# The categories of corings and coalgebras
over a ring are locally countably presentable

Leonid Positselski  Institute of Mathematics, Czech Academy of Sciences
Žitná 25, 115 67 Praha 1
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

For any commutative ring $R$, we show that the categories of $R$-coalgebras
and cocommutative $R$-coalgebras are locally $\aleph_{1}$-presentable, while
the categories of $R$-flat $R$-coalgebras are $\aleph_{1}$-accessible.
Similarly, for any associative ring $R$, the category of $R$-corings is
locally $\aleph_{1}$-presentable, while the category of $R$-$R$-bimodule flat
$R$-corings is $\aleph_{1}$-accessible. The cardinality of the ring $R$ can be
arbitrarily large. We also discuss $R$-corings with surjective counit and flat
kernel. The proofs are straightforward applications of an abstract category-
theoretic principle going back to Ulmer. For right or two-sided $R$-module
flat $R$-corings, our cardinality estimate for the accessibility rank is not
as good.

###### Contents

  1. 1 Category-Theoretic Preliminaries
  2. 2 Coalgebras over a Commutative Ring
  3. 3 Flat Coalgebras over a Commutative Ring
  4. 4 Corings over an Associative Ring
  5. 5 Bimodule-Flat Corings
  6. 6 Corings with Flat Kernel
  7. 7 Module-Flat Corings

## Introduction

The aim of this paper is to close a gaping hole in contemporary literature
concerning the local presentability rank of the categories of coalgebras and
corings over a fixed ring $R$. There is little really new material in this
paper, as our results go back to the unpublished 1977 preprint of Ulmer [20].
The results of this paper concerning various classes of _flat_ coalgebras and
corings are relatively more original.

Some ideas of the preprint [20] were taken up and developed in a different
form in the 1984 dissertation of Bird [3]; but that remained unpublished, too.
In the end, it appears that Ulmer’s ideas were almost completely forgotten and
not incorporated into the contemporary body of knowledge. This created the
gaping hole mentioned in the previous paragraph. We hope that this paper will
fare better than Ulmer’s preprint and people will remember the idea now.

One approach used in contemporary literature goes back to the 1974 paper of
Barr [2, Theorem 3.1]. This theorem claims that, for any commutative ring $R$,
any $R$-coalgebra $C$ is the directed union of its subcoalgebras
$C^{\prime}\subset C$ such that $C^{\prime}$ is a pure $R$-submodule in $C$
and the cardinality of $C^{\prime}$ does not exceed the cardinality of $R$
plus $\aleph_{0}$. The necessity to require the subcoalgebra $C^{\prime}$ to
be a pure $R$-submodule in $C$ arises from the well-known difficulty with
nonexactness of the tensor product in the theory of coalgebras or corings over
a ring. Basically, the very notion of a subcoalgebra does not make much sense
without the purity condition. It was emphasized in the present author’s recent
preprint [13] that the purification-based approach to accessibility often
leads to suboptimal cardinality bounds.

The references to [2] in connection with the claim that the category of
cocommutative $R$-coalgebras is locally $\kappa$-presentable for any infinite
cardinal $\kappa>|R|+\aleph_{0}$ can be found in [15, Theorem 2.2] and [16,
Section 3.4]. The question of finding the best possible cardinality estimate
for the local presentability rank of this category was posed in [15, Remark
2.3].

A category-theoretic approach to corings and comodules was suggested by Porst
in the paper [9]. Unfortunately, that paper does not reflect the understanding
of the locally presentable/accessible category theory involved that could be
found in Ulmer [20]. The result is that [9, Theorem 9] claims local
presentability of the categories of coalgebras and corings without any
explicit cardinality bound.

The paper [18, Section 3] treats the categories of $R$-flat cocommutative
$R$-coalgebras for Prüfer domains $R$. In particular, [18, Lemma 3.2] claims
that the category of $R$-flat $R$-coalgebras is locally presentable, with the
argument based on [9]. Once again, there is no explicit cardinality bound.

The classical case of coalgebras over a field $k$ is much easier than the
general case of a ground ring $R$. It is well-known that any coassociative
coalgebra over $k$ is the directed union of its finite-dimensional
subcoalgebras [19, Theorem 2.2.1]. It follows that the categories of
coassociative coalgebras (with or without cocommutativity or counitality) are
locally finitely presentable, with finitely presentable objects being
precisely all the finite-dimensional coalgebras.

The analogous assertion is _not_ true for Lie coalgebras over a field $k$ (e.
g., in characteristic $0$, the coalgebra dual to the topological Lie algebra
$k[[z]]d/dz$ of vector fields on the one-dimensional formal disk contains _no_
nonzero proper subcoalgebras). Still, one can easily show that any Lie
coalgebra over $k$ is the $\aleph_{1}$-directed union of its subcoalgebras of
at most countable dimension. Hence the category of Lie coalgebras over $k$ is
locally $\aleph_{1}$-presentable, and its $\aleph_{1}$-presentable objects are
precisely all the at most countably-dimensional Lie coalgebras.

Let us now describe the content of the present paper. Given a commutative ring
$R$, we consider the category of coassociative, counital $R$-coalgebras
$R{\operatorname{\mathsf{--Coalg}}}$ and the category of cocommutative,
coassociative, counital $R$-coalgebras $R{\operatorname{\mathsf{--Cocom}}}$.
We show that both the categories are locally $\aleph_{1}$-presentable. The
$\aleph_{1}$-presentable objects are the coalgebras that are
$\aleph_{1}$-presentable _as $R$-modules_. Furthermore, we point out that
similar results hold for noncounital or noncoassociative coalgebras, for Lie
coalgebras, for conilpotent coalgebras, and for DG-coalgebras.

We also consider the full subcategories
$R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Coalg}}}$ and
$R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Cocom}}}$ consisting of the coalgebras that are
_flat as $R$-modules_. We show that both the categories
$R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}$ and
$R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}$ are
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of these
categories are the coalgebras that are flat and $\aleph_{1}$-presentable as
$R$-modules. The similar assertions hold for the categories of noncounital or
noncoassociative coalgebras, Lie coalgebras, conilpotent coalgebras, and DG-
coalgebras.

Given a commutative ring $k$ and an associative $k$-algebra $R$, we consider
the category $R_{k}{\operatorname{\mathsf{--Corings}}}$ of coassociative,
counital corings over $R$ (i. e., comonoid objects in the monoidal category of
$R$-$R$-bimodules over $k$). We show that the category
$R_{k}{\operatorname{\mathsf{--Corings}}}$ is locally
$\aleph_{1}$-presentable. The $\aleph_{1}$-presentable objects are the corings
that are $\aleph_{1}$-presentable _as $R$-$R$-bimodules over $k$_ (i. e., as
modules over the ring $R\otimes_{k}R^{\mathrm{op}}$). Furthermore, we consider
the category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$ of
$R$-corings that are _flat as $R\otimes_{k}R^{\mathrm{op}}$-modules_. We show
that the category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$
is $\aleph_{1}$-accessible, and its $\aleph_{1}$-presentable objects are the
corings that are flat and $\aleph_{1}$-presentable as
$R\otimes_{k}R^{\mathrm{op}}$-modules.

As a variation on the previous result, we consider _corings with flat kernel_
, i. e., the $R$-corings $C$ such that the counit map $\epsilon\colon
C\longrightarrow R$ is surjective and its kernel $\overline{C}$ is a flat
$R$-$R$-bimodule over $k$. Such corings appear in connection with the
Burt–Butler theory of bocses [5, §3], [8, Definition 4.19 and Theorem 4.20].
Assuming that the ring $R\otimes_{k}R^{\mathrm{op}}$ is countably Noetherian
(e. g., just Noetherian in the usual sense), our result tells that the
category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\overline{\mathsf{bifl}}}$
of $R$-corings with flat kernel is $\aleph_{1}$-accessible, and its
$\aleph_{1}$-presentable objects are the corings with flat kernel that are
$\aleph_{1}$-presentable (equivalently, $<\aleph_{1}$-generated) as
$R\otimes_{k}R^{\mathrm{op}}$-modules.

More natural flatness conditions on an $R$-coring $C$ are that $C$ be flat as
a right $R$-module, or both as a right $R$-module and as a left $R$-module
(but not necessarily as a bimodule). The former condition characterizes the
$R$-corings $C$ for which the category of left $C$-comodules is abelian with
an exact forgetful functor to the category of left $R$-modules [4, Sections
18.6, 18.14, and 18.16], [10, Proposition 2.12(a)], [11, Lemma 2.1]; hence its
obvious importance. In this context, our results are less impressive, in that
the cardinality estimate is not any better than the one obtainable with the
purification-based approach. The problem is to obtain a good bound for the
accessibility rank of the additive category
$R{\operatorname{\mathsf{--Bimod_{fl}--}}}R$ of right $R$-flat
$R$-$R$-bimodules and the additive category
$R{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}R$ of left and right $R$-flat
$R$-$R$-bimodules.

Still, we explain how our category-theoretic approach can be applied to the
questions from the previous paragraph, if only for illustrative purposes. The
conclusion is that the category
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}$ of right $R$-flat
$R$-corings over $k$ and the category
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{lfrl}}$ of left and right
$R$-flat $R$-corings over $k$ are $\kappa$-accessible for any regular cardinal
$\kappa>|R|+\aleph_{0}$. The $\kappa$-presentable objects are the corings
(with the respective flatness property) whose underlying set has cardinality
less than $\kappa$.

We do not discuss comodules in this paper. An extensive treatment of
accessibility properties of comodule categories can be found in the paper [11,
Sections 2–6]; see in particular [11, Theorem 3.1 and Remark 3.2].

### Acknowledgement

The author is supported by the GAČR project 23-05148S and the Czech Academy of
Sciences (RVO 67985840).

##  1\. Category-Theoretic Preliminaries

We use the book [1] as the background reference source on locally presentable
and accessible categories. We refer to [1, Definition 1.4, Theorem and
Corollary 1.5, Definition 1.13(1), and Remark 1.21] for a relevant discussion
of _$\kappa$ -directed_ vs.  _$\kappa$ -filtered colimits_ for a regular
cardinal $\kappa$. Let us only recall that a poset is said to be _$\kappa$
-directed_ if every its subset of the cardinality smaller than $\kappa$ has an
upper bound.

Let $\kappa$ be a regular cardinal and $\mathsf{K}$ be a category with
$\kappa$-directed (equivalently, $\kappa$-filtered) colimits. An object
$S\in\mathsf{K}$ is said to be _$\kappa$ -presentable_ [1, Definitions 1.1 and
1.13(2)] if the functor
$\operatorname{Hom}_{\mathsf{K}}(S,{-})\colon\mathsf{K}\longrightarrow\mathsf{Sets}$
preserves $\kappa$-directed colimits. We denote the full subcategory of
$\kappa$-presentable objects by $\mathsf{K}_{<\kappa}\subset\mathsf{K}$.

The category $\mathsf{K}$ is called _$\kappa$ -accessible_ [1, Definition 2.1]
if there is a _set_ of $\kappa$-presentable objects
$\mathsf{S}\subset\mathsf{K}$ such that all the objects of $\mathsf{K}$ are
$\kappa$-directed colimits of objects from $\mathsf{S}$. If this is the case,
then the $\kappa$-presentable objects $\mathsf{K}$ are precisely all the
retracts of the objects from $\mathsf{S}$. A $\kappa$-accessible category
where all colimits exist is called _locally $\kappa$-presentable_ [1,
Definition 1.17 and Theorem 1.20].

$\aleph_{0}$-presentable objects are called _finitely presentable_ ,
$\aleph_{0}$-accessible categories are called _finitely accessible_ [1, Remark
2.2(1)], and locally $\aleph_{0}$-presentable categories are called _locally
finitely presentable_ [1, Definition 1.9 and Theorem 1.11]. We call
$\aleph_{1}$-presentable objects _countably presentable_ ,
$\aleph_{1}$-accessible categories _countably accessible_ , and locally
$\aleph_{1}$-presentable categories _locally countably presentable_.

######  Proposition 1.1.

Let $\kappa$ be a regular cardinal and $(\mathsf{K}_{\xi})_{\xi\in\Xi}$ be a
family of $\kappa$-accessible categories, indexed by a set $\Xi$ of the
cardinality smaller than $\kappa$. Then the Cartesian product
$\mathsf{K}=\prod_{\xi\in\Xi}\mathsf{K}_{\xi}$ is also a $\kappa$-accessible
category. The $\kappa$-presentable objects of $\mathsf{K}$ are precisely all
the collections of objects $(S_{\xi}\in\mathsf{K}_{\xi})_{\xi\in\Xi}$ such
that $S_{\xi}\in(\mathsf{K}_{\xi})_{<\kappa}$ for every $\xi\in\Xi$.

###### Proof.

This is a corrected version of [1, proof of Proposition 2.67]. See [12,
Proposition 2.1] for the details. ∎

In the following three theorems, we consider a regular cardinal $\kappa$ and a
smaller infinite cardinal $\lambda<\kappa$ (so $\kappa$ has to be
uncountable). A _$\lambda$ -indexed chain_ (of objects and morphisms) in a
category $\mathsf{K}$ is a directed diagram $(K_{i}\to K_{j})_{0\leq
i<j<\lambda}$ indexed by the ordered set $\lambda$. In the applications in the
subsequent sections of this paper, we will be mostly interested in the case
$\lambda=\aleph_{0}$ and $\kappa=\aleph_{1}$.

Let $\mathsf{K}$ and $\mathsf{L}$ be two categories, and let $F$,
$G\colon\mathsf{K}\rightrightarrows\mathsf{L}$ be a pair of parallel functors.
The _inserter category_ [1, Section 2.71] of the pair of parallel functors
$F$, $G$ is the category $\mathsf{D}$ whose objects are pairs $(K,\phi)$,
where $K\in\mathsf{K}$ is an object and $\phi\colon F(K)\longrightarrow G(K)$
is a morphism in $\mathsf{L}$. The morphisms in $\mathsf{D}$ are defined in
the obvious way.

######  Theorem 1.2.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}$ and $\mathsf{L}$ be $\kappa$-accessible categories
where colimits of $\lambda$-indexed chains exist. Let $F$,
$G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be a pair of parallel functors
preserving $\kappa$-directed colimits and colimits of $\lambda$-indexed
chains. Assume further that the functor $F$ takes $\kappa$-presentable objects
to $\kappa$-presentable objects. Then the inserter category $\mathsf{D}$ of
the pair of functors $F$, $G$ is $\kappa$-accessible. The $\kappa$-presentable
objects of $\mathsf{D}$ are precisely all the pairs $(S,\phi)\in\mathsf{D}$
with $S\in\mathsf{K}_{<\kappa}$.

###### Proof.

This result goes back to [20, Theorem 3.8, Corollary 3.9, and Remark
3.11(II)]. For a recent exposition, see [12, Theorem 4.1]. ∎

Let $\mathsf{K}$ and $\mathsf{L}$ be two categories, $F$,
$G\colon\mathsf{K}\rightrightarrows\mathsf{L}$ be a pair of parallel functors,
and $\phi$, $\psi\colon F\rightrightarrows G$ be a pair of parallel natural
transformations. The _equifier category_ [1, Lemma 2.76] of the pair of
parallel natural transformations $\phi$ and $\psi$ is the full subcategory
$\mathsf{E}\subset\mathsf{K}$ consisting of all objects $E\in\mathsf{K}$ for
which the morphisms $\phi_{E}$ and $\psi_{E}\colon F(E)\rightrightarrows G(E)$
are equal to each other in $\mathsf{L}$, that is $\phi_{E}=\psi_{E}$.

######  Theorem 1.3.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}$ and $\mathsf{L}$ be $\kappa$-accessible categories
where colimits of $\lambda$-indexed chains exist. Let $F$,
$G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be a pair of parallel functors
preserving $\kappa$-directed colimits and colimits of $\lambda$-indexed
chains. Assume further that the functor $F$ takes $\kappa$-presentable objects
to $\kappa$-presentable objects. Let $\phi$ and $\psi\colon F\longrightarrow
G$ be a pair of parallel natural transformations. Then the equifier category
$\mathsf{E}$ of the pair of natural transformations $\phi$, $\psi$ is
$\kappa$-accessible. The $\kappa$-presentable objects of $\mathsf{E}$ are
precisely all the objects of $\mathsf{E}$ that are $\kappa$-presentable in
$\mathsf{K}$.

###### Proof.

This also goes back to [20, Theorem 3.8, Corollary 3.9, and Remark 3.11(II)].
For a recent exposition, see [12, Theorem 3.1]. ∎

Let $\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ be three categories,
and $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ and
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be two functors. The
_pseudopullback category_ of the pair of functors $F_{1}$, $F_{2}$ is the
category $\mathsf{C}$ whose objects are triples $(K_{1},K_{2},\theta)$, where
$K_{1}\in\mathsf{K}_{1}$ and $K_{2}\in\mathsf{K}_{2}$ are objects, and
$\theta\colon F_{1}(K_{1})\simeq F_{2}(K_{2})$ is an isomorphism in
$\mathsf{L}$. The morphisms in $\mathsf{C}$ are defined in the obvious way.

######  Theorem 1.4.

Let $\kappa$ be a regular cardinal and $\lambda<\kappa$ be a smaller infinite
cardinal. Let $\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ be
$\kappa$-accessible categories where colimits of $\lambda$-indexed chains
exist. Let $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ and
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be two functors
preserving $\kappa$-directed colimits and colimits of $\lambda$-indexed
chains. Assume further that the functors $F_{1}$ and $F_{2}$ take
$\kappa$-presentable objects to $\kappa$-presentable objects. Then the
pseudopullback $\mathsf{C}$ of the pair of functors $F_{1}$, $F_{2}$ is
$\kappa$-accessible. The $\kappa$-presentable objects of $\mathsf{C}$ are
precisely all the triples $(S_{1},S_{2},\theta)\in\mathsf{C}$ with
$S_{1}\in(\mathsf{K}_{1})_{<\kappa}$ and $S_{2}\in(\mathsf{K}_{2})_{<\kappa}$.

###### Proof.

This result, going back to [20, Remark 3.2(I), Theorem 3.8, Corollary 3.9, and
Remark 3.11(II)], can be found in [17, Pseudopullback Theorem 2.2] with the
proof in [6, Proposition 3.1]. For another exposition, see [12, Corollary
5.1]. ∎

Before this section is finished, let us collect a couple of well-known module-
theoretic results concerning (local) presentability and accessibility. Given
an associative ring $R$, we denote by $R{\operatorname{\mathsf{--Mod}}}$ the
abelian category of left $R$-modules and by
$R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Mod}}}$ the full subcategory of flat left
$R$-modules.

######  Proposition 1.5.

Let $R$ be an associative ring and $\kappa$ be a regular cardinal. Then the
category $R{\operatorname{\mathsf{--Mod}}}$ of left $R$-modules is locally
$\kappa$-presentable. The $\kappa$-presentable objects of
$R{\operatorname{\mathsf{--Mod}}}$ are precisely all the $R$-modules that can
be presented as the cokernel of a morphism of free $R$-modules with less than
$\kappa$ generators. ∎

######  Proposition 1.6.

Let $R$ be an associative ring and $\kappa$ be a regular cardinal. Then the
category $R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$ of flat left
$R$-modules is $\kappa$-accessible. The $\kappa$-presentable objects of
$R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$ are precisely all the flat
$R$-modules that are $\kappa$-presentable in
$R{\operatorname{\mathsf{--Mod}}}$.

###### Proof.

See [12, Proposition 10.2] or [11, Lemma 1.2] for some discussion. ∎

##  2\. Coalgebras over a Commutative Ring

Let $R$ be a commutative ring. A (_coassociative, counital_) _coalgebra_ over
$R$ is a comonoid object in the monoidal category of $R$-modules
$R{\operatorname{\mathsf{--Mod}}}$ with respect to the operation of tensor
product over $R$. In other words, a coalgebra $C$ is an $R$-module endowed
with two maps of _comultiplication_ $\mu\colon C\longrightarrow C\otimes_{R}C$
and _counit_ $\epsilon\colon C\longrightarrow R$, which must be $R$-linear
maps satisfying the conventional coassociativity and counitality axioms.
Specifically, the two compositions

$C\longrightarrow C\otimes_{R}C\rightrightarrows C\otimes_{R}C\otimes_{R}C$

of the comultiplication map with two maps induced by the comultiplication map
must be equal to each other, that is
$(\mu\otimes\mathrm{id}_{C})\circ\mu=(\mathrm{id}_{C}\otimes\mu)\circ\mu$, and
the two compositions

$C\longrightarrow C\otimes_{R}C\rightrightarrows C$

of the comultiplication map with two maps induced by the counit map must be
equal to the identity map, that is
$(\epsilon\otimes\mathrm{id}_{C})\circ\mu=\mathrm{id}_{C}=(\mathrm{id}_{C}\otimes\epsilon)\circ\mu$.

An $R$-coalgebra $C$ is said to be _cocommutative_ if
$\sigma_{C}\circ\mu=\mu$,

$C\longrightarrow C\otimes_{R}C\longrightarrow C\otimes_{R}C,$

where $\sigma_{C}\colon C\otimes_{R}C\longrightarrow C\otimes_{R}C$ is the map
permuting the tensor factors. The notion of a cocommutative $R$-coalgebra
makes sense because $R{\operatorname{\mathsf{--Mod}}}$ is naturally a
symmetric monoidal category.

Morphisms of coalgebras $C\longrightarrow D$ are defined in the obvious way as
$R$-linear maps compatible with the comultiplication and counit. We denote the
category of $R$-coalgebras by $R{\operatorname{\mathsf{--Coalg}}}$ and the
full subcategory of cocommutative $R$-coalgebras by
$R{\operatorname{\mathsf{--Cocom}}}\subset
R{\operatorname{\mathsf{--Coalg}}}$.

######  Lemma 2.1.

All colimits exist in the categories $R{\operatorname{\mathsf{--Coalg}}}$ and
$R{\operatorname{\mathsf{--Cocom}}}$. The forgetful functors
$R{\operatorname{\mathsf{--Coalg}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ and
$R{\operatorname{\mathsf{--Cocom}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ preserve colimits.

###### Proof.

Let us discuss the colimits in $R{\operatorname{\mathsf{--Coalg}}}$; the
situation in $R{\operatorname{\mathsf{--Cocom}}}$ is similar. It suffices to
check that coproducts and coequalizers exist in
$R{\operatorname{\mathsf{--Coalg}}}$ and are preserved by the forgetful
functor to $R{\operatorname{\mathsf{--Mod}}}$. Given a family of
$R$-coalgebras $(C_{\xi})_{\xi\in\Xi}$, it is easy to construct a coalgebra
structure on the direct sum $\bigoplus_{\xi\in\Xi}C_{\xi}$ taken in
$R{\operatorname{\mathsf{--Mod}}}$ and show that the resulting coalgebra is
the coproduct of $C_{\xi}$ in $R{\operatorname{\mathsf{--Coalg}}}$. Let us
explain the coequalizers in more detail.

Suppose given a pair of parallel morphisms of $R$-coalgebras $f$, $g\colon
C\rightrightarrows D$. We claim that the image $I$ of the $R$-module map
$f-g\colon C\longrightarrow D$ is a coideal in $D$. This means that the counit
map $\epsilon_{D}\colon D\longrightarrow R$ vanishes in restriction to $I$ and
the image of the composition $I\longrightarrow D\longrightarrow D\otimes_{R}D$
of the inclusion map $I\longrightarrow D$ with the comultiplication map
$\mu_{D}\colon D\longrightarrow D\otimes_{R}D$ is contained in the image of
the map $I\otimes_{R}D\oplus D\otimes_{R}I\longrightarrow D\otimes_{R}D$
induced by the inclusion map $I\longrightarrow D$.

Indeed, one clearly has
$\epsilon_{D}(f(c)-g(c))=\epsilon_{D}(f(c))-\epsilon_{D}(g(c))=\epsilon_{C}(c)-\epsilon_{C}(c)=0$
for all $c\in C$. Now let the element $\mu_{C}(c)\in C\otimes_{R}C$ be equal
to $\sum_{i=1}^{n}c^{\prime}_{i}\otimes c^{\prime\prime}_{i}$, where
$c^{\prime}_{i}$, $c^{\prime\prime}_{i}\in C$. Then we have
$\mu_{D}(f(c)-g(c))=(f\otimes f-g\otimes
g)(\mu_{C}(c))=\sum_{i=1}^{n}(f(c^{\prime}_{i})\otimes
f(c^{\prime\prime}_{i})-g(c^{\prime}_{i})\otimes
g(c^{\prime\prime}_{i}))=\sum_{i=1}^{n}(f(c^{\prime}_{i})-g(c^{\prime}_{i}))\otimes
f(c^{\prime\prime}_{i})+\sum_{i=1}^{n}g(c^{\prime}_{i})\otimes(f(c^{\prime\prime}_{i})-g(c^{\prime\prime}_{i}))\in\operatorname{im}(I\otimes_{R}D\oplus
D\otimes_{R}I)\subset D\otimes_{R}D$, as desired.

It follows that there is a unique $R$-coalgebra structure on the quotient
$R$-module $E=D/I$ such that the natural surjective map $D\longrightarrow E$
is a coalgebra morphism. One can readily check that the morphism
$D\longrightarrow E$ is the coequalizer of the morphisms $f$ and $g$ in
$R{\operatorname{\mathsf{--Coalg}}}$. ∎

The following theorem is the main result of this section.

######  Theorem 2.2.

Let $R$ be a commutative ring.

(a) The category of coassociative, counital $R$-coalgebras
$R{\operatorname{\mathsf{--Coalg}}}$ is locally $\aleph_{1}$-presentable. The
$\aleph_{1}$-presentable objects of $R{\operatorname{\mathsf{--Coalg}}}$ are
precisely all the $R$-coalgebras whose underlying $R$-modules are countably
presentable as objects of $R{\operatorname{\mathsf{--Mod}}}$.

(b) The category of coassociative, cocommutative, counital $R$-coalgebras
$R{\operatorname{\mathsf{--Cocom}}}$ is locally $\aleph_{1}$-presentable. The
$\aleph_{1}$-presentable objects of $R{\operatorname{\mathsf{--Cocom}}}$ are
precisely all the cocommutative $R$-coalgebras whose underlying $R$-modules
are countably presentable as objects of $R{\operatorname{\mathsf{--Mod}}}$.

###### Proof.

This is [20, Example 4.3]. Let us spell out a proof of part (b) based on the
techniques from [12] collected in Section 1. In view of Lemma 2.1, it suffices
to show that the category $R{\operatorname{\mathsf{--Cocom}}}$ is
$\aleph_{1}$-accessible and describe its full subcategory of
$\aleph_{1}$-presentable objects.

Firstly we apply Theorem 1.2 (for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$). Put $\mathsf{K}=R{\operatorname{\mathsf{--Mod}}}$ and
$\mathsf{L}=R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}$. Let
$F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking an
$R$-module $C$ to the pair of $R$-modules $(C,C)\in
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}$, and
let $G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking the
$R$-module $C$ to the pair of $R$-modules $(C\otimes_{R}C,\>R)\in
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}$. Then
the inserter category $\mathsf{D}$ is the category of $R$-modules $C$ endowed
with two $R$-linear maps $\mu\colon C\longrightarrow C\otimes_{R}C$ and
$\epsilon\colon C\longrightarrow R$.

By Propositions 1.5 and 1.1, both the categories $\mathsf{K}$ and $\mathsf{L}$
are $\aleph_{1}$-accessible (in fact, locally $\aleph_{1}$-presentable). The
assumptions of Theorem 1.2 are satisfied, and we can conclude that the
category $\mathsf{D}$ is $\aleph_{1}$-accessible. Theorem 1.2 also provides a
description of the full subcategory of $\aleph_{1}$-presentable objects in
$\mathsf{D}$.

Now we apply Theorem 1.3 (again for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$). Put $\mathsf{K}=\mathsf{D}$ and
$\mathsf{L}=(R{\operatorname{\mathsf{--Mod}}})^{4}=R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}$. Let
$F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking a triple
$(C,\mu,\epsilon)\in\mathsf{D}$ to the quadruple of $R$-modules
$(C,C,C,C)\in\mathsf{L}$, and let $G\colon\mathsf{K}\longrightarrow\mathsf{L}$
be the functor taking the same object $(C,\mu,\epsilon)\in\mathsf{D}$ to the
quadruple of $R$-modules
$(C\otimes_{R}C\otimes_{R}C,\>C,\>C,\>C\otimes_{R}C)\in\mathsf{L}$.

Let $\phi\colon F\longrightarrow G$ be the natural transformation acting by
the following quadruple of morphisms $\phi_{1}\colon C\longrightarrow
C\otimes_{R}C\otimes_{R}C$, $\phi_{2}\colon C\longrightarrow C$,
$\phi_{3}\colon C\longrightarrow C$, and $\phi_{4}\colon C\longrightarrow
C\otimes_{R}C$. The morphism $\phi_{1}$ is the composition
$(\mu\otimes\mathrm{id}_{C})\circ\mu\colon C\longrightarrow
C\otimes_{R}C\otimes_{R}C$. The morphism $\phi_{2}$ is the composition
$(\epsilon\otimes\mathrm{id}_{C})\circ\mu\colon C\longrightarrow C$. The
morphism $\phi_{3}$ is the composition
$(\mathrm{id}_{C}\otimes\epsilon)\circ\mu\colon C\longrightarrow C$. The
morphism $\phi_{4}$ is the composition $\sigma_{C}\circ\mu\colon
C\longrightarrow C\otimes_{R}C$.

Let $\psi\colon F\longrightarrow G$ be the natural transformation acting by
the following quadruple of morphisms $\psi_{1}\colon C\longrightarrow
C\otimes_{R}C\otimes_{R}C$, $\psi_{2}\colon C\longrightarrow C$,
$\psi_{3}\colon C\longrightarrow C$, and $\psi_{4}\colon C\longrightarrow
C\otimes_{R}C$. The morphism $\psi_{1}$ is the composition
$(\mathrm{id}_{C}\otimes\mu)\circ\mu\colon C\longrightarrow
C\otimes_{R}C\otimes_{R}C$. The morphisms $\psi_{2}$ and $\psi_{3}$ are the
identity morphisms $\mathrm{id}_{C}\colon C\longrightarrow C$. The morphism
$\psi_{4}$ is the morphism $\mu\colon C\longrightarrow C\otimes_{R}C$. Then
the equifier category $\mathsf{E}$ is the category of coassociative, counital,
cocommutative coalgebras $C$ over $R$, that is
$\mathsf{E}=R{\operatorname{\mathsf{--Cocom}}}$.

By Propositions 1.5 and 1.1, and by the discussion above, both the categories
$\mathsf{K}=\mathsf{D}$ and $\mathsf{L}$ are $\aleph_{1}$-accessible. The
assumptions of Theorem 1.3 are satisfied, and we can conclude that the
category $\mathsf{E}=R{\operatorname{\mathsf{--Cocom}}}$ is
$\aleph_{1}$-accessible. Theorem 1.3 also provides the desired description of
the full subcategory of $\aleph_{1}$-presentable objects in $\mathsf{E}$.

Essentially the same argument is applicable in the case of the
noncocommutative coalgebras (part (a) of the theorem). One only needs to drop
the elements related to cocommutativity in the proof above (i. e., the
morphisms $\phi_{4}$ and $\psi_{4}$, and the related components of the
category $\mathsf{L}$ and the functors $F$ and $G$ in the context of Theorem
1.3). So one takes
$\mathsf{L}=(R{\operatorname{\mathsf{--Mod}}})^{3}=R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}$ when
applying Theorem 1.3 for the proof of part (a). ∎

######  Remark 2.3.

Numerous variations on the theme of Theorem 2.2 are possible, producing other
examples of locally $\aleph_{1}$-presentable categories of $R$-coalgebras. Let
us mention some of them.

(1) Noncounital and/or noncoassociative $R$-coalgebras also form locally
$\aleph_{1}$-presentable categories. All one needs to do in these cases is to
drop the related elements from the proof of Theorem 2.2 above.

(2) The category of Lie coalgebras over an arbitrary commutative ring $R$ is
locally $\aleph_{1}$-presentable. In order to obtain a proof, one needs to
(drop the unitality and) replace the tensor power functors $C\longmapsto
C\otimes_{R}C$ and $C\longmapsto C\otimes_{R}C\otimes_{R}C$ with the exterior
power functors $C\longmapsto\bigwedge^{2}_{R}C$ and
$C\longmapsto\bigwedge^{3}_{R}C$ in the proof of Theorem 2.2.

(3) A coassociative, noncounital coalgebra $D$ over a commutative ring $R$ is
said to be _conilpotent_ if for every element $d\in D$ there exists an integer
$n\geq 1$ such that the element $d$ is annihilated by the iterated
comultiplication map $\mu^{(n)}\colon D\longrightarrow D^{\otimes n+1}$, that
is $\mu^{(n)}(d)=0$. In order to show that the category of conilpotent
$R$-coalgebras is locally $\aleph_{1}$-presentable, one can consider the
following inserter category $\mathsf{D}$. Put
$\mathsf{K}=\mathsf{L}=R{\operatorname{\mathsf{--Mod}}}$, let
$F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the identity functor, and let
$G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor assigning to every
$R$-module $D\in R{\operatorname{\mathsf{--Mod}}}$ the $R$-module
$\bigoplus_{n=1}^{\infty}D^{\otimes n}=D\oplus D\otimes_{R}D\oplus
D\otimes_{R}D\otimes_{R}D\oplus\dotsb$. Then the suitable equifier full
subcategory $\mathsf{E}\subset\mathsf{D}$ is the category of conilpotent
$R$-coalgebras.

(4) A _DG-coalgebra_ $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$R$ is a comonoid object in the (symmetric) monoidal category of _complexes_
of $R$-modules with respect to the operation of tensor product $\otimes_{R}$.
In order to show that the category of DG-coagebras over $R$ is locally
$\aleph_{1}$-presentable, one needs to know that the category of complexes of
$R$-modules is locally finitely presentable and its $\aleph_{1}$-presentable
objects are precisely all the complexes of $\aleph_{1}$-presentable
$R$-modules [13, Lemma 1.5]. Then one argues similarly to the proof of Theorem
2.2, replacing modules by complexes everywhere.

In each of the cases (1–4), the description of the full subcategory of
$\aleph_{1}$-presentable objects in the respective locally
$\aleph_{1}$-presentable category of coalgebras is similar to the one in
Theorem 2.2.

######  Remark 2.4.

Let us offer a sketchy and approximate explanation of the workings of the
proof of Theorem 2.2, based as it is on Theorems 1.2 and 1.3. Following Remark
2.3(1), we ignore the counit and consider the category of coassociative,
noncocommutative, noncounital coalgebras $C$ over a fixed commutative ring
$R$.

(1) In order to illustrate the workings of the proof of Theorem 1.2 (as
spelled out in [12, proof of Theorem 4.1]) in the situation at hand, let us
start with coalgebras that are not even coassociative. So we are interested in
$R$-modules $C$ endowed with an $R$-module map $\nu_{C}\colon C\longrightarrow
C\otimes_{R}C$. How does one approach $C$ with countably presentable
$R$-modules $U$ endowed with $R$-module maps $\nu_{U}\colon U\longrightarrow
U\otimes_{R}U$ ?

Suppose given an $R$-module $C$ endowed with an $R$-linear map $\nu_{C}\colon
C\longrightarrow C\otimes_{R}C$, a countably presentable $R$-module $S$, and
an $R$-module map $S\longrightarrow C$. Let us explain, roughly following [12,
proof of Lemma 4.5], how to construct a countably presentable $R$-module $U$
with an $R$-module map $\nu_{U}\colon U\longrightarrow U\otimes_{R}U$, an
$R$-module map $U\longrightarrow C$ compatible with $\nu_{U}$ and $\nu_{C}$,
and an $R$-module map $S\longrightarrow U$ making the triangular diagram
$S\longrightarrow U\longrightarrow C$ commutative.

Let $C=\varinjlim_{\xi\in\Xi}T_{\xi}$ be a representation of the $R$-module
$C$ as a colimit of countably presentable $R$-modules $T_{\xi}$ indexed by an
$\aleph_{1}$-directed poset $\Xi$. Notice that we do _not_ assume any
comultiplication maps $\nu$ on the $R$-modules $T_{\xi}$. However, we know
that $C\otimes_{R}C=\varinjlim_{\xi\in\Xi}T_{\xi}\otimes_{R}T_{\xi}$, as the
tensor product functor $\otimes_{R}$ commutes with (directed) colimits.

Since the $R$-module $S$ is countably presentable and the poset $\Xi$ is
$\aleph_{1}$-directed, the $R$-module map $S\longrightarrow C$ factorizes as
$S\longrightarrow T_{\xi_{0}}\longrightarrow C$ for some index
$\xi_{0}\in\Xi$. Now consider the composition $T_{\xi_{0}}\longrightarrow
C\longrightarrow C\otimes_{R}C$. Since the $R$-module $T_{\xi_{0}}$ is
countably presentable, the poset $\Xi$ is $\aleph_{1}$-directed, and
$C\otimes_{R}C=\varinjlim_{\xi\in\Xi}T_{\xi}\otimes_{R}T_{\xi}$, we can find
an index $\xi_{1}\geq\xi_{0}$ in $\Xi$ such that the composition
$T_{\xi_{0}}\longrightarrow C\longrightarrow C\otimes_{R}C$ factorizes through
the $R$-module morphism $T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}\longrightarrow
C\otimes_{R}C$. So we obtain an $R$-module map $T_{\xi_{0}}\longrightarrow
T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}$.

Next we consider the composition $T_{\xi_{1}}\longrightarrow C\longrightarrow
C\otimes_{R}C$. As in the previous paragraph, we can find an index
$\xi^{\prime}_{2}\geq\xi_{1}$ in $\Xi$ such that the composition
$T_{\xi_{1}}\longrightarrow C\longrightarrow C\otimes_{R}C$ factorizes through
the $R$-module map
$T_{\xi^{\prime}_{2}}\otimes_{R}T_{\xi^{\prime}_{2}}\longrightarrow
C\otimes_{R}C$. So we obtain an $R$-module map $T_{\xi_{1}}\longrightarrow
T_{\xi^{\prime}_{2}}\otimes_{R}T_{\xi^{\prime}_{2}}$. Now, at this point, the
leftmost square of the diagram

$\textstyle{{T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi^{\prime}_{2}}\otimes_{R}T_{\xi^{\prime}_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C\otimes_{R}C}}$

_need not_ be commutative, while the rightmost and the outer squares are
commutative. Since the $R$-module $T_{\xi_{0}}$ is countably presentable, the
poset $\Xi$ is $\aleph_{1}$-directed, and
$C\otimes_{R}C=\varinjlim_{\xi\in\Xi}T_{\xi}\otimes_{R}T_{\xi}$, we can find
an index $\xi_{2}\geq\xi^{\prime}_{2}$ in $\Xi$ such that the whole diagram

$\textstyle{{T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{2}}\otimes_{R}T_{\xi_{2}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C\otimes_{R}C}}$

becomes commutative.

Proceeding in this way, we construct an $\aleph_{0}$-indexed chain of indices
$\xi_{0}\leq\xi_{1}\leq\xi_{2}\leq\dotsb$ in $\Xi$ and a compatible sequence
of $R$-module maps $T_{\xi_{i}}\longrightarrow
T_{\xi_{i+1}}\otimes_{R}T_{\xi_{i+1}}$. It remains to put
$U=\varinjlim_{i\in\aleph_{0}}T_{\xi_{i}}$.

(2) Now, in order to illustrate the workings of the proof of Theorem 1.3 (as
spelled out in [12, proof of Theorem 3.1]), suppose that we have managed to
prove that every noncoassociative $R$-coalgebra $(D,\nu_{D})$ is an
$\aleph_{1}$-directed colimit of noncoassociative $R$-coalgebras $(T,\nu_{T})$
with countably presentable underlying $R$-modules $T$. It is easy to see that
any such $R$-coalgebra $(T,\nu_{T})$ is an $\aleph_{1}$-presentable object of
the category of noncoassociative $R$-coalgebras [12, Proposition 4.2]. Let
$(C,\mu_{C})$ be a coassociative $R$-coalgebra. How does one approach $C$ with
coassociative $R$-coalgebras $(U,\mu_{U})$ with countably presentable
underlying $R$-modules $U$ ?

Suppose given a coassociative $R$-coalgebra $(C,\mu_{C})$, a noncoassociative
$R$-coalgebra $(S,\nu_{S})$ with a countably presentable underlying $R$-module
$S$, and an $R$-coalgebra morphism $S\longrightarrow C$. Let us explain,
following [12, proof of Proposition 3.2], how to construct a factorization of
the morphism $S\longrightarrow C$ through a coassociative $R$-coalgebra
$(U,\mu_{U})$ with a countably presentable underlying $R$-module $U$.

Let $C=\varinjlim_{\xi\in\Xi}T_{\xi}$ be a representation of the $R$-coalgebra
$C$ as a colimit of noncoassociative $R$-coalgebras $(T_{\xi},\nu_{T_{\xi}})$,
with countably presentable underlying $R$-modules $T_{\xi}$, indexed by an
$\aleph_{1}$-directed poset $\Xi$. Then the $R$-coalgebra map
$(S,\nu_{S})\longrightarrow(C,\mu_{C})$ factorizes through the $R$-coalgebra
map $(T_{\xi_{0}},\nu_{T_{\xi_{0}}})\longrightarrow(C,\mu_{C})$ for some
$\xi_{0}\in\Xi$.

Now the two compositions

$\textstyle{{T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}}$

_need not_ be equal to each other, as the coalgebra $T_{\xi_{0}}$ is not
coassociative. However, the two compositions

$\textstyle{{T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{C\otimes_{R}C\otimes_{R}C}}$

_are_ equal to each other, as the coalgebra $C$ is coassociative and
$T_{\xi_{0}}\longrightarrow C$ is an $R$-coalgebra morphism. Since the
$R$-module $T_{\xi_{0}}$ is countably presentable, the poset $\Xi$ is
$\aleph_{1}$-directed, and
$C\otimes_{R}C\otimes_{R}C=\varinjlim_{\xi\in\Xi}T_{\xi}\otimes_{R}T_{\xi}\otimes_{R}T_{\xi}$,
we can find an index $\xi_{1}\geq\xi_{0}$ in $\Xi$ such that the two
compositions

$\textstyle{{T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}\otimes_{R}T_{\xi_{0}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}\otimes_{R}T_{\xi_{1}}}}$

are equal to each other.

Proceeding in this way, we construct an $\aleph_{0}$-indexed chain of indices
$\xi_{0}\leq\xi_{1}\leq\xi_{2}\leq\dotsb$ in $\Xi$ such that, for every $i\geq
0$, the two compositions

$\textstyle{{T_{\xi_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{i}}\otimes_{R}T_{\xi_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{i}}\otimes_{R}T_{\xi_{i}}\otimes_{R}T_{\xi_{i}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{T_{\xi_{i+1}}\otimes_{R}T_{\xi_{i+1}}\otimes_{R}T_{\xi_{i+1}}}}$

are equal to each other. It remains to put
$U=\varinjlim_{i\in\aleph_{0}}T_{\xi_{i}}$ and
$\mu_{U}=\varinjlim_{i\in\aleph_{0}}\nu_{T_{\xi_{i}}}$.

##  3\. Flat Coalgebras over a Commutative Ring

All the results of the previous Section 2, with the exception of Lemma 2.1,
remain valid for the full subcategories in the respective coalgebra categories
consisting of the coalgebras that are _flat as $R$-modules_. The only
difference is that the local $\aleph_{1}$-presentability claims need to be
replaced by the $\aleph_{1}$-accessibility. Let us spell out some details.

We denote by $R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Coalg}}}$ and
$R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Cocom}}}$ the full subcategories of coalgebras that
are flat as $R$-modules. So $R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}$
is the category of $R$-flat coassociative, counital $R$-coalgebras and
$R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}\subset
R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}$ is the full subcategory of
$R$-flat cocommutative $R$-coalgebras.

######  Theorem 3.1.

Let $R$ be a commutative ring.

(a) The category of coassociative, counital $R$-flat $R$-coalgebras
$R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}$ is $\aleph_{1}$-accessible.
The $\aleph_{1}$-presentable objects of
$R{\operatorname{\mathsf{--Coalg}}}_{\mathsf{fl}}$ are precisely all the
$R$-flat $R$-coalgebras whose underlying $R$-modules are countably presentable
as objects of $R{\operatorname{\mathsf{--Mod}}}$.

(b) The category of coassociative, cocommutative, counital $R$-flat
$R$-coalgebras $R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}$ is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of
$R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}$ are precisely all the
$R$-flat cocommutative $R$-coalgebras whose underlying $R$-modules are
countably presentable as objects of $R{\operatorname{\mathsf{--Mod}}}$.

###### Proof.

The argument is similar to the proof of Theorem 2.2, with suitable changes.
Let us spell out the proof of part (b).

First we apply Theorem 1.2 (for $\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$).
Put $\mathsf{K}=R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$ and
$\mathsf{L}=R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}$. Let
$F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking a flat
$R$-module $C$ to the pair of $R$-modules $(C,C)\in
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}$, and
let $G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking the
$R$-module $C$ to the pair of $R$-modules $(C\otimes_{R}C,\>R)\in
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}$. Then
the inserter category $\mathsf{D}$ is the category of flat $R$-modules $C$
endowed with two $R$-linear maps $\mu\colon C\longrightarrow C\otimes_{R}C$
and $\epsilon\colon C\longrightarrow R$.

By Propositions 1.5–1.6 and 1.1, both the categories $\mathsf{K}$ and
$\mathsf{L}$ are $\aleph_{1}$-accessible. The assumptions of Theorem 1.2 are
satisfied, so we can conclude that the category $\mathsf{D}$ is
$\aleph_{1}$-accessible. We also obtain a description of the full subcategory
of $\aleph_{1}$-presentable objects in $\mathsf{D}$.

Now we apply Theorem 1.3 (for $\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$).
Put $\mathsf{K}=\mathsf{D}$ and
$\mathsf{L}=(R{\operatorname{\mathsf{--Mod}}})^{4}=R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}\times R{\operatorname{\mathsf{--Mod}}}\times
R{\operatorname{\mathsf{--Mod}}}$. Let
$F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking a triple
$(C,\mu,\epsilon)\in\mathsf{D}$ to the quadruple of $R$-modules
$(C,C,C,C)\in\mathsf{L}$, and let $G\colon\mathsf{K}\longrightarrow\mathsf{L}$
be the functor taking the same object $(C,\mu,\epsilon)\in\mathsf{D}$ to the
quadruple of $R$-modules
$(C\otimes_{R}C\otimes_{R}C,\>C,\>C,\>C\otimes_{R}C)\in\mathsf{L}$. The
natural transformations $\phi$ and $\psi$ are the same as in the proof of
Theorem 2.2.

Then the equifier category $\mathsf{E}$ is the category of $R$-flat,
coassociative, counital, cocommutative coalgebras $C$ over $R$, that is
$\mathsf{E}=R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}$. The assumptions
of Theorem 1.3 are satisfied, and we can conclude that the category
$\mathsf{E}=R{\operatorname{\mathsf{--Cocom}}}_{\mathsf{fl}}$ is
$\aleph_{1}$-accessible. Theorem 1.3 also provides the desired description of
the full subcategory of $\aleph_{1}$-presentable objects in $\mathsf{E}$.

The same argument is applicable in the case of noncocommutative coalgebras
(part (a) of the theorem). One just needs to drop the elements related to
cocommutativity in the proof above.

Alternatively, one can argue similarly to the proof of Theorem 5.1 below,
using Theorem 2.2 as a black box and applying Theorem 1.4. ∎

######  Remark 3.2.

Similarly to Remark 2.3, numerous variations on the theme of Theorem 3.1 are
possible, producing other examples of $\aleph_{1}$-accessible categories of
$R$-flat $R$-coalgebras. Let us mention some of them.

(1) Noncounital and/or noncoassociative $R$-flat $R$-coalgebras also form
$\aleph_{1}$-accessible categories. All one needs to do in these cases is to
drop the related elements from the proof of Theorem 3.1.

(2) The category of $R$-flat Lie coalgebras over an arbitrary commutative ring
$R$ is $\aleph_{1}$-accessible. This is similar to Remark 2.3(2).

(3) The category of $R$-flat conilpotent coalgebras over any commutative ring
$R$ is $\aleph_{1}$-accessible. This is similar to Remark 2.3(3).

(4) Concerning flat DG-coalgebras, there are two natural versions of such a
concept. One can consider DG-coalgebras
$C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $R$ whose underlying
complex of $R$-modules is _termwise_ flat (i. e., every $R$-module $C^{n}$ if
flat, for $n\in\mathbb{Z}$), or one can consider the DG-coalgebras whose
underlying complex $C^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
_homotopy flat_ complex of flat $R$-modules.

In the former context, one needs to know that the category of complexes of
flat $R$-modules is $\aleph_{1}$-accessible, and the complexes of flat
countably presentable $R$-modules are precisely all the
$\aleph_{1}$-presentable objects of this category [12, Corollaries 10.3–10.4],
[14, Proposition 1.4].

In the latter context, one needs the use the result that the category of
homotopy flat complexes of flat $R$-modules is $\aleph_{1}$-accessible, and
the homotopy flat complexes of countably presentable flat $R$-modules are
precisely all the $\aleph_{1}$-presentable objects of this category. This is
based on [7, Theorem 1.1]; see [14, Proposition 1.7].

With these preliminary observations in mind, the arguments are similar to the
proof of Theorem 3.1. One just needs to replace the category of flat
$R$-modules $R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$ by the category
of complexes of flat $R$-modules or homotopy flat complexes of flat
$R$-modules, as desired; and replace all mentions of the category of
$R$-modules $R{\operatorname{\mathsf{--Mod}}}$ by the category of complexes of
$R$-modules.

##  4\. Corings over an Associative Ring

Let $R$ be an associative ring (which may or may not be commutative). A
(_coassociative, counital_) _coring_ over $R$ is a comonoid object in the
monoidal category of $R$-$R$-bimodules $R{\operatorname{\mathsf{--Bimod--}}}R$
with respect to the operation of tensor product over $R$. In other words, a
coring $C$ is an $R$-$R$-bimodule endowed with two maps of _comultiplication_
$\mu\colon C\longrightarrow C\otimes_{R}C$ and _counit_ $\epsilon\colon
C\longrightarrow R$, which must be $R$-$R$-bimodule maps satisfying the
coassociativity and counitality axioms.

We suppress the explicit diagrams and equations, as they are written down
exactly the same as in the definition of a coassociative, counital
$R$-coalgebra in Section 2. The difference between the two definitions is that
$C$ was an $R$-module (for a commutative ring $R$) in Section 2, while $C$ is
an $R$-$R$-bimodule (for an associative ring $R$) in the present section.

Assume further that $R$ is an associative, unital algebra over a fixed
commutative ring $k$. An _$R$ -$R$-bimodule over $k$_ is an $R$-$R$-bimodule
in which the left and right actions of $k$ agree. If a coring $C$ over $R$ is
an $R$-$R$-bimodule over $k$, we will say that $C$ is an _$R$ -coring over
$k$_ (then it follows automatically that the comultiplication and counit $\mu$
and $\epsilon$ are $k$-linear maps). From now on, we will assume that all our
$R$-$R$-bimodules are $R$-$R$-bimodules over $k$. The category of
$R$-$R$-bimodules over $k$ will be denoted by
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$.

Morphisms of $R$-corings over $k$ are defined in the obvious way as
$R$-$R$-bimodule maps compatible with the comultiplication and counit. We
denote the category of $R$-corings over $k$ by
$R_{k}{\operatorname{\mathsf{--Corings}}}$.

######  Lemma 4.1.

All colimits exist in the category $R_{k}{\operatorname{\mathsf{--Corings}}}$.
The forgetful functor $R_{k}{\operatorname{\mathsf{--Corings}}}\longrightarrow
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$ preserves colimits.

###### Proof.

Similar to the proof of Lemma 2.1, with $R$-module maps replaced by
$R$-$R$-bimodule maps. ∎

######  Theorem 4.2.

Let $k$ be a commutative ring and $R$ be an associative, unital $k$-algebra.
Then the category $R_{k}{\operatorname{\mathsf{--Corings}}}$ of coassociative,
counital $R$-corings over $k$ is locally $\aleph_{1}$-presentable. The
$\aleph_{1}$-presentable objects of $R_{k}{\operatorname{\mathsf{--Corings}}}$
are precisely all the $R$-corings whose underlying $R$-$R$-bimodules are
countably presentable as objects of
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$.

###### Proof.

This is a particular case of [20, Example 4.10]. Our argument is similar to
the proof of Theorem 2.2(a). In view of Lemma 4.1, it suffices to show that
the category $R_{k}{\operatorname{\mathsf{--Corings}}}$ is
$\aleph_{1}$-accessible and describe its full subcategory of
$\aleph_{1}$-presentable objects.

Firstly we apply Theorem 1.2 (for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$). Put
$\mathsf{K}=R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$ and
$\mathsf{L}=(R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R)\times(R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R)$.
Let $F\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking an
$R$-$R$-bimodule $C$ to the pair of $R$-$R$-bimodules $(C,C)\in
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R\times
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$, and let
$G\colon\mathsf{K}\longrightarrow\mathsf{L}$ be the functor taking the
$R$-$R$-bimodule $C$ to the pair of $R$-$R$-bimodules $(C\otimes_{R}C,\>R)\in
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R\times
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$. Then the inserter category
$\mathsf{D}$ is the category of $R$-$R$-bimodules $C$ over $k$ endowed with
two $R$-$R$-bimodule maps $\mu\colon C\longrightarrow C\otimes_{R}C$ and
$\epsilon\colon C\longrightarrow R$.

By Proposition 1.5 (for the ring $R\otimes_{k}R^{\mathrm{op}}$) and
Proposition 1.1, both the categories $\mathsf{K}$ and $\mathsf{L}$ are
$\aleph_{1}$-accessible (in fact, locally $\aleph_{1}$-presentable). The
assumptions of Theorem 1.2 are satisfied, and we can conclude that the
category $\mathsf{D}$ is $\aleph_{1}$-accessible. Notice that, in the
assumptions of Theorem 1.2, the functor $G$ _need not_ take
$\kappa$-presentable objects to $\kappa$-presentable objects; and in the
situation at hand it doesn’t. The functor $F$ _must_ take $\kappa$-presentable
objects to $\kappa$-presentable objects; and in the situation at hand it does.
Theorem 1.2 also provides a description of the full subcategory of
$\aleph_{1}$-presentable objects in $\mathsf{D}$.

Secondly we apply Theorem 1.3, continuing to argue exactly as in the proof of
Theorem 2.2(a) with the category $R{\operatorname{\mathsf{--Mod}}}$ replaced
by $R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$ everywhere. So we put
$\mathsf{K}=\mathsf{D}$ and
$\mathsf{L}=(R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R)^{3}$, etc. Once
again, for applicability of Theorem 1.3, the functor $F$ has to take
$\aleph_{1}$-presentable objects to $\aleph_{1}$-presentable objects (and it
does), while the functor $G$ need not have this property (and doesn’t). So the
assumptions of Theorem 1.3 are satisfied, and we conclude that the category
$\mathsf{E}=R_{k}{\operatorname{\mathsf{--Corings}}}$ is
$\aleph_{1}$-accessible. Theorem 1.3 also provides the desired description of
the full subcategory of $\aleph_{1}$-presentable objects in $\mathsf{E}$. ∎

######  Remark 4.3.

Similarly to Remark 2.3, one can think of some variations on the theme of
Theorem 4.2, producing locally $\aleph_{1}$-presentable categories of corings
with the simplest expected description of the full subcategories of
$\aleph_{1}$-presentable objects by virtue of essentially the same argument.
This applies to noncoassociative and/or noncounital $R$-corings, DG-corings
over $R$, etc.

##  5\. Bimodule-Flat Corings

Let $k$ be a commutative ring and $R$ be an associative, unital $k$-algebra.
As in the previous section, we consider $R$-corings $C$ over $k$, i. e.,
$R$-$R$-bimodules over $k$ endowed with a coassociative, counital coring
structure.

We will say that an $R$-coring $C$ is _$R$ -$R$-bimodule flat_ (or “bimodule
flat” for brevity) if $C$ is flat as a module over the ring
$R\otimes_{k}R^{\mathrm{op}}$. Let us denote the category of $R$-$R$-bimodule
flat corings over $k$ by
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$.

######  Theorem 5.1.

Let $k$ be a commutative ring and $R$ be an associative, unital $k$-algebra.
Then the category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$
of $R$-$R$-bimodule flat coassociative, counital $R$-corings over $k$ is
$\aleph_{1}$-accessible. The $\aleph_{1}$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$ are precisely all
the bimodule flat $R$-corings whose underlying $R$-$R$-bimodules are countably
presentable as objects of
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$.

###### Proof.

One can argue similarly to the proofs of Theorems 3.1(a) and 4.2; but we
prefer to present a differently structured argument based on Theorem 1.4 and
using Theorem 4.2 as a black box.

Put $\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Corings}}}$,
$\mathsf{K}_{2}=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$,
and
$\mathsf{L}=R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}\nobreak
R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$. Let
$F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the forgetful functor
assigning to every $R$-coring $C$ over $k$ its underlying $R$-$R$-bimodule
$C$, and let $F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the
identity inclusion functor. Then the pseudopullback category $\mathsf{C}$ of
the functors $F_{1}$ and $F_{2}$ is equivalent to the desired category of
bimodule flat corings
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{bifl}}$.

Propositions 1.5–1.6 and Theorem 4.2 tell that the categories
$\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ are
$\aleph_{1}$-accessible, and describe their full subcategories of
$\aleph_{1}$-presentable objects. Theorem 1.4 (for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$) is applicable to the pair of functors $F_{1}$ and
$F_{2}$. Theorem 1.4 tells that the category $\mathsf{C}$ is
$\aleph_{1}$-accessible, and provides the desired description of its full
subcategory of $\aleph_{1}$-presentable objects. ∎

######  Remark 5.2.

Similarly to Remarks 3.2(1) and 4.3, one can consider the categories of
noncoassociative and/or noncounital bimodule flat corings. The obvious
versions of Theorem 5.1 hold in these contexts as well.

##  6\. Corings with Flat Kernel

As in the previous two sections, we consider $R$-corings $C$ over $k$, where
$k$ is a commutative ring and $R$ is an associative, unital $k$-algebra.
Inspired by [5, §3] and [8, Definition 4.19 and Theorem 4.20], we say that an
$R$-coring $C$ over $k$ _has flat kernel_ if the following two conditions are
satisfied:

  1. (1)

the counit map $\epsilon\colon C\longrightarrow R$ is surjective;

  2. (2)

the kernel $\overline{C}$ of the counit map $\epsilon$ is a flat module over
the ring $R\otimes_{k}R^{\mathrm{op}}$.

If a coring $C$ satisfies only condition (1) but not necessarily (2), we will
say that $C$ _has surjective counit_. Let us denote the full subcategory of
$R$-corings with surjective counits by
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}$ and the full subcategory of
$R$-corings with flat kernels by
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\overline{\mathsf{bifl}}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}$.

For the purposes of this section, we will need some further preliminary
material, continuing the discussion in Section 1. Given a category
$\mathsf{K}$, let us denote by $\mathsf{K}^{\to}$ the category of morphisms in
$\mathsf{K}$ (with commutative squares in $\mathsf{K}$ as morphisms in
$\mathsf{K}^{\to}$). Denote by
$\mathsf{K}^{\mathsf{epi}}\subset\mathsf{K}^{\to}$ the full subcategory in
$\mathsf{K}^{\to}$ whose objects are the epimorphisms in $\mathsf{K}$ (so the
morphisms in $\mathsf{K}^{\mathsf{epi}}$ are the commutative squares in
$\mathsf{K}$ in which one pair of morphisms are epimorphisms, while the other
pair is formed by arbitrary morphisms).

######  Proposition 6.1.

Let $S$ be an associative ring and $\kappa$ be a regular cardinal. Then the
category $S{\operatorname{\mathsf{--Mod}}}^{\to}$ of morphisms of left
$S$-modules is locally $\kappa$-presentable. The $\kappa$-presentable objects
of $S{\operatorname{\mathsf{--Mod}}}^{\to}$ are precisely all the morphisms of
$\kappa$-presentable left $S$-modules. ∎

###### Proof.

This can be viewed as a particular case of Proposition 1.5 for the suitable
ring of uppertriangular $2\times 2$-matrices
$R=\left(\begin{smallmatrix}S&S\\\ 0&S\end{smallmatrix}\right)$. See [12,
Lemma 10.6] for a much more general assertion. ∎

######  Proposition 6.2.

Let $S$ be an associative ring and $\kappa$ be a regular cardinal. Then the
category $S{\operatorname{\mathsf{--Mod}}}^{\mathsf{epi}}$ of epimorphisms of
left $S$-modules is locally $\kappa$-presentable. The $\kappa$-presentable
objects of $S{\operatorname{\mathsf{--Mod}}}^{\mathsf{epi}}$ are precisely all
the epimorphisms of $S$-modules $T\longrightarrow U$, where $T$ and $U$ are
$\kappa$-presentable left $S$-modules.

###### Proof.

This is [12, Lemma 10.7] or [13, Lemma 1.6]. ∎

Given a regular cardinal $\kappa$ and a ring $S$, an $S$-module is said to be
_$ <\kappa$-generated_ if it admits a set of generators of the cardinality
smaller than $\kappa$. In particular, $<\aleph_{1}$-generated modules are said
to be _countably generated_.

A ring $S$ is said to be _left countably Noetherian_ if every left ideal in
$S$ is (at most) countably generated, or equivalently, every submodule of a
countably generated left $S$-module is countably generated. In the context of
this section, we are interested in the countable Noetherianity property of the
ring $S=R\otimes_{k}R^{\mathrm{op}}$. As the ring $S$ is isomorphic to its
opposite ring, $S\simeq S^{\mathrm{op}}$, the left and right countable
Noetherianity conditions are equivalent in this case. So we will simply say
that the ring $R\otimes_{k}R^{\mathrm{op}}$ is (assumed to be) countably
Noetherian.

######  Theorem 6.3.

Let $k$ be a commutative ring and $R$ be an associative, unital $k$-algebra.
Assume that the ring $R\otimes_{k}R^{\mathrm{op}}$ is countably Noetherian. In
this setting:

(a) The category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$ of
$R$-corings over $k$ with surjective counits is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$ are precisely all
the $R$-corings $C$ with surjective counits whose underlying $R$-$R$-bimodules
$C$ are countably presentable as objects of
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$,
or equivalently, countably generated as modules over
$R\otimes_{k}R^{\mathrm{op}}$.

(b) The category
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\overline{\mathsf{bifl}}}$ of
$R$-corings over $k$ with flat kernels is $\aleph_{1}$-accessible. The
$\aleph_{1}$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\overline{\mathsf{bifl}}}$ are
precisely all $R$-corings $C$ with flat kernels whose underlying
$R$-$R$-bimodules $C$ are countably presentable as objects of
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$, or equivalently, countably
generated as modules over $R\otimes_{k}R^{\mathrm{op}}$.

###### Proof.

Notice first of all that, over a countably left Noetherian ring $S$, a left
module is countably generated if and only if it is countably presentable. In
particular, in the situation at hand, the countable Noetherianity condition
guarantees that the $R\otimes_{k}R^{\mathrm{op}}$-module $R$ is countably
presentable. This is important for part (a).

Part (a): it is worth pointing out that, in view of Lemma 4.1, the full
subcategory $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$ is
closed under colimits _of nonempty diagrams_ in
$R_{k}{\operatorname{\mathsf{--Corings}}}$. So all colimits of nonempty
diagrams exist in $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$;
however, the category
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$ has _no_ initial
object.

We use Theorem 4.2 as a black box and apply Theorem 1.4. Put
$\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Corings}}}$,
$\mathsf{K}_{2}=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}^{\mathsf{epi}}$,
and
$\mathsf{L}=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}^{\to}$.
Let $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the functor
assigning to an $R$-coring $C$ its counit map $\epsilon\colon C\longrightarrow
R$, and let $F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the
identity inclusion functor. Then the pseudopullback category $\mathsf{C}$ of
the functors $F_{1}$ and $F_{2}$ is equivalent to the desired category of
corings with surjective counits
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$.

Propositions 6.1–6.2 and Theorem 4.2 tell that the categories
$\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ are
$\aleph_{1}$-accessible, and describe their full subcategories of
$\aleph_{1}$-presentable objects. Theorem 1.4 (for $\kappa=\aleph_{1}$ and
$\lambda=\aleph_{0}$) is applicable to the pair of functors $F_{1}$ and
$F_{2}$. Here one needs to know that $R$ is a countably presentable module
over $R\otimes_{k}R^{\mathrm{op}}$ in order to claim that the functor $F_{1}$
takes $\aleph_{1}$-presentable objects to $\aleph_{1}$-presentable objects. It
is clear from Lemma 4.1 that the functor $F_{1}$ preserves directed colimits.
Theorem 1.4 tells that the category $\mathsf{C}$ is $\aleph_{1}$-accessible,
and provides the desired description of its full subcategory of
$\aleph_{1}$-presentable objects.

Part (b): we apply Theorem 1.4 again, using part (a) as a black box. Put
$\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{sur}}$,
$\mathsf{K}_{2}=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}$,
and $\mathsf{L}=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$.
Let $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the functor
assigning the kernel $\overline{C}$ of the counit map $\epsilon\colon
C\longrightarrow R$ to an $R$-coring $C$ with surjective counit. Let
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the identity inclusion
functor. Then the pseudopullback category $\mathsf{C}$ of the functors $F_{1}$
and $F_{2}$ is equivalent to the desired category of corings with flat kernels
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\overline{\mathsf{bifl}}}$.

Propositions 1.5–1.6 and part (a) of the present theorem tell that the
categories $\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ are
$\aleph_{1}$-accessible, and describe their full subcategories of
$\aleph_{1}$-presentable objects. Once again we claim that Theorem 1.4 (for
$\kappa=\aleph_{1}$ and $\lambda=\aleph_{0}$) is applicable to the pair of
functors $F_{1}$ and $F_{2}$. Here we need to know that that the kernel
$\overline{C}$ of the counit map $\epsilon\colon C\longrightarrow R$ is
countably presentable as a module over $R\otimes_{k}R^{\mathrm{op}}$ whenever
so is the $R$-$R$-bimodule $C$. This is where the countable Noetherianity
condition is used again. It is also important that the functor $F_{1}$
preserves directed colimits. Theorem 1.4 tells that the category $\mathsf{C}$
is $\aleph_{1}$-accessible, and provides the desired description of its full
subcategory of $\aleph_{1}$-presentable objects. ∎

##  7\. Module-Flat Corings

We continue to consider $R$-corings $C$ over $k$, where $k$ is a commutative
ring and $R$ is an associative, unital $k$-algebra. We say that a coring $C$
is _right $R$-module flat_ (or “right flat” for brevity) if $C$ is a flat
right $R$-module. The importance of this condition is explained by the results
of [4, Sections 18.6, 18.14, and 18.16], [10, Proposition 2.12(a)], and [11,
Lemma 2.1], telling that the category of left $C$-comodules
$C{\operatorname{\mathsf{--Comod}}}$ is abelian _and_ the forgetful functor
$C{\operatorname{\mathsf{--Comod}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ is exact if and only if $C$ is a flat right
$R$-module.

Furthermore, we say that a coring $C$ is _left and right $R$-module flat_ (or
just “left and right flat”) if $C$ is a flat left $R$-module _and_ a flat
right $R$-module. This condition should not be confused with the condition of
flatness of $C$ as a module over $R\otimes_{k}R^{\mathrm{op}}$ (which was
discussed in Section 5). When $R$ is a flat $k$-module, one can say that any
$R$-$R$-bimodule flat $R$-coring is left and right $R$-module flat, but _not_
vice versa. We denote the full subcategory of right $R$-module flat
$R$-corings over $k$ by
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}$ and the full subcategory of left and
right $R$-module flat $R$-corings by
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{lfrl}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}\subset
R_{k}{\operatorname{\mathsf{--Corings}}}$.

We will also need terminology and notation for the related categories of
$R$-$R$-bimodules over $k$ with $R$-module flatness conditions. So let
$R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R\subset
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$ denote the full subcategory
of $R$-$R$-bimodules that are flat as right $R$-modules, and let
$R_{k}{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}{}_{k}R\subset
R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R\subset
R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R$ denote the full subcategory
of $R$-$R$-bimodules that are flat both as left $R$-modules and as right
$R$-modules. The objects of
$R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R$ will be called _right
$R$-module flat $R$-$R$-bimodules_ (or simply “right flat bimodules”), while
the objects of $R_{k}{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}{}_{k}R$
will be called _left and right $R$-module flat $R$-$R$-bimodules_ (or simply
“left and right flat bimodules”).

Finally, let us denote by ${\operatorname{\mathsf{Mod--}}}R$ the category of
right $R$-modules, and by
${\operatorname{\mathsf{Mod_{fl}--}}}R\subset{\operatorname{\mathsf{Mod--}}}R$
the full subcategory of flat right $R$-modules.

For any set $X$, we denote by $|X|$ the cardinality of $X$. Notice that, for
any ring $S$ and any regular cardinal $\kappa>|S|$, an $S$-module $M$ is
$\kappa$-presentable if and only if $M$ is $<\kappa$-generated, and if and
only if $|M|<\kappa$.

######  Proposition 7.1.

Let $k$ be a commutative ring, $R$ be an associative, unital $k$-algebra, and
$\kappa>|R|+\aleph_{0}$ be a regular cardinal. In this setting:

(a) The category $R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R$ of
right $R$-module flat $R$-$R$-bimodules over $k$ is $\kappa$-accessible. The
$\kappa$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R$ are precisely all the
right flat bimodules of the cardinality smaller than $\kappa$.

(b) The category $R_{k}{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}{}_{k}R$
of left and right $R$-module flat $R$-$R$-bimodules over $k$ is
$\kappa$-accessible. The $\kappa$-presentable objects of
$R_{k}{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}{}_{k}R$ are precisely
all the left and right flat bimodules of the cardinality smaller than
$\kappa$.

###### Proof.

Part (a): apply Theorem 1.4 to the following pair of functors $F_{1}$ and
$F_{2}$. Put
$\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$,
$\mathsf{K}_{2}={\operatorname{\mathsf{Mod_{fl}--}}}R$, and
$\mathsf{L}={\operatorname{\mathsf{Mod--}}}R$. Let
$F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the forgetful functor
assigning to an $R$-$R$-bimodule its underlying right $R$-module, and let
$F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the identity inclusion
functor. Then the pseudopullback category $\mathsf{C}$ of the functors $F_{1}$
and $F_{2}$ is equivalent to the desired category of right $R$-module flat
$R$-$R$-bimodules $R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R$.

Propositions 1.5–1.6 tell that the categories $\mathsf{K}_{1}$,
$\mathsf{K}_{2}$, and $\mathsf{L}$ are $\kappa$-accessible, and describe their
full subcategories of $\kappa$-presentable objects. We claim that Theorem 1.4
(for the given cardinal $\kappa$ and $\lambda=\aleph_{0}$) is applicable to
the pair of functors $F_{1}$ and $F_{2}$. The condition that $\kappa>|R|$ is
needed here for the functor $F_{1}$ to take $\kappa$-presentable objects to
$\kappa$-presentable objects. Theorem 1.4 tells that the category $\mathsf{C}$
is $\kappa$-accessible, and provides the desired description of its full
subcategory of $\kappa$-presentable objects.

Part (b): once again, we apply Theorem 1.4. Put
$\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}\nobreak
R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$,
$\mathsf{K}_{2}=R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}\times{\operatorname{\mathsf{Mod_{fl}--}}}R$,
and
$\mathsf{L}=R{\operatorname{\mathsf{--Mod}}}\times{\operatorname{\mathsf{Mod--}}}R$.
Let $F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the forgetful
functor assigning to an $R$-$R$-bimodule $B$ the pair of its underlying left
and right $R$-modules $(B,B)\in
R{\operatorname{\mathsf{--Mod}}}\times{\operatorname{\mathsf{Mod--}}}R$, and
let $F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the Cartesian
product of the identity inclusion functors
$R{\operatorname{\mathsf{--Mod}}}_{\mathsf{fl}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ and
${\operatorname{\mathsf{Mod_{fl}--}}}R\longrightarrow{\operatorname{\mathsf{Mod--}}}R$.

Propositions 1.5–1.6 together with Proposition 1.1 tell that the categories
$\mathsf{K}_{1}$, $\mathsf{K}_{2}$, and $\mathsf{L}$ are $\kappa$-accessible,
and describe their full subcategories of $\kappa$-presentable objects. Once
again, the condition that $\kappa>|R|$ implies that the functor $F_{1}$ takes
$\kappa$-presentable objects to $\kappa$-presentable objects. So Theorem 1.4
is applicable for the given cardinal $\kappa$ and $\lambda=\aleph_{0}$,
providing the desired conclusions. ∎

The following proposition is a straightforward generalization of Theorem 4.2.

######  Proposition 7.2.

Let $k$ be a commutative ring, $R$ be an associative, unital $k$-algebra, and
$\kappa$ be an uncountable regular cardinal. Then the category
$R_{k}{\operatorname{\mathsf{--Corings}}}$ of coassociative, counital
$R$-corings over $k$ is locally $\kappa$-presentable. The $\kappa$-presentable
objects of $R_{k}{\operatorname{\mathsf{--Corings}}}$ are precisely all the
$R$-corings whose underlying $R$-$R$-bimodules are $\kappa$-presentable as
objects of
$R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$.

###### Proof.

The same as the proof of Theorem 4.2, except that one needs to apply Theorems
1.2 and 1.3 for the given cardinal $\kappa$ and $\lambda=\aleph_{0}$. ∎

######  Theorem 7.3.

Let $k$ be a commutative ring, $R$ be an associative, unital $k$-algebra, and
$\kappa>|R|+\aleph_{0}$ be a regular cardinal. In this setting:

(a) The category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}$ of
right $R$-module flat $R$-corings over $k$ is $\kappa$-accessible. The
$\kappa$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}$ are precisely all
the right flat $R$-corings of the cardinality smaller than $\kappa$.

(b) The category $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{lfrl}}$ of
left and right $R$-module flat $R$-corings over $k$ is $\kappa$-accessible.
The $\kappa$-presentable objects of
$R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{lfrl}}$ are precisely all
the left and right flat $R$-corings of the cardinality smaller than $\kappa$.

###### Proof.

This is similar to Theorem 5.1. We sketch the argument based on Theorem 1.4.

Part (a): put $\mathsf{K}_{1}=R_{k}{\operatorname{\mathsf{--Corings}}}$,
$\mathsf{K}_{2}=R_{k}{\operatorname{\mathsf{--Bimod_{fl}--}}}{}_{k}R$, and
$\mathsf{L}=R_{k}{\operatorname{\mathsf{--Bimod--}}}{}_{k}R=(R\otimes_{k}\nobreak
R^{\mathrm{op}}){\operatorname{\mathsf{--Mod}}}$. Let
$F_{1}\colon\mathsf{K}_{1}\longrightarrow\mathsf{L}$ be the forgetful functor
assigning to every $R$-coring $C$ over $k$ its underlying $R$-$R$-bimodule
$C$, and let $F_{2}\colon\mathsf{K}_{2}\longrightarrow\mathsf{L}$ be the
identity inclusion functor. Then the pseudopullback category $\mathsf{C}$ of
the functors $F_{1}$ and $F_{2}$ is equivalent to the desired category of
right flat corings $R_{k}{\operatorname{\mathsf{--Corings}}}_{\mathsf{rfl}}$.

Propositions 1.5, 7.1(a), and 7.2 tell that the categories $\mathsf{K}_{1}$,
$\mathsf{K}_{2}$, and $\mathsf{L}$ are $\kappa$-accessible, and describe their
full subcategories of $\kappa$-presentable objects. Theorem 1.4 (for the given
cardinal $\kappa$ and $\lambda=\aleph_{0}$) is applicable to the pair of
functors $F_{1}$ and $F_{2}$. Theorem 1.4 tells that the category $\mathsf{C}$
is $\kappa$-accessible, and provides the desired description of its full
subcategory of $\kappa$-presentable objects.

The proof of part (b) is similar, except that one needs to take
$\mathsf{K}_{2}=R_{k}{\operatorname{\mathsf{--{}_{fl}Bimod_{fl}--}}}{}_{k}R$
and use Proposition 7.1(b). ∎

######  Remark 7.4.

In the context of this section, the purification-based approach in the spirit
of [2, Theorem 3.1] actually gives a better result than our Theorem 7.3, in
that it represents a coring $C$ with a flatness condition as a
$\kappa$-directed _union_ of its $R$-$R$-bimodule pure _subcorings_
$C^{\prime}$ satisfying the same flatness condition and having cardinalities
smaller than $\kappa$. Similarly, the purification-based approach gives a
better result than Proposition 7.1, in that it represents a bimodule $B$ with
a flatness condition as a $\kappa$-directed union of its pure _subbimodules_
$B^{\prime}$ satisfying the same flatness condition and having cardinalities
smaller than $\kappa$. Notice that, for any ring homomorphism
$R\longrightarrow S$ and any $S$-module $M$, any pure $S$-submodule of $M$ is
also pure as an $R$-submodule of $M$; in particular, any pure submodule over
$R\otimes_{k}R^{\mathrm{op}}$ is a pure left and right $R$-submodule as well.
So this last section is included in this paper only for illustrative and
comparison purposes, as well as to point out an apparent limitation of our
methods.

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