline{\mathcal{C}}{}^{T(X)}$. Since the canonical morphism $\Mimo \widehat{\module{M}}\to \widehat{\module{M}}$ is an isomorphism in $\overline{\mathcal{C}}{}^{T(X)}$, and there is a canonical isomorphism $\widehat{\module{M}}\cong \module{M}$ in $\overline{\mathcal{C}}{}^{T(X)}$, the composite of (<ref>) with the functor $\Mono(X)\to \overline{\Mono}(X)$ must be an inverse to (<ref>). In particular, the composite is injective, so (<ref>) must be injective. To see that (<ref>) preserves indecomposables, note that its composite with the functor $\Mono(X)\to \overline{\Mono}(X)$ preserves indecomposables, since it is an inverse (<ref>). Furthermore,  $\Mimo \widehat{\module{M}}$ has no nonzero injective summands by Lemma <ref>, and must therefore be indecomposable in $\Mono(X)$ if it is indecomposable in $\overline{\Mono}(X)$. This proves the claim.

We now characterize the indecomposables in $\Mono(X)$. This is a very useful result, in particular when $\overline{\mathcal{C}}$ is abelian, so that the indecomposable objects in $\overline{\mathcal{C}}{}^{T(X)}$ are easier to compute. We illustrate its power in the next section for quiver representations of different classes of Artin algebras.

We have bijections

\begin{align}
\renewcommand{\arraystretch}{1.8}
\begin{array}{ccc}
\renewcommand{\arraystretch}{1.1}
\begin{Bmatrix}
\text{Indecomposable injective} \\
\text{objects in $\mathcal{C}$}
\end{Bmatrix}^{\cong}
\xrightarrow{\cong}
\renewcommand{\arraystretch}{1.2}
\begin{Bmatrix}
\text{Indecomposable injective} \\
\text{objects in $\Mono(X)$}
\end{Bmatrix}^{\cong} \quad I \mapsto  f_!(I).  \\
\end{array}\label{Bijection:Injective}
\renewcommand{\arraystretch}{1}
\end{align}

\begin{align}
\renewcommand{\arraystretch}{1.8}
\begin{array}{ccc}
\renewcommand{\arraystretch}{1.1}
\begin{Bmatrix}
\text{Indecomposable} \\
\text{objects in $\overline{\mathcal{C}}{}^{T(X)}$}
\end{Bmatrix}^{\cong}
\xrightarrow{\cong}
\renewcommand{\arraystretch}{1.2}
\begin{Bmatrix}
\text{Indecomposable non-injective} \\
\text{objects in $\Mono(X)$}
\end{Bmatrix}^{\cong} \quad \module{M} \mapsto  \Mimo \module{M}.\\
\end{array}\label{Bijection:Non-injective}
\renewcommand{\arraystretch}{1}
\end{align}

By Proposition <ref> and Corollary <ref> the injective objects in $\Mono(X)$ are precisely the objects of the form $f_!(J)$ with $J$ injective in $\mathcal{C}$. Furthermore, $f_!(J)$ is indecomposable if and only if $J$ is indecomposable by Corollary <ref>. This implies that the map (<ref>) is well-defined and surjective. For injectivity, note that $f_!(J)\cong f_!(J')$ implies $J\cong \Kopf f_!(J)\cong \Kopf f_!(J')\cong J'$.

Next, consider the map in (<ref>). It is well-defined and injective by Proposition <ref>. To see that it is surjective, let $\module{M}'\in \Mono(X)$ be an arbitrary indecomposable non-injective object, and let $\module{M}$ be the image of $\module{M}'$ in $\overline{\mathcal{C}}{}^{T(X)}$. Since composing (<ref>) with the functor $\Mono(X)\to \overline{\Mono}(X)$ gives an inverse to (<ref>) by Proposition <ref>, we get that $\module{M}'\cong \Mimo \widehat{\module{M}}$ in $\overline{\Mono}(X)$. Hence, there exist inverse isomorphisms

\[
\begin{pmatrix}
\phi_1&\phi_2\\
\phi_3&\phi_4
\end{pmatrix}\colon \module{M}'\oplus f_!(I)\to \Mimo \widehat{\module{M}}\oplus f_!(J)
\quad \text{and} \quad \begin{pmatrix}
\psi_1&\psi_2\\
\psi_3&\psi_4
\end{pmatrix}\colon \Mimo \widehat{\module{M}}\oplus f_!(J)\to \module{M}'\oplus f_!(I)
\]

in $\Mono(X)$ for some injective objects $I$ and $J$ in $\mathcal{C}$. In particular, we have that

\[
1_{\Mimo \module{M}'}=\phi_1\circ \psi_1+\phi_2\circ \psi_3.
\]

Now by Lemma <ref> the morphism $\phi_2$ is in the radical of $\Mono(X)$, so $\phi_1\circ \psi_1=1_{\Mimo \module{M}'}-\phi_2\circ \psi_3$ is an isomorphism. Hence $\phi_1\colon \module{M}'\to \Mimo \widehat{\module{M}}$ is a split epimorphism, and since $\module{M}'$ is indecomposable it must be an isomorphism. This shows that (<ref>) is surjective.

Theorem <ref> recovers <cit.> in the setting of Remark <ref>.

Let $Q$ be a finite and acyclic quiver and $\mathfrak{B}$ a modulation by abelian categories $\mathcal{B}_\mathtt{i}$ and exact functors $F_\alpha$ preserving injective objects, see Example <ref>. Set $\mathcal{C}=\prod_{\mathtt{i}\in Q_0}\mathcal{B}_\mathtt{i}$. Then $\mathcal{C}$ has injective envelopes resp. maximal injective summands if and only if $\mathcal{B}_\mathtt{i}$ has injective envelopes resp. maximal injective summands, for all $\mathtt{i}\in Q_0$. The Mimo map in Proposition <ref> goes from $\operatorname{rep}\overline{\mathfrak{B}}$ to the monomorphism category of $\mathfrak{B}$, where $\overline{\mathfrak{B}}$ is the modulation by stable categories as in Example <ref>. Explicitly, it sends $(B_\mathtt{i},B_\alpha)$ to $\Mimo (\widehat{B}_\mathtt{i},B'_\alpha)$ where $\widehat{B}_\mathtt{i}$ is any object in $\mathcal{B}_\mathtt{i}$ with no nonzero injective summands and which is isomorphic to $B_{\mathtt{i}}$ in $\overline{\mathcal{B}}_\mathtt{i}$, and $B'_\alpha\colon \widehat{B}_{\mathtt{i}}\to \widehat{B}_{\mathtt{i}}$ is any lift of $B_\alpha$ to $\mathcal{B}_\mathtt{i}$, and $\Mimo (\widehat{B}_\mathtt{i},B'_\alpha)$ is the Mimo-construction described in Example <ref>.

§ APPLICATIONS TO QUIVER REPRESENTATIONS OVER ARTIN ALGEBRAS

The goal of this section is to study monomorphism categories of quivers over Artin algebras, using our results. Throughout the section $Q$ denotes a finite acyclic quiver, $\mathbbm{k}$ is a commutative artinian ring, and $\Lambda$ is an artinian $\mathbbm{k}$-algebra. The categories of representations of $Q$ are denoted by $\rep (Q,\Mod \Lambda)$ and $\rep (Q,\operatorname{mod} \Lambda)$, see Example <ref>. They can be identified with the module categories $\operatorname{Mod}\Lambda Q^{\op}$ and $\operatorname{mod}\Lambda Q^{\op}$, respectively. If we set $\Lambda Q_0\coloneqq\prod_{\mathtt{i}\in Q_0}\Lambda$ then we get

\[
\operatorname{Mod}\Lambda Q_0=\prod_{\mathtt{i}\in Q_0}\operatorname{Mod}\Lambda \quad \text{and} \quad \operatorname{mod}\Lambda Q_0=\prod_{\mathtt{i}\in Q_0}\operatorname{mod}\Lambda.
\]

We have adjoint functors

\[
f_!\colon \operatorname{Mod}\Lambda Q_0\to \rep (Q,\Mod \Lambda) \quad \text{and} \quad f^*\colon \rep (Q,\Mod \Lambda)\to \operatorname{Mod}\Lambda Q_0
\]

which restrict to

\[
f_!\colon \operatorname{mod}\Lambda Q_0\to \rep (Q,\operatorname{mod} \Lambda) \quad \text{and} \quad f^*\colon \rep (Q,\operatorname{mod} \Lambda)\to \operatorname{mod}\Lambda Q_0.
\]

The monomorphism subcategories of $\rep (Q,\Mod \Lambda)$ and $\rep (Q,\operatorname{mod} \Lambda)$ are denoted by $\Mono_Q(\Lambda)$ and $\mono_Q(\Lambda)$, respectively. They consist of representations $(M_\mathtt{i},M_\alpha)_{\mathtt{i}\in Q_0, \alpha\in Q_1}$ for which

\[
M_{\mathtt{i},\operatorname{in}}\colon \bigoplus_{\substack{\alpha\in Q_1\\t(\alpha)=\mathtt{i}}}M_{s(\alpha)}\xrightarrow{(M_\alpha)} M_\mathtt{i}
\]

is a monomorphism for all $\mathtt{i}\in Q_0$, see Example <ref>. Next we recall the bijection in Theorem <ref> in this context. For $M\in \operatorname{mod}\Lambda$ we let $M(\mathtt{i})\in \operatorname{mod}\Lambda Q_0$ denote the object given by