# A model structure and Hopf-cyclic theory on the category of coequivariant
modules over a comodule algebra

Mariko Ohara  Center for Liberal Arts and Sciences, Faculty of Engineering,
Toyama Prefectural University, 5180, Kurokawa, Imizu City, Toyama Prefecture,
JAPAN, 939-0398.
Tel : +81-766-56-7500<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be a coFrobenius Hopf algebra over a field $k$. Let $A$ be a right
$H$-comodule algebra over $k$.

We recall that the category $\mathcal{M}^{H}$ of right $H$-comodules admits a
certain model structure whose homotopy category is equivalent to the stable
category of right $H$-comodules given in [Fari]. In the first part of this
paper, we show that the category $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ of left
$A$-module objects in $\mathcal{M}^{H}$ admits a model structure, which
becomes a model subcategory of the category of $A\\#H^{\ast}$-modules endowed
with a model structure given in [Ohara] if $H$ is finite dimensional with a
certain assumption. Note that $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ is not a
Frobenius category in general. We also construct a functorial cofibrant
replacement by proceeding the similar argument as in [Qi].

Hopf-cyclic theory is refered as a theory of cyclic homology of (co)module
(co)algebra over a Hopf algebra $H$ whose coefficients in Hopf $H$-modules. In
the latter half of this paper, we see that cyclic $H$-comodules which give
Hopf-cyclic (co)homology with coefficients in Hopf $H$-modules are
contructible in the homotopy category of right $H$-comodules, and we
investigate a Hopf-cyclic (co)homology in slightly modified setting by
assuming $A$ a right $H$-comodule $k$-Hopf algebra with $H$-colinear bijective
antipode in stable category of right $H$-comodules and give an analogue of the
characteristic map.

###### Key words and phrases:

Hopf algebra, derived category, model category

###### 2020 Mathematics Subject Classification:

16T15, 18G20, 57T05 55N25 (primary), 16E30, 55U10 (secondary)

##  1\. Introduction

Hopf algebras play the central role of representation theory and theory of
decategorification. Recently, Khovanov [Khov], Qi [Qi] and Farinati [Fari]
defined the derived category of $H$-modules, $A\\#H$-modules for an $H$-module
algebra $A$ and $H$-comodules, respectively. They calculated the Grothendieck
groups $K_{0}$ and $G_{0}$ of the derived categories of $A\\#H$-modules and
right $H$-comodules, respectively.

In [Ohara], the author showed that the category of $A\\#H$-modules admits a
certain model structure under certain assumption, and the derived category of
$A\\#H$-modules in [Qi] is arising from the model structure.

For $H$-comodules, Hovey gave a model structure on the category of chain
complexes on $H$-comodules. Hess and Shipley generalized the stable model
structure on the category of chain complexes of $H$-modules as a left induced
model structure and related with the model structure on the category of
simplicial sets and spectra, respectively. Hess, Kȩdziorek, Riehl and Shipley
observed a necessary and sufficient condition for induced model structure
which is combinatorial and studied the category of differential graded
comodules.

The category of modules over a Frobenius algebra admits a model structure
whose weak equivalences are the stable equivalences by Hovey [Hov]. Li
generalized the model structure to an exact Frobenius category. The model
structure is well-accepted among experts but the proof is written in his paper
for the first time [Li].

A Hopf algebra $H$, such that its linear dual $H^{\ast}$ has a left integral,
is called coFrobenius as in Definition 2.2. Let $H$ be a coFrobenius Hopf
algebra. Let $A$ be a right $H$-comodule algebra in the category
$\mathcal{M}^{H}$ of right $H$-comodules. In this paper, we recall that the
category of left $A$-module objects in $\mathcal{M}^{H}$ is endowed with the
stable model structure with $\lambda\circ S\circ\eta\neq 0$. Here
$\lambda:H\to k$ is a cointegral, $S$ the antipode and $\eta$ the unit map. In
the case of $A\\#H$-module as in [Qi] and [Ohara], $\mathrm{Hom}_{A}(-,-)$ is
naturally an $H$-module. However, in the case of $H$-coequivariant
$A$-modules, $\mathrm{Hom}_{A}(-,-)$ is not an $H$-comodule in general. We
also modified suspention and desuspention defined in [Fari] to be compatible
with $A$-module structure.

We have the following.

######  Theorem 1.1 (Proposition 3.7).

Let $H$ be a coFrobenius Hopf algebra and $\mathcal{M}^{H}$ the category of
right $H$-comodules together with $H$-colinear maps endowed with the stable
model structure. Let $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ be the category of
left $A$-modules of $\mathcal{M}^{H}$ for an $H$-comodule algebra $A$. Let us
denote the forgetful functor by
$U:\mathrm{LMod}_{A}(\mathcal{M}^{H})\to\mathcal{M}^{H}$.

Then, $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ admits a model structure with
respect to the three classes of maps; a map $f$ in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$ is a weak equivalence if $U(f)$ is a weak
equivalence, a fibration if $U(f)$ is a fibration and a cofibration if $f$ has
the left lifting property with respect to trivial fibrations, respectively.

We remark that the model structure in the main theorem and the right-induced
model structure in the paper of Hess and Shipley [HeS1] are introduced in
different situation, respectively, but the model structure on the category of
left $A$-module objects in $\mathcal{M}^{H}$ in this paper has the similar
structure of right induced model structure [HeS1, Definition 4.1].

If $H$ is finite dimensional, we have a comparison with the model structure in
[Ohara] via the functor $i$ given in [Hov].

######  Corollary 1.2.

Let $H$ be a finite dimensional coFrobenius commutative Hopf algebra and
$\mathcal{M}_{H^{\ast}}$ the category of left $H^{\ast}$-modules. Let us take
a right $H$-comodule algebra $A$, which can be regarded as a left
$A\\#H^{\ast}$-module. Let $i:\mathcal{M}^{H}\to\mathcal{M}_{H}^{\ast}$ be the
inclusion functor and $\tilde{i}$ the induced functor on the category of left
$A$-modules. Then, we have the following commutative diagram of functors.

$\textstyle{\mathrm{LMod}_{A}(\mathcal{M}^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{\tilde{i}}$$\textstyle{\mathrm{LMod}_{A}(\mathcal{M}_{H^{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{\mathcal{M}^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{M}_{H^{\ast}}}$

Here, we denote by $U$ the forgetful functors.

Assume that $H$ is coFrobenius and that
$\mathrm{LMod}_{A}(\mathcal{M}_{H^{\ast}})$ inherits a model structure which
is defined as in [Ohara]. Then, $\tilde{i}$ and $i$ are left and right Quillen
functor and $U$ are right Quillen functors.

Let $k$ be a field and $A$ a $k$-algebra. Let $C_{\ast}(A)$ the Hochschild
chain complex, with $C(A)_{n}=A^{\otimes n+1}$. This induces a bicomplex
$B(A)_{\ast\ast}$ with $B(A)_{pq}=A^{\otimes(q-p+1)}$ if $q\geq p$ and $0$
otherwise, whose the vertical differential $b$ is the Hochschild differential
and the horizontal differential is given by the Connes’ operator
$B_{n}:A^{\otimes n+1}\to A^{\otimes n+2}$. The cyclic homology of $A$ is
defined by $HC_{\ast}(A)=H_{\ast}(TotB_{\ast\ast}(A))$. This definition makes
sense for a mixed complex $(M,b,B)$, where $M$ is a graded $k$-module with
differentials $b$ and $B$ such that $bB+Bb=0$. An alternative definition of
cyclic homology is $HC_{\ast}(M)=H_{\ast}(k\otimes_{\Lambda}^{\mathbb{L}}M)$
for a mixed complex $(M,b,B)$, where $(-)\otimes_{\Lambda}^{\mathbb{L}}(-)$
stands for the derived tensor product.

Let us define the notation $\Lambda(x,y)$ to be the signed Hopf algebra
$k\\{x,y\\}/(x^{2},y^{2},xy+yx)$. In [Fari], the Hopf algebra
$k[\mathbb{Z}]\\#\Lambda(x,y)$ is defined to be $k\\{g^{\pm
1},x,y\\}/(gx=-xg,gy=-yg,0=x^{2}=y^{2}=xy+yx)$ together with the
comultiplication map given by $\Delta(g)=g\otimes g$, $\Delta(x)=x\otimes
g+1\otimes x$ and $\Delta(y)=y\otimes g^{-1}+1\otimes y$. In this case, $x$
and $y$ are corresponding to the differential of degree $+1$ and $-1$,
respectively. He also calculated the Hopfological cohomology of a mixed
complexes, which is in turn slightly different from the cyclic homology [Fari,
Section 5.3].

Hopf-cyclic homology is introduced by Connes and Moscovich [Hopf1]. It has
been studied by Kaygun and Khalkhali [KK], Rangipour and many other
researchers and resently generalized to certain Hopf algebroids by Balodi,
which can be refered as a theory of cyclic homology of (co)module (co)algebra
over a Hopf algebra $H$ with coefficients in stable Hopf $H$-modules. For
example, Hopf-cyclic theory appears in Hopf-Galois extension. For a right
$H$-comodule algebra $A$ and the coinvariant subalgebra $B=A^{coH}$, the
extension $B\subset A$ is called Galois if the natural map $A\otimes_{B}A\to
A\otimes H$ is bijective. By the result of Jara and Ştefan, the cyclic
structure on the relative cyclic complex of $B\subset A$ gives rise to a
cyclic module via the natural map, depending only on $H$ and stable anti-
Yetter Drinfeld module $A/[A,B]$.

We recall the properties when a relative $(A,H)$-Hopf module becomes a
projective $A$-module and the Hopf-cyclic modules with a stable $H$-module /
comodule is equivalent to zero in the homotopy category of right
$H$-comodules. Here, a stable $H$-module / comodule is an $H$-comodule wich is
also $H$-module together with $m\circ\Delta=id_{H}$. So we would like to
consider a slightly modified cyclic homology assuming $A$ a Hopf algebra $A$,
induce an analogue of the characteristic map and compare it with one in the
category of chain complexes of $H$-comodules.

######  Theorem 1.3 (Corollary 4.13).

The Hopf-cyclic homology of $H$-comodule $k$-Hopf algebra $A$ together with
bijective $H$-colinear antipode $S:A\to A$ whose coefficients in stable
$A$-module / comodule in the category of $H$-comodules satisfies the universal
approximation as in [Kay]. We also obtain analogue of Hopf-cyclic and
equivariant Hopf-cyclic theory. There is a map $Q^{A}_{\bullet}(A,k)\to
Q^{A}_{\bullet}(A,A)$ of right $H$-comodules in $\mathcal{M}^{H}$ induced from
the unit map.

We remark that, as an expansion of an idea of taking $k$ as in
$Q^{A}_{\bullet}(A,k)$, if we take an $A$-coinvariant part of $M$ assuming
that $M$ is a Hopf $A$-module in $\mathcal{M}^{H}$, we have the degree shift
of $Q_{\bullet}^{A}(A,M)$ as in Remark 4.14.

#### Acknowledgement

The author would like to thank Professor Dai Tamaki to suggest studying Hopf
algebras and a lot of comments over the topic of Hopf-cyclic theory during the
author’s stay in Shinshu University. The author also would like to thank
Professor Takeshi Torii to advise comparing right $H$-comodules with the
category of certain dg-modues endowed with the several model structure. The
author really would like to thank Professor Marco A. Farinati for comments on
earlier version of this paper, especially around mapping cylinder and
suspention. We also owe a significant debt to him, who directed us to the
proper injective embedding and give helpful comments for general wisdom about
injective resolution.

##  2\. The category of $H$-comodules and left module objectss over a monoid
$A$

Let $k$ be a field. Let $H$ be a Hopf algebra over $k$. We write $\otimes$ for
$\otimes_{k}$. Throughout this paper, $H$ will be assumed to be coFrobenius.
The condition is given in Definition 2.2.

###  2.1. The category of right $H$-comodules

A right $H$-comodule is a $k$-vector space $M$ together with a linear map
$\rho:M\to M\otimes H$ satisfying two conditions
$(id\otimes\Delta)\circ\rho=(\rho\otimes id)\circ\rho$ and
$(id\otimes\varepsilon)\circ\rho=id$. Here $\Delta$ and $\varepsilon$ are the
comultiplication map and the counit map of $H$, respectively.

The following lemma is a basic fact (cf. [Fari, Lemma 3.14]).

######  Lemma 2.1.

For any right $H$-comodule $M$, let $V$ be the underlying $k$-vector space of
$M$. Then, we have an isomorphism $M\otimes H\cong V\otimes H$ of right
$H$-comodules given by $m\otimes h\mapsto m_{(0)}\otimes m_{(1)}h$. The
inverse map is given by $m\otimes h\mapsto m_{(0)}\otimes S(m_{(1)})h$, which
are coming to a right $H$-comodule map. Here, we regard $M\otimes H$ and
$V\otimes H$ as right $H$-comodules via the diagonal map and the map
$id\otimes\Delta$, respectively.

If $M$ is a right $H$-comodule, then $M$ can be regarded as a right
$H^{\ast}$-module. Here $H^{\ast}$ stands for the linear dual
$\mathrm{Hom}_{k}(H,k)$ of $H$. A right $H^{\ast}$-module is not necessarily a
right $H$-comodule in general.

The right $H$-comodule structure of $M\otimes_{k}N$ consisting of two right
$H$-comodules $M$ and $N$ is given by diagonal; $(id\otimes
m)\circ(id\otimes\tau\otimes
id)\circ(\rho_{M}\otimes\rho_{N}):M\otimes_{k}N\to
M\otimes_{k}H\otimes_{k}N\otimes_{k}H\to M\otimes_{k}N\otimes H\otimes_{k}H\to
M\otimes_{k}N\otimes_{k}H$. Here, $\rho_{M}$ and $\rho_{N}$ is the right
$H$-comodule structure of $M$ and $N$, respectively, and $\tau$ is the
braiding. As a special case, we regard a right $H$-comodule of the form
$M\otimes H$ as a right $H$-comodule via the structure map $id\otimes\Delta$.
By Lemma 2.1, it is isomorphc as right $H$-comodules to $M\otimes H$ via the
diagonal coaction.

We say a map between right $H$-comodules an $H$-colinear map if it is
compatible with right $H$-comodule structures. We denote by
$\mathrm{Hom}^{H}(M,N)$ the morphism space of maps of $H$-colinear right
$H$-comodules from $M$ to $N$.

Let us denote the category of right $H$-comodules by $\mathcal{M}^{H}$.

######  Definition 2.2 ([AC2013], [Fari]).

Let $k$ be a field and $H$ a Hopf algebra over $k$. A left integral
$\Lambda:H\to k$ is a map satisfying $(id\otimes\Lambda)\Delta h=\Lambda(h)1$
for any $h\in H$. In other words, we have
$h_{(1)}\Lambda(h_{(2)})=\Lambda(h)1$. For a Hopf algebra $H$, if its linear
dual $H^{\ast}=\mathrm{Hom}_{k}(H,k)$ admits a left integral $\Lambda$, we
refer to $H$ as a coFrobenius Hopf algebra.

######  Remark 2.3.

A Hopf algebra $H$ being right coFrobenius amounts to being left coFrobenius,
so that it is left-right symmetric for Hopf algebras. Note that it is not the
case for coalgebras.

Throughout this paper, we assume that $H$ is coFrobenius.

Let $H$ be a coFrobenius Hopf algebra. If $M$ is a right $H$-comodule, we can
think of $M$ as a right $H^{\ast}$-module via the structure map $M\otimes
H^{\ast}\to M\otimes H\otimes H^{\ast}\to M$, where the last map is the
evaluation map $H\otimes H^{\ast}\to k$. This defines fully faithful functor
from the category of right $H$-comodules to the category of right
$H^{\ast}$-modules [Sweedler]. We say that a $H^{\ast}$-module $M$ is rational
(or tame) if, for all $m\in M$, the submodule generated by $m$ is finite-
dimensional. The essential image of the fully faithful functor consists of
rational $H^{\ast}$-modules.

If $H$ is coFrobenius, then every finite dimensional $H$-comodule is a
quotient of a finite dimensional projective $H$-comodule and has an embedding
to a finite dimensional injective $H$-comodule. Especially, the category of
right $H$-comodules has enough projectives and enough injectives. Also the
category $\mathcal{M}^{H}$ of right $H$-comodules becomes a Frobenius category
[AC2013, Theorem 2.8], [Fari, Theorem 2.2].

We know that any injective right $H$-comodule is the direct summand of the
form $M\otimes H$ [Fari, Lemma 3.13 (2)].

Let $H$ be a coFrobenius Hopf algebra over a field $k$. The category of right
$H$-comodules has enough projectives if and only if $H$ is semiperfect, i.e.,
every simple $H$-comodule has an injective hull which is finite-dimensional as
an $H$-vector space. This result is attributed to [Lin].

The category of right $H$-comodules has all small limits and colimits and is
locally presentable [Porst].

######  Definition 2.4 (Stable equivalence).

Let $X$ and $Y$ be two $H$-comodules in $\mathcal{M}^{H}$. A map $f,g:X\to Y$
in $\mathcal{M}^{H}$ is a stably equivalent if $f-g$ factors through an
injective $H$-comodule. This is an equivalence relation that is compatible
with composition. A map $f,g:X\to Y$ of $H$-comodules is a stable equivalence
if it is an isomorphism after taking stable equivalence classes of right
$H$-comodule maps.

######  Definition 2.5.

Let $M$ and $N$ be right $H$-comodules. Let $\mathrm{Hom}^{H}(M,N)_{0}$ be a
$k$-submodule of $\mathrm{Hom}^{H}(M,N)$ consisting of those maps which
factors through an injective right $H$-comodule. A set of maps
$\underline{\mathrm{Hom}^{H}(M,N)}$ is defined to be
$\mathrm{Hom}^{H}(M,N)/\mathrm{Hom}^{H}(M,N)_{0}$.

Let $C$ be an additive full subcategory of an abelian category which is
extension closed and $E$ is a class of short exact sequences in $C$. A map
$A\to B$ of a short exact sequence $A\to B\to C$ in $E$ is called a
cofibration and $B\to C$ called a fibration. Since the category of
$H$-comodules becomes a Frobenius category, e.g., the class of enough
injectives coincides with the class of enough projectives. Therefore the
category of $H$-comodules becomes a Frobenius category. From [Li, Theorem
1.1], an abelian Frobenius category admits the model structure given by the
following three classes of maps; a cofibration is a monomorphism, a fibration
is a epimorphism and a weak equivalence is a stable equivalence.

######  Proposition 2.6 ([Li], Theorem 1.1).

For a coFrobenius Hopf algebra $H$ over a field $k$, the category of right
$H$-comodules becomes an abelian Frobenius category, so that it inherits the
stable model structure, e.g., the class of cofibrations consisting of
monomorphisms, fibrations consisting of epimorphisms and weak equivalences
consisting of stable equivalences.

∎

We denote by $\underline{\mathcal{M}^{H}}$ the homotopy category with respect
to the stable model structure. Since all objects are cofibrant, the category
$\underline{\mathcal{M}^{H}}$ is categorically equivalent to the homotopy
category defined in [Fari]. Moreover, we can construct the derived category of
$\mathcal{M}^{H}$, by inverting weak equivalences, which is arising from the
model structure and is categorically equivalent to the derived category given
in [Fari].

Next, we recall suspention and desuspention of a right $H$-comodule as in
[Fari]. Let $M$ be a right $H$-comodule with the structure map $\rho$. Let
$M\otimes_{k}H$ be a right $H$-comodule with the structure map
$id\otimes_{k}\Delta$. By definition of a right $H$-comodule, we have
$(id\otimes\Delta)\rho=(\rho\otimes id)\rho$ for a structure map $\rho:M\to
M\otimes_{k}H$ of $M$. Then, the suspension of $M$ is defined by
$T(M)=(M\otimes H)/\rho(M)$, where the right $H$-comodule structure of
$M\otimes H$ is given by $id\otimes\Delta$. On the other hand, we have
$(\Lambda^{\prime}\otimes
id)\Delta(h)=\Lambda(S(h_{(1)}))h_{(2)}=\Lambda(S(h_{(1)}))S^{-1}S(h_{(2)})=S^{-1}(\Lambda(S(h_{(1)})S(h_{(2)})))=S^{-1}(\Lambda(Sh)1)=\Lambda^{\prime}(h)1$,
so that $\Lambda^{\prime}$ is compatible with the structure map $\Delta$ as a
right $H$-comodule of $H$ and the trivial $H$-coaction on $k$.

For a left integral $\Lambda:H\to k$, $\Lambda^{\prime}=\Lambda\circ S$ is
right integral, so that ${\mathrm{Ker}}(\Lambda^{\prime})$ becomes a right
$H$-comodule. The desuspension of $M$ is defined by
$T^{-1}(M)=M\otimes{\mathrm{Ker}}(\Lambda^{\prime})$.

By Lemma 2.1, we have an $H$-colinear isomorphism on $M\otimes_{k}H$ between
the right $H$-comodule structure via diagonal coaction and via
$id\otimes\Delta$.

By definition, we have the following exaxt sequences;

$0\to M\to M\otimes H\to T(M)\to 0$

and

$0\to T^{-1}(M)\to M\otimes H\to M\to 0.$

We replace $M$ as $T^{\prime}M$ in the first short exact sequence, then we
have $TT^{\prime}\simeq id$ and $T^{\prime}T\simeq id$ in the homotopy
category. Thus, the homotopy category $\underline{\mathcal{M}^{H}}$ and the
derived category becomes triangulated.

These suspention $T$ and desuspention $T^{-1}$ are useful when chasing exact
sequence of cohomology as in [Fari]. However, we will define a slightly
different suspention and desuspention as following definition since we will
realize them as left $A$-module objects for a right $H$-comodule algebra $A$
later in section $2.2$.

######  Definition 2.7.

Let $M$ be a right $H$-comodule. We will choice an injective embedding as
follows.

  1. (i)

First, since $H$ is coFrobenius, $\Lambda^{\prime}$ is not a zero map, so that
there exists $x\in H$ such that $\Lambda^{\prime}(x)$ is not zero.

  2. (ii)

There exists finite dimensional $H$-subcomodule in $H$ that contains $x$. (we
can take $x=\Sigma\epsilon(x_{0})x_{1}$, where $x_{1}$ is regarded as a basis
of $k$-vector space). We take one of these $k$-basis $x_{1}$s whose image
under $\Lambda^{\prime}$ is not zero. We write it as $x^{1}$.

  3. (iii)

If we take another $x^{2}$ as (ii), we see that $(x^{1})\otimes
H\cong(x^{2})\otimes H$ as $H$-comodules since we have an $H$-comodule
isomorphism $M\otimes H\cong V\otimes H$ as Lemma 2.1, where $V$ is the
underlying $k$-module. Thus we have $kx^{1}\otimes H\cong kx^{2}\otimes H$.
Here, $kx^{1}$ and $kx^{2}$ are both $k$-vector space generated by $x^{1}$ and
$x^{2}$, respectively

  4. (iv)

Then, we take an injective embedding: $id\otimes x^{1}:M\to M\otimes H$ : the
right component is always $x^{1}$ of (ii).

######  Remark 2.8 (The existence of a functorial injective embedding).

For the trivial module $k$, if we choose a monomorphism $k\to B$ where $B$ is
an injective $H$-comodule, by tensoring with this map, we obtain an injective
embedding $M\to M\otimes B$ instead of $M\to M\otimes H$. When $H$ is
coFrobenius, we can choose those injective $B$ to be finite dimensional.

######  Definition 2.9.

We define $\Sigma(M)$ by the pushout diagram in $\mathcal{M}^{H}$

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\otimes
x}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma(M)}$

Here, $id\otimes x$ is an injective embedding we take as in Definition 2.8.
Note that $\Sigma(M)$ is equivalent to $T(M)$ in
$\underline{\mathcal{M}^{H}}$.

As in [Fari], we define $\Sigma^{-1}(M)$ by the pullback in $\mathcal{M}^{H}$

$\textstyle{\Sigma^{-1}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\otimes\Lambda^{\prime}}$$\textstyle{M}$

###  2.2. The category of left modules over a monoid object $A$ in
$\mathcal{M}^{H}$

We say that a right $H$-comodule $A$ is an $H$-comodule algebra $A$ if it is a
$k$-algebra $A$ together with a right $H$-comodule structure such that the
multiplication map $A\otimes A\to A$ and the unit $k\to A$ are $H$-colinear,
i.e., compatible with $H$-comodule structure.

Let $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ be the category of left $A$-module
objects in $\mathcal{M}^{H}$. We say an object $M$ in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$ an $H$-coequivariant $A$-module.

For an $H$-coequivariant $A$-module $M$ and right $H$-comodule $V$, we have an
$H$-coequivariant $A$-module $M\otimes V$, where $A$-module structure in
$M\otimes V$ is the one coming from $M$ and the right $H$-comodule structure
is the diagonal one.

As in [Khov] and [Qi], we consider the restriction functor

$\mathrm{LMod}_{A}(\mathcal{M}^{H})\to\mathcal{M}^{H}$

and define $M$ in $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ to be contructible if
$M$ is injective as right $H$-comodule. We say that a map $f:M\to N$ in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$ is a stable equivalence if it becomes an
isomorphism in $\underline{\mathcal{M}^{H}}$.

Let $M$ and $N$ be $H$-coequivariant $A$-modules, i.e., objects in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$. We say that a map $f:M\to N$ of
$A$-modues is $H$-colinear if the following diagram commutes.

$\textstyle{A\otimes
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\otimes
f}$$\scriptstyle{m_{M}}$$\scriptstyle{\rho_{M}}$$\textstyle{A\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{N}}$$\scriptstyle{m_{N}}$$\textstyle{A\otimes
M\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{M}\otimes
id}$$\textstyle{A\otimes N\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{N}\otimes
id}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\rho_{M}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{N}}$$\textstyle{M\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes
id}$$\textstyle{N\otimes H}$

For $H$-coequivariant $A$-modules $M$ and $N$, we denote by
$\mathrm{Hom}_{A}^{H}(M,N)$ the set of maps that are $H$-colinear and also
$A$-linear.

If $H$ is a coFrobenius Hopf algebras, for an $H$-comodule algebra $A$ and
$H$-coequivariant $A$-modules $M$ and $N$, $\mathrm{Hom}_{A}(M,N)$ does not
have an $H$-comodule structure in general [Fari, Section 7.1]. Therefore, we
regard right $H$-comodules $M$ and $N$ as (right) $H^{\ast}$-modules via the
evaluation map $H\otimes H^{\ast}\to k$, we obtain a left $A$-module right
$H^{\ast}$-module $M$.

######  Definition 2.10.

An $H^{\ast}$-action of $x\in H^{\ast}$ on $f:M\to N$ is given by
$xf=(1\otimes m^{\ast}_{H}(x))(\rho_{N}\otimes 1)(f\otimes S)\rho_{M}$. In
other words,

$(xf)(m)=x(f(m_{(0)})_{1}S(m_{(1)}))f(m_{(0)})_{0}.$

Then, $f$ is $H$-colinear if and only if it is $H^{\ast}$-linear, and $f$ is
$H$-colinear if and only if $xf=\epsilon(x)f=x(1)f$ for any $x\in H^{\ast}$
[Fari, Proposition 7.6].

This definition also recovers the action on morphism spaces in [Qi] when $H$
is finite dimensional. The standard action of an element $x\in H^{\ast}$ on a
map $f$, also acting on an element $m\in M$ is given by

$(xf)(m)=x_{(1)}f(S(x_{(2)})m).$

The set $\mathrm{Hom}_{A}$ itself may not be an $H$-comodule but admits
$H^{\ast}$-action. So we can take ”invariant of a morphism space of
$H^{\ast}$-linear maps”.

######  Proposition 2.11.

$\mathrm{Hom}_{A}(M,N)$ is an $H^{\ast}$-module. Explicitly, for an $A$-module
map $f:M\to N$ between $H$-coequivariant $A$-modules, it is $H^{\ast}$-linear
if and only if it is $H$-colinear.

Moreover, any map $f\in\mathrm{Hom}_{A}^{H}(M,N)$ between $H$-coequivariant
$A$-modules $M$ and $N$ can be regarded as a map in $\mathrm{Hom}_{A}(M,N)$
whose $H^{\ast}$-action as in Definition 2.10 is the action given by the
counit $\epsilon\in H^{\ast}$. Conversely, any map $f\in\mathrm{Hom}_{A}(M,N)$
inherits an $H^{\ast}$-action as in Definition 2.10, and it extends to a map
in $\mathrm{Hom}_{A}^{H}(M,N)$ if the $H^{\ast}$-action is given by the counit
$\epsilon\in H^{\ast}$.

###### Proof.

The first statement is well-known as in [Qi] and followed by the fact that $f$
is $H$-colinear if and only if $xf=\epsilon(x)f=x(1)f$ for any $x\in
H^{\ast}$.

For the second statement, if $L$ is an $H^{\ast}$-module, one may define
$L^{H^{\ast}}=\\{m\in L|xm=x(1)m\\}\cong\mathrm{Hom}_{H^{\ast}}(k,L)$. Note
that if $L$ is a right $H$-comodule, we have $L^{coH}=\\{m\in
L|\rho(m)=m\otimes 1\\}=L^{H^{\ast}}$. If $L$ stands for
$\mathrm{Hom}_{A}(M,N)$, then $L^{H^{\ast}}$ consists of $A$-linear and
$H^{\ast}$-linear maps, so it is $\mathrm{Hom}_{A}^{H}(M,N)$. ∎

If $M$ and $N$ are objects in $\mathrm{LMod}_{A}(\mathcal{M}^{H})$, we denote
by $I_{A}(M,N)$ the subset of $\mathrm{Hom}^{H}_{A}(M,N)$ consisting of maps
that factors through an $H$-coequivariant $A$-modules that are injective as
right $H$-comodules. The stable homotopy category denoted by
$\underline{\mathrm{LMod}}_{A}(\mathcal{M}^{H})$ is defined as the category
with same objects as $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ but morphism

$\mathrm{Hom}_{\underline{\mathrm{LMod}}_{A}(\mathcal{M}^{H})}(M,N)=\mathrm{Hom}_{A}^{H}(M,N)/I_{A}(M,N).$

The derived category $D_{H}(A)$ is defined by formally inverting stable
equivalences in the homotopy category
$\underline{\mathrm{RMod}}_{A}(\mathcal{M}^{H})$.

Especially, the projective objects are preserved via the exact functor

$\underline{\mathrm{U}}:\underline{\mathrm{LMod}}_{A}(\mathcal{M}^{H})\to\underline{\mathcal{M}^{H}}$

induced from the forgetful functor.

The suspention $\Sigma(M)$ and desuspention $\Sigma^{-1}(M)$ in
$\mathcal{M}^{H}$ actually gives objects in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$.

From now on, for a right $H$-comodule map $\Lambda^{\prime}:H\to k$, that is
the composition of an integral $\Lambda:H\to k$ with the antipode $S:H\to H$.

######  Remark 2.12.

Let $X$ be an $H$-coequivariant $A$-module. In [Fari], the structure map
$\rho:X\to X\otimes H$ is regarded as an injective embedding. If we regard the
field $k$ as the trivial $H$-comodule via $\eta$ and the Hopf algebra $H$ as
the right $H$-comodule via the comultiplication $\Delta$, the map $\eta$ is
coming to be a right $H$-comodule map.

Note that $id_{X}\otimes\eta:X\to X\otimes H$,
$id_{X}\otimes\varepsilon:X\otimes H\to X$ and the identity map on $X$ are
left $A$-module maps but $id\otimes\varepsilon$ is not a right $H$-comodule
map. Here $X\otimes H$ is endowed with a right $H$-comodule structure by
$id\otimes\Delta$.

Assume that $X\otimes H$ is a right $H$-comodule by $id\otimes\Delta$. Let us
consider the following exact sequences on $k$-vector space level.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{X}\otimes\varepsilon}$$\textstyle{X.}$

Note that $\rho$ is a map of right $H$-comodules. Since $H$ is a bialgebra,
this is a split exact sequence on $k$-vector space level. On the other hand,
we have an exact sequence of $k$-vector space

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{X}\otimes\eta}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{X}\otimes\varepsilon}$$\textstyle{X.}$

This is also a split exact sequence on $k$-vector space level and also splits
as an exact sequence of left $A$-modules when we leave the $H$-comodule
structures. Note that $A$ acts only on each left component and $id\otimes\eta$
is a map of $H$-coequivariant left $A$-modules.

We define a mapping cylinder and a mapping cocylinder.

######  Definition 2.13.

We define an $H$-coequivariant $A$-module $C_{f}$ by the pushout

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{id\otimes
x}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{f}.}$

Here, we regard $X\otimes H$ as a right $H$-comodule via the structure map
$id\otimes\Delta$.

As the following diagram and Definition 2.9, we obtain the map $C_{f}\to\Sigma
X$ by the universality.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{id\otimes
x}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{C_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma X}$

We have the commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{id_{X}\otimes
x}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Y}}$$\textstyle{X\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes\Lambda^{\prime}}$$\textstyle{C_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\otimes
k\cong Y.}$

Similarly, we define an $H$-coequivariant $A$-module $P_{f}$ by the pullback

$\textstyle{P_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Y}\otimes\Lambda^{\prime}}$$\textstyle{Y\otimes_{k}k\cong
Y.}$

As the following diagram and Definition 2.9, we obtain the map
$\Sigma^{-1}Y\to P_{f}$ by the universality.

$\textstyle{\Sigma^{-1}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{Y\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Y}\otimes\Lambda^{\prime}}$$\textstyle{Y}$

We have the commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{X}}$$\scriptstyle{f\otimes
1}$$\textstyle{P_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id_{Y}\otimes\Lambda^{\prime}}$$\textstyle{Y.}$

As a variant of [Qi, Lemma 4.3] and also [Fari, Lemma 2.10], we have the
following lemma of standard triangles.

######  Lemma 2.14.

For a map $f:X\to Y$ in $\mathrm{LMod}_{A}(\mathcal{M}^{H})$, the maps $Y\to
C_{f}$ and $P_{f}\to X$ are $A$-split.

Especially, two short exact sequences

$0\to Y\to C_{f}\to\Sigma X\to 0$

and

$0\to\Sigma^{-1}Y\to P_{f}\to X\to 0$

are $A$-split.

Moreover, the both two exact sequences induce a triangle $X\to Y\to
C_{f}\to\Sigma X$ in the homotopy category
$\underline{\mathrm{LMod}}_{A}(\mathcal{M}^{H})$.

###### Proof.

Since $id_{X}\otimes x:X\to X\otimes H$ and
$id_{Y}\otimes\Lambda^{\prime}:Y\otimes H\to Y$ are $A$-split, the first
assertion follows.

The last claim follows since we have a short exact sequence $0\to X\otimes
H\to C_{f}\to\mathrm{Coker}(f)\to 0$, and so $C_{f}\simeq\mathrm{Coker}(f)$ in
the homotopy category.

∎

Thus, $\underline{\mathrm{LMod}}_{A}(\mathcal{M}^{H})$ is a triangulated
category.

##  3\. A model structure on the category of $H$-coequivariant $A$-modules

Now, we use the adjunction

$A\otimes_{k}(-):\mathcal{M}^{H}\rightleftarrows\mathrm{LMod}_{A}(\mathcal{M}^{H}):U,$

where $U$ is the functor which regards $H$-coequivariant $A$-modules as
$H$-comodules and $A\otimes_{k}(-)$ is the left adjoint, which is given by the
free functor.

By using the stable model structure on $\mathcal{M}^{H}$ given in Definition
2.6 and this adjunction, we take the following three classes of maps in the
category $\mathrm{LMod}_{A}(\mathcal{M}^{H})$.

######  Definition 3.1.

Let $f:X\to Y$ be a map in $\mathrm{LMod}_{A}(\mathcal{M}^{H})$.

  1. (i)

We say that $f$ is a weak equivalence if $Uf$ is a weak equivalence.

  2. (ii)

We say that $f$ is a fibration if $Uf$ is a fibration.

  3. (iii)

We say that $f$ is a cofibration if $f$ has the left lifting property with
respect to trivial fibrations.

We will see that the three classes of maps as in Definition 3.1 defines a
model structure on the category $\mathrm{LMod}_{A}(\mathcal{M}^{H})$. Note
that the category $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ is abelian so it is
bicomplete. The proof will have many parallels with the case of
$A\\#H$-modules in [Ohara]

######  Lemma 3.2 (2 out of 3).

Let $f$, $g$ and $g\circ f$ be morphisms in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$. If two of the three morphisms are weak
equivalences in $\mathrm{LMod}_{A}(\mathcal{M}^{H})$, then, so is the third.

###### Proof.

Since $U$ is a covaiant functor, the composition $U(f)\circ U(g)$ is $U(f\circ
g)$. The assertions follow from restricting these maps on $H$-comodules under
the functor $U$. ∎

######  Lemma 3.3.

Let $f$ and $g$ be maps of $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ such that $f$
is a retract of $g$, i.e., $f$ and $g$ satisfies the following commutative
diagram.

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$

where the horizontal composites are identities.

If $g$ is a weak equivalence, cofibration, or fibration, respectively, then so
is the third.

###### Proof.

When $g$ is a weak equivalence or fibration, respectively, we can see that a
retract $f$ of $g$ is also a weak equivalence or fibration, respectively, by
regarding the diagram as a diagram of $H$-modules via the functor $U$. If the
map $g$ is a cofibration, for any trivial fibration $X\to Y$, the map $g$ has
the left lifting property as follows

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y,}$

so we have the left lifting of $f:M\to N$ with respect to $X\to Y$. ∎

The lifting propeties for trivial fibrations are obvious and, for pairs of
trivial cofibrations and fibrations, notice that the cokernel of a trivial
cofibration as $H$-coequivariant $A$-module is an injective $H$-comodule. We
obtain the certain form of direct sum of $H$-comodules and obtain the desired
$H$-coequivariant $A$-module map by Proposition 2.11.

######  Proposition 3.4 (Factorization).

For any map $f:X\to Y$, we have a factorization $f:X\to E\to Y$ for some $E$,
where $X\to E$ is a cofibration and $E\to Y$ is a trivial fibration. Also we
have a factorization $f:X\to E^{\prime}\to Y$, where $X\to E^{\prime}$ is a
trivial cofibration, and $E^{\prime}\to Y$ is a fibration.

###### Proof.

It suffices to show that, if we have a cofibrant replacement and fibrant
replacement, respectively, we obtain the desired factorization of this lemma
as follows. A cofibrant replacement is constructed in the following
Proposition 3.6 and all objects are fibrant.

For a cofibrant replacement $QC_{f}\to C_{f}$ of $C_{f}$, where the map
$QC_{f}\to C_{f}$ is a trivial fibration, we have the following diagram :

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{P_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{QC_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{u}$$\textstyle{\Sigma
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
X.}$

Two vertial maps in the middle square is surjective and two horizontal maps in
the middle square is $A$-split. We define the map $u$ as a map the right
square commutes. We construct the map $P_{u}\to Y$ by using $A$-splittings,
Lemma 2.14 and Proposition 2.11. We have the map $\Sigma^{-1}\Sigma X\to
P_{u}$ by the universality of desuspension. The map $P_{u}\to Y$ is surjective
since it is composed of two surjections. Passing to the quotient triangulated
category, the map $P_{u}\to Y$ is a weak equivalence.

The map $X\to P_{u}$ is a cofibration. Consider the following diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N}$$\textstyle{QC_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $M\to N$ is a trivial fibration and $P_{u}\to QC_{f}$ is $A$-split.
Since $QC_{f}$ is cofibrant, we have the dotted arrow. By using $A$-splitting
endowed with trivial $H$-coaction, we have the lifting $P_{u}\to M$.

For a fibrant replacement $P_{f}\to RP_{f}$, where the map $P_{f}\to RP_{f}$
is a trivial cofibration, we have the following diagram :

$\textstyle{\Sigma^{-1}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{P_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\Sigma^{-1}Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{RP_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y.}$

Note that two horizontal maps in the middle square is $A$-split. The map $v$
is defined to be the left square commutes. We construct the map $X\to C_{v}$
by using $A$-splittings, Lemma 2.14 and Proposition 2.11. We have the map
$C_{v}\to\Sigma\Sigma^{-1}Y$ by the universality of suspension. The map $X\to
C_{v}$ is a stable equivalence since it is composed of two stable
equivalences.

Note that $X\to C_{v}$ is also cofibration since $A$-split monomorphisms are
included in the class of cofibrations.

The map $C_{v}\to Y$ is a fibration. Consider the following diagram

$\textstyle{RP_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

where $M\to N$ is a trivial cofibration and $RP_{f}\to C_{v}$ is $A$-split.
Since $RP_{f}$ is fibrant, we have the dotted arrow. By using $A$-splitting
endowed with trivial $H$-coaction, we have the lifting $N\to C_{v}$. ∎

###  3.1. Functorial cofibrant replacement for $H$-coequivariant $A$-modules

######  Definition 3.5.

Let $M$ be an $H$-coequivariant $A$-module. $M$ is said to be cofibrant if for
any surjective stable equivalence $L\to N$ of $H$-coequivariant $A$-modules,
the induced map of $k$-vector spaces

$\mathrm{Hom}_{A}^{H}(M,L)\to\mathrm{Hom}_{A}^{H}(M,N)$

is surjective.

As the similar argument in [Qi], we will construct a functorial cofibrant
replacement.

######  Proposition 3.6 (Functorial cofibrant replacement).

For each $H$-coequivariant $A$-module $M$, there is a short exact sequence in
$\mathrm{LMod}_{A}(\mathcal{M}^{H})$ which is split exact as a sequence of
$A$-modules, $0\to M\to aM\to\bar{p}M\to 0$ where $\bar{p}M$, and also
$p(M)=\Sigma^{-1}(\bar{p}(M))$ is cofibrant and aM is an contructible and a
left $A$-module. The construction of the short exact sequence is functorial in
$M$.

###### Proof.

This construction is an analogue of normalized chain complex of simplicial
resolution. We construct the bar resolution for $M=A$. The proof also gives
rise to the short exact sequence $0\to M\to aM\to\bar{p}M\to 0$ claimed in the
theorem.

One can take such a simplicial module {$C_{\bullet}$} that the differential
$\delta:C_{n}\to C_{n+1}$ is the alternative sum of the face maps
$\delta=\Sigma_{0\leq i\leq n-1}(-1)^{i}d_{i}$. Then, $(C_{\bullet},\delta)$
becomes a complex.

We use an expedient notation $A\otimes H/(x)$ for the cokernel of $id\otimes
x:A\to A\otimes H$. If $A$ is a right $H$-comodule algebra, all the face and
degeneration maps are $H$-comodule maps. Therefore, $\delta=\Sigma_{0\leq
i\leq n-1}(-1)^{i}d_{i}$ is also an $H$-colinear map [Qi, Example 2.8]. We
take the mapping cylinder of $\delta_{0}:A\otimes A\to A$, and we have
$C_{\delta_{0}}\cong A\otimes A\otimes(H/(x))\oplus A$ as left $A$-modules, so
that we have an $A$-split filtration $\\{0\\}\subset A\subset C_{\delta_{0}}$,
the isomorphism viewed as a left $A$-module map. Here, we regard the
$A$-action on $(H/(x))$ as trivial $A$-action.

We take $\delta_{1}$ and we have the diagram

$\textstyle{A\otimes A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}}$$\scriptstyle{id\otimes
x}$$\textstyle{A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes
A\otimes A\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\delta_{1}}.}$

Then, we have a map $\bar{\delta_{1}}:Coker(\rho_{1})\to C_{\delta_{0}}$ from
the following diagram

$\textstyle{A\otimes A\otimes A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{2}}$$\scriptstyle{id\otimes
x}$$\textstyle{A\otimes A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{id\otimes
x}$$\scriptstyle{\delta_{1}}$$\textstyle{A\otimes
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$$\scriptstyle{id\otimes
x}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes
A\otimes A\otimes A\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{2}\otimes
id_{H}}$$\textstyle{A\otimes A\otimes A\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}\otimes
id_{H}}$$\textstyle{A\otimes A\otimes
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Coker(id\otimes
x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\delta_{2}\otimes
id_{H}}}$$\textstyle{Coker(id\otimes
x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\delta_{1}}}$$\textstyle{C_{\delta_{0}}}$

and we know that the composition
$\bar{\delta_{1}}\circ\overline{\delta_{2}\otimes id_{H}}=0$.

Then, we obtain the cylinder $C_{\bar{\delta_{1}}}$ of the map
$\bar{\delta_{1}}$.

Since $id\otimes x$ is an $A$-split map, the cylinder $C_{\bar{\delta_{1}}}$
has an $A$-split filtration $0\subset A\subset Cone(\delta_{0})\subset
C_{\bar{\delta_{1}}}$ whose subquotients are isomorphic, as left $A$-modules,
to $A$, $A^{\otimes 2}\otimes(H/(x))$, and $A^{\otimes
3}\otimes(H/(x))^{\otimes 2}$, respectively.

Thus, we proceed to construct bar resolution inductively.

Let $C_{0}=C_{\delta_{0}}$. Inductively, we obtain the following

(i) $C_{n}\cong\text{mapping cylinder
of}(\bar{\delta_{n}}:A^{\otimes(n+2)}\otimes(H/(x))^{\otimes n}\to C_{n-1})$.
Here we use the expedient notation $A^{\otimes(n+2)}\otimes(H/(x))^{\otimes
n}$ for the object obtained by iterated taking cokernel procedure.

(ii) $\delta_{n}(\overline{\delta_{n+1}\otimes id})=0$

This assumption implies that $C_{n-1}$ is a submodule of $C_{n}$ and $C_{n}$
has an $A$-split filtration

$0=F^{-1}\subset F^{0}\subset\cdots\subset F^{p-1}\subset
F^{p}\subset\cdots\subset F^{n+1}=C_{n},$

whose subquotients $F^{n}/F^{n-1}$ are isomorphic to
$A^{\otimes(n+2)}\otimes(H/(x))^{\otimes n}$.

Note that $\mathrm{Hom}_{A}(P,K)$ is contructible for a projective $A$-module
$P$ and an injective $H$-comodule $K$ by decomposing $P$ into finite
dimensional ones and that $F^{n}/F^{n-1}$ is $A$-free. Apply
$\mathrm{Hom}_{A}(-,K)$ to the $A$-split short exact sequence $0\to F^{n-1}\to
F^{n}\to F^{n}/F^{n-1}\to 0$. Induction for $F_{n-1}$ and $F_{n}/F_{n-1}$, we
can see that $\mathrm{Hom}_{A}(F_{n},K)$ is contructible by [Fari, Corollary
3.18]. Thus, $\mathrm{Hom}_{A}(C_{n},K)$ is contructible for an injective
$H$-comodule $K$.

Recall that the definition of cofibrant objects as in Definition 3.5. For a
surjective stable equivalence $L\to M$, the kernel is an injective
$H$-comodule $K$. We have an exact sequence
$0\to\mathrm{Hom}_{A}(C_{n},K)\to\mathrm{Hom}_{A}(C_{n},L)\to\mathrm{Hom}_{A}(C_{n},M)\to
0$. Since $\mathrm{Hom}_{A}(C_{n},K)$ is contructible, we have
$\mathrm{Hom}_{A}(C_{n},L)\cong\mathrm{Hom}_{A}(C_{n},K)\oplus\mathrm{Hom}_{A}(C_{n},M)$.
By taking $H^{\ast}$-invariant part of both hand sides, we have a surjection
$\mathrm{Hom}_{A}^{H}(C_{n},L)\to\mathrm{Hom}_{A}^{H}(C_{n},M)$. Thus, $C_{n}$
is cofibrant.

Now we define $aA=\cup_{0\leq n\leq\infty}C_{n}$, which fits into a short
exact sequence, $0\to A\to aA\to\bar{p}A\to 0$ since each $C_{n}$ has the
factor $A$ in the $A$-split filtration. Now we are going to show that $aA$ in
the above short exact sequence is contractible as a right $H$-comodule.

We will show this for $aA$, and the general case follows by the same argument.
The homotopy $s:A^{\otimes n}\to A^{\otimes(n+1)}$ as an extra degeneracy is a
right $H$-comodule map since $A$ is an $H$-comodule algebra. Thus, a mapping
cylinder becomes a retract of cone of the form $A^{\otimes(n+1)}\otimes H$, so
that each $C_{n}$ is an injective right $H$-comodule.

It is then easily seen that the map $A\otimes A\otimes k\cong A\otimes A\to A$
extends to a surjection $pA\to A$.

∎

Finally, we obtain the following proposition.

######  Proposition 3.7.

The category $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ inherits a model structure
with respect to the three classes of maps defined as in Definition 3.1.

###### Proof.

By the lemmas and the fact that all limits and colimits exists since it is a
module category. ∎

######  Corollary 3.8.

The forgetful functor $U:\mathrm{LMod}_{A}(\mathcal{M}^{H})\to\mathcal{M}^{H}$
is the right Quillen functor. Here, we regard $\mathcal{M}^{H}$ as the stable
model category.

∎

######  Remark 3.9.

Although the model structure in Proposition 3.7 and right-induced model
structure in the paper of Hess and Shipley [HeS1] are introduced in different
situations, but the model structure on the category of left $A$-module objects
in $\mathcal{M}^{H}$ in this paper has the similar structure of right induced
model structure [HeS1, Definition 4.1].

###  3.2. Relation to the category of chain complexes of comodules

In [Hov, Section 2], Hovey constructed a Quillen adjunction
$F:\mathcal{M}_{H^{\ast}}\to Ch(H)$ for a finite dimensional commutative Hopf
algebra $H$, where $\mathcal{M}_{H^{\ast}}$ is endowed with the model
structure on Frobenius category while the weak equivalences in $Ch(H)$ are
”homotopy isomorphisms” as in [Hov]. Let $Tk$ be a Tate resolution of the base
field $k$. Recall that a Tate resolution $Tk$ of $k$ is a complex of
projectives that are also injectives with no homology, such that $Z_{0}Tk=k$.
We take a projective resolution $P_{\ast}\to k$ and an injective resolution
$k\to I_{\ast}$, and let $(Tk)_{n}=P_{n-1}$ if $n>0$ and $(Tk)_{n}=I_{-n}$ if
$n\leq 0$. In particular, the cycles in degree $0$ are just $k$. Then, we
define $FM=Tk\otimes M$. The right adjoint $U:Ch(H)\to\mathcal{M}_{H^{\ast}}$
of $F$ is then defined by $UX=Z_{0}\mathrm{Hom}(Tk,X)$. Since the tensor
product is over a field, $F$ preserves injections, and hence cofibrations.
Note that the complex $Tk\otimes H$ is chain homotopy equivalent to $0$.

By composition with the forgetful functor $U$, we have the composition of
functors

$\textstyle{\mathrm{LMod}_{A}(\mathcal{M}^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{\mathcal{M}^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{M}_{H^{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{Ch(H)}$

Here, $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ inherits the model structure as in
Proposition 3.7. Note that $\mathrm{LMod}_{A}(\mathcal{M}^{H})$ may not a
Frobenius category in general. We denote by $\mathcal{M}_{H^{\ast}}$ the
category of $H^{\ast}$-modules, the forgetful functor $U$ is the right adjoint
and the inclusion $i$ is the Quillen functor and the fonctor $F=(-)\otimes Tk$
given by tensoring a Tate resolution of the base field is the left Quillen
adjoint since $F$ preserves cofibrations and also colimits.

##  4\. Hopf cyclic cohomology revisited

We will need several related combinatorial categories $\Delta$,
$\Lambda_{\infty}$, $\Lambda_{\mathbb{N}}$ and $\Lambda$.

We recall the definition of the para-cyclic category $\Lambda_{\infty}$. It is
the full subcategory $\Lambda_{\infty}\subset\mathbb{Z}Poset$ consisting of
all objects isomorphic to $\frac{1}{n}\mathbb{Z}$ for $n>1$. Here, $Poset$ is
the category or partially ordered sets and non-decreasing maps,
$\mathbb{Z}Poset$ is the category of objects in $Poset$ equipped with a
$\mathbb{Z}$-action, and we consider $\frac{1}{n}\mathbb{Z}$ as an object with
its natural ordering and the $\mathbb{Z}$-action given by addition.

There is an action of $\mathbb{Z}$ on the morphism spaces of
$\Lambda_{\infty}$, with the generator $\sigma\in\mathbb{Z}$ sending a
morphism $f:\frac{1}{n}\mathbb{Z}\to\frac{1}{m}\mathbb{Z}$ to
$\sigma(f):\frac{1}{n}\mathbb{Z}\to\frac{1}{m}\mathbb{Z}$ given by
$\sigma(f)=f+1$. By restricting $B\mathbb{Z}$ to $B\mathbb{N}$, we obtain
pseudo-para-cyclic category $\Lambda_{\mathbb{N}}$. For any integer $p\geq 1$,
we define $\Lambda_{p}=\Lambda_{\infty}/B(p\mathbb{Z})$. By definition, there
is a $BC_{p}=B\mathbb{Z}/B(p\mathbb{Z})$-action on the category $\Lambda_{p}$,
and we refer to $\Lambda=\Lambda_{p}/BC_{p}$ as the cyclic category. The
categories $\Lambda_{\infty}$ and $\Lambda$ are self-dual.

######  Remark 4.1.

Elmendorf investigated a larger category $L$ in order to understand the self-
duality of $\Lambda$. The same category as $L$ is denoted by
$\Lambda_{\infty}$ by Getzler and Jones and is called the para-cyclic
category. The point is that an object $[n]$ in $\Lambda$ can be regarded as
$\mathbb{Z}/(n+1)\mathbb{Z}$ as sets. The isomorphism $L\to\Lambda_{\infty}$
is given by $\mathbb{Z}\times\\{n\\}\mapsto\frac{1}{n+1}\mathbb{Z}$. Nikolaus
and Scholze recalled $L$ as a full subcategory of the category of posets with
$\mathbb{Z}$-actions. The subcategory generated by $d^{i}_{n}$ and $s^{j}_{n}$
for $n\geq 0$ in $L$ is isomorphic to $\Delta$. Any morphism in $L$ can be
factored as compositions of $d_{n}^{i}$, $s_{n}^{i}$, $t_{n}$ and
$t_{n}^{-1}$. This says that $L$ is obtained from $\Delta$ by adding operators
$t_{n}$.

Especially, $\Lambda_{\infty}$ contains $\Delta$. The category
$\Lambda_{\infty}$ is obtained by adding isomorphisms $t_{n}:[n]\to[n]$ to
$\Delta$ for $n\in\mathbb{Z}_{\geq 0}$ with relations

$\displaystyle t_{n}\circ d^{i}_{n-1}=d^{i-1}_{n-1}\circ t_{n-1}$
$\displaystyle(i>0)$ $\displaystyle t_{n}\circ d^{0}_{n-1}=d_{n-1}^{n}$
$\displaystyle t_{n}\circ s^{i}_{n+1}=s^{i-1}_{n+1}\circ t_{n+1}$
$\displaystyle(i>0)$ $\displaystyle t_{n}\circ s^{0}_{n+1}=s^{n+1}_{n+1}\circ
t^{2}_{n+1}$

Roughly speaking, it is obtained by dropping the condition $t_{n}^{n+1}=1$
from the cyclic category.

The subcategory of $\Lambda_{\infty}$ generated by $\Delta$ and $t_{n}^{j}$
for $j\geq 0$ is denoted by $\Lambda_{\mathbb{N}}$.

######  Definition 4.2.

A contravariant functor $X:\Lambda^{op}_{\infty}\to\mathcal{C}$ is called a
para-cyclic object in $\mathcal{C}$. Covariant functors
$\Lambda_{\infty}\to\mathcal{C}$ are called para-cocyclic objects.

A para-cyclic object in $\mathcal{C}$ is a simplicial object $X$ in
$\mathcal{C}$ together with morphisms $t_{n}:X_{n}\to X_{n}$ for $n\geq 0$
satisfying the following identities:

$\displaystyle d^{i}_{n}\circ t_{n}=t_{n-1}\circ d^{n-1}_{i-1}$ $\displaystyle
d_{0}^{n}\circ t_{n}=d_{n}^{n}$ $\displaystyle s_{i}^{n}\circ
t_{n}=t_{n+1}\circ s_{i-1}^{n}$ $\displaystyle s_{0}^{n}\circ
t_{n}=(t_{n+1})^{2}\circ s_{n}^{n}$

Dually a para-cocycic object in $\mathcal{C}$ is a cosimplicial object $Y$
together with morphisms $t_{n}:Y^{n}\to Y^{n}$ for $n\geq 0$ satisfying the
following identities:

$\displaystyle t_{n}\circ d^{i}=d^{i-1}\circ t_{n-1}$ $\displaystyle
t_{n}\circ d^{0}=d^{n}$ $\displaystyle t_{n}\circ s^{i}=s^{i-1}\circ t_{n+1}$
$\displaystyle t_{n}\circ s^{0}=s^{n}\circ t^{2}_{n+1}$

Let $\Lambda_{+}$ be the subcategory of $\Lambda_{\mathbb{N}}$ generated by
$d^{n}_{i}$ and $s^{n}_{i}$ with only $0\leq i\leq n$. In other words,
$\Lambda_{+}$ is the subcategory of the category $\Lambda$ leaving out the
cyclic morphisms and the last face maps $d^{n}_{n+1}$ at each degree $n\geq
0$.

A pseudo-para-cyclic object in a category $\mathcal{C}$ is a functor
$\Lambda^{op}_{+}\to\mathcal{C}$. Dually a pseudo-para-cocyclic object in a
category $\mathcal{C}$ is a covariant functor $\Lambda_{+}\to\mathcal{C}$.
Roughly speaking, this object is obtained by dropping the condition that
$d_{n}$ (or $d_{0}$) and $t_{n}$ are $A$-module maps.

A cyclic object in a category $\mathcal{C}$ is defined to be a functor
$X:\Lambda^{op}\to\mathcal{C}$. Morphisms between cyclic objects are natural
transformations. Dually a functor $X:\Lambda\to\mathcal{C}$ is called a
cocyclic object in $\mathcal{C}$.

The category $\Lambda$ is generated by $d^{i}:[n]\to[n+1]$,
$s^{i}:[n]\to[n-1]$ and $t_{n}:[n]\to[n]$ subject to certain relations.
Especially, $\Lambda$ includes $\Delta$ as a subcategory. A cyclic object in a
category $\mathcal{C}$ is a simplicial object $M$ equipped with additional
morphisms $t_{n}:M_{n}\to M_{n}$ subject to the relations

$\displaystyle\begin{cases}d_{i}\circ t_{n}=t_{n-1}\circ d_{i-1}&1\leq i\leq
n\\\ d_{0}\circ t_{n}=d_{n}&\\\ s_{i}\circ t_{n}=t_{n+1}\circ s_{i-1}&1\leq
i\leq n\\\ s_{0}\circ t_{n}=t^{2}_{n+1}\circ s_{n}&\\\
t_{n}^{n+1}=1&\end{cases}$

######  Example 4.3.

For a $k$-algebra $A$ over a commutative ring $k$, the cyclic bar construction
$N_{\bullet}^{cyc}$ is a cyclic $k$-module.

######  Lemma 4.4.

Let $X$ be a simplicial object together with invertible morphisms
$t_{n}:X_{n}\to X_{n}$ for all $n\geq 0$ satisfying the following conditions:

$\displaystyle d_{i}=t^{i}_{n-1}\circ d_{0}\circ t^{-i}_{n}$ $\displaystyle
s_{i}=t^{i}_{n+1}\circ s_{0}\circ t^{-i}_{n}$

Then $X$ is a para-cyclic object.

###### Proof.

For $i>0$, we have the equations

$\displaystyle d_{i}\circ t_{n}$ $\displaystyle=t^{i}_{n-1}\circ d_{0}\circ
t_{n}^{-i}\circ t_{n}$ $\displaystyle=t_{n-1}\circ(t_{n-1}^{i-1})\circ
d_{0}\circ t_{n}^{-(i-1)}$ $\displaystyle=t_{n-1}\circ d_{i-1}$ $\displaystyle
s_{i}\circ t_{n}$ $\displaystyle=t_{n+1}^{i}\circ s_{0}\circ t_{n}^{-i}t_{n}$
$\displaystyle=t_{n+1}\circ t_{n+1}^{i-1}\circ s_{0}\circ t^{-(i-1)}_{n}$
$\displaystyle=t_{n+1}\circ s_{i-1}$

When $i=0$, we have the equations

$\displaystyle d_{0}\circ t_{n}$ $\displaystyle=t^{-n}_{n-1}\circ
t^{n}_{n-1}\circ d_{0}\circ t_{n}^{-n}\circ t^{n}_{n}$
$\displaystyle=d_{n}\circ t_{n}^{n}$ $\displaystyle s_{0}\circ t_{n}$
$\displaystyle=t_{n+1}^{n+2}\circ s_{0}\circ t_{n}$
$\displaystyle=t^{2}_{n+1}\circ t^{n}_{n+1}\circ s_{0}\circ
t_{n}^{-(n+1)}\circ tn$ $\displaystyle=t_{n+1}^{2}\circ s_{n}$

Thus $X$ is a para-cyclic object. ∎

###  4.1. Vanishing of Hopf-cyclic theory

We recall Doi’s results in [Doi2] and [Doi] that determined the form of
projective $A$-modules, e.g., a projective $A$-module with right $H$-comodule
structure is a direct summand of $A\otimes_{k}V$, where $V$ is a right
$H$-comodule. He also considerd a certain algebra map $H\to A$ called a total
integral, and projectivity was related to the existence of total integral as
follows.

  1. (i)

Let $H$ be a Hopf algebra over a field $k$ and $A$ a right $H$-comodule
algebra. If a map $\phi:H\to A$ of right $H$-comodules with $\phi(1)=1$, i.e.,
if there exists total integral, then any right relative $(A,H)$-Hopf module is
an injective $H$-comodule.

  2. (ii)

Furthermore, if the above right $H$-comodule map $\phi:H\to A$ is an algebra
map, then the fundamental theorem for right relative $(A,H)$-Hopf modules is
true.

  3. (iii)

Let $H$ be a Hopf algebra over a commutative ring $R$ and $A$ a right
$H$-comodule algebra. Doi [Doi] gave a sufficient condition such that an
epimorphism of $(A,H)$-Hopf module splits if it splits $A$-linearly. As an
application in the case when $R$ is a field, he got that an $(A,H)$-Hopf
module is finitely generated projective as an $A$-module if and only if it is
a Hopf module direct summand of $A\otimes M$ for some finite dimensional
$H$-comodule $M$.

In our case, Hopf cyclic theory will be considered in the base category
$\mathcal{M}^{H}$. If we consider an $H$-module in $\mathcal{M}^{H}$, it is
automatically compatible with the right $H$-comodule structure, so that it is
contructible by Doi’s result. By fundamental theorem of Hopf modules, any
Hopf-cyclic objects in $\mathcal{M}^{H}$ whose coefficients in a Hopf module
is zero in $\underline{\mathcal{M}^{H}}$.

######  Corollary 4.5.

A Hopf (co)cyclic homology with coefficient in a $H$-module / comodule
vanishes in the stable homotopy category $\underline{\mathcal{M}^{H}}$.

###  4.2. Hopf cyclic for $H$-comodule bialgebra $A$

######  Definition 4.6.

For a bialgebra $B$, a stable $B$-module / comodule is a $B$-module and
$B$-comodule with respect to the structure maps $m_{M}:B\otimes_{k}M\to M$ and
$\rho_{M}:M\to B\otimes_{k}M$, respectively, such that
$m_{M}\circ\rho_{M}:M\to B\otimes_{k}M\to M$ is the identity.

We let $A$ a bialgebra in $\mathcal{M}^{H}$ with respect to $\otimes_{k}$.

######  Definition 4.7.

Let $A$ be a right $H$-comodule bialgebra and $M$ a stable left
$A$-module/comodule. Define $T_{n}(A,M)=A^{\otimes(n+1)}\otimes M$ for $n\geq
0$. An element $a_{0}\otimes a_{1}\otimes\cdots\otimes a_{n}\otimes m$ of
$T_{n}(A,M)$ is denoted by $[a_{0}|a_{1}|\cdots|a_{n}]m$.

Define face, degeneracy, and cyclic operators by

$\displaystyle d_{i}([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=\begin{cases}[a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{n}]m,&i\neq
n\\\ [m^{(-1)}a_{n}a_{0}|a_{1}|\cdots|a_{n-1}]m^{(0)},&i=n\end{cases}$
$\displaystyle s_{i}([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=[a_{0}|a_{1}|\cdots|a_{i}|1|a_{i+1}|\cdots|a_{n}]m$
$\displaystyle t_{n}([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=[m^{(-1)}a_{n}|a_{0}|\cdots|a_{n-1}]m^{(0)}.$

######  Definition 4.8.

Define a left $H$-comodule structure on $T_{n}(A,M)$ by the diagonal coaction

######  Proposition 4.9.

The above maps define a structure of pseudo-para-cyclic right $H$-comodule on
$T_{\bullet}(A,M)$.

###### Proof.

Note that the structure maps of bialgebra $A$ and $A$-module/comodule $M$ are
all compatible with right $H$-comodule structure, so that $T_{\bullet}(A,M)$
is a right $H$-comodule. Let us verify some nontrivial part of the simplicial
identities. For $i<j$

$\displaystyle(d_{i}\circ d_{j})([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=\begin{cases}d_{i}([a_{0}|a_{1}|\cdots|a_{j}a_{j+1}|\cdots|a_{n}]m),&j<n\\\
d_{i}([m^{(-1)}a_{n}a_{0}|a_{1}|\cdots|a_{n-1}]m^{(0)},&j=n\end{cases}$
$\displaystyle=\begin{cases}[a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{j}a_{j+1}|\cdots|a_{n}]m,&i+1<j<n\\\
[a_{0}|a_{1}|\cdots|a_{i}a_{i+1}a_{i+2}|\cdots|a_{n}]m,&i+1=j<n\\\
[m^{(-1)}a_{n}a_{0}a_{1}|a_{2}|\cdots|a_{n-1}]m^{(0)},&i=0,j=n\\\
[m^{(-1)}a_{n}a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{n-1}]m^{(0)},&1<i+1<j=n\\\
[m^{(-1)}a_{n-1}m^{(-2)}a_{n}a_{0}|a_{1}|\cdots|a_{n-2}]m^{(0)},&i+1=j=n\end{cases}$
$\displaystyle(d_{j-1}\circ d_{i})([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=d_{j-1}([a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{n}]m)$
$\displaystyle=\begin{cases}[a_{0}|a_{1}|\cdots|a_{i}a_{i+1}a_{i+2}|\cdots|a_{n}]m&i+1=j<n\\\
[a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{j}a_{j+1}|\cdots|a_{n}]m&i+1<j<n\\\
[m^{(-1)}a_{n}a_{0}a_{1}|a_{2}|\cdots|a_{n-1}]m^{(0)}&i=0,j=n\\\
[m^{(-1)}a_{n}a_{0}|a_{1}|\cdots|a_{i}a_{i+1}|\cdots|a_{n-1}]m^{(0)},&1<i+1<j=n\\\
[m^{(-1)}a_{n-1}m^{(-2)}a_{n}a_{0}|a_{1}|\cdots|a_{n-2}]m^{(0)},&i=j=n.\end{cases}$

Let us check the relations between $t_{n}$ and $d_{i}$.

$\displaystyle(d_{i}\circ t_{n})([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=d_{i}([m^{(-1)}a_{n}|a_{0}|\cdots|a_{n-1}]m^{(0)})$
$\displaystyle=\begin{cases}[m^{(-1)}a_{n}|a_{0}|\cdots|a_{i}a_{i+1}|\cdots|a_{n-1}]m^{(0)},&i<n\\\
[m^{(-1)}a_{n-1}m^{(-2)}a_{n}|a_{0}|\cdots|a_{n-2}]m^{(0)},&i=n\end{cases}$
$\displaystyle(t_{n-1}\circ d_{i-1})([a_{0}|a_{1}|\cdots|a_{n}]m)$
$\displaystyle=\begin{cases}[m^{(-1)}a_{n}|a_{0}|\cdots|a_{i}a_{i+1}|\cdots|a_{n-1}]m^{(0)},&i<n\\\
[m^{(-1)}m^{(-2)}a_{n-1}a_{n}|a_{0}|\cdots|a_{n-2}]m^{(0)},&i=n\end{cases}$

Here, $m^{(-1)}m^{(-2)}a_{n-1}a_{n}$ may not be equal to
$m^{(-1)}a_{n-1}m^{(-2)}a_{n}$ but at least we checked that $T_{\bullet}(A,M)$
is pseudo-para-cyclic.

For a left $H$-comodule structure on $T_{\bullet}(A,M)$, it can be verified
since $d_{i}$, $s_{i}$ and $t_{n}$ are $H$-colinear maps. ∎

The reasons why the above pseudo-para-cyclic $T_{\bullet}(A,M)$ is not para-
cyclic are that $d_{n}$ does not commute with $t_{n}$ and that the map $t_{n}$
is not invertible.

Note that an $H$-module algebra $A$ means an $H$-module and $k$-algebra with
$H$-linear unit and multiplication. This is equivalent to original definition
of $H$-module algebra that is an $H$-module and $k$-algebra with compatibility
[Mont, Section 4.1]. The same is true for $H$-comodule algebra, and also
$H$-(co)module bialgebra. However, for an $H$-(co)module $k$-Hopf algebra, the
antipode would be a $k$-linear map.

######  Lemma 4.10.

If $A$ is a right $H$-comodule bialgebra which is also $k$-Hopf algebra with
bijective $H$-colinear antipode $S$, then the map $t_{n}$ is an $H$-colinear
isomorphism. The inverse map is given by
$t^{-1}_{n}([a_{1}|a_{2}|\cdots|a_{n}]m)=([a_{1}|a_{2}|\cdots|a_{n}|S(m^{(-1)})a_{0}]m^{(0)})$.

Moreover, we have $t_{n}^{n+1}=id$ as $H$-comodule maps.

###### Proof.

The first part follows from Lemma 4.4.

Since $M$ is stable and by using Lemma 2.1, we obtain the equation
$t^{n+1}_{n}([a_{0}|a_{1}|\cdots|a_{n}]m)=([(a_{0}|a_{1}|\cdots|a_{n})]\epsilon(m^{(-1)})m^{(0)})=[a_{0}|a_{1}|\cdots|a_{n}]m$.
Therefore, we have $t_{n}^{n+1}=id$. ∎

######  Definition 4.11 (cf.[Kay], p347).

Let $H$, $A$, and $M$ be as above. Define $Q^{A}_{\bullet}(A,M)$ by the
largest pseudo-para-cyclic subcomodule of $T_{n}(A,M)$ which is a cyclic right
$H$-comodule.

In [Kay], $Q^{A}_{\bullet}(A,M)$ is called the co-approximation of
$T_{n}(A,M)$. Note that $Q^{A}_{n}(A,M)$ is also a para-cyclic $H$-comodule,
and actually a cyclic $H$-comodule.

We use the universality of Kan extension. We recall that $Q^{A}_{n}(A,M)$ is
the right Kan extension.

Assume that $A$ is a right $H$-comodule bialgebra which is also $k$-Hopf
algebra with bijective $H$-colinear antipode $S$ and that $M$ is stable. We
obtain a cyclic right $H$-comodule $T_{\bullet}(A,M)$ in $\mathcal{M}^{H}$. On
the other hand, we also have a cyclic right $H$-comodule
$Q^{A}_{\bullet}(A,M)$ which is the right Kan extension along
$\Lambda_{+}\to\Lambda$.

######  Proposition 4.12.

There is always a map $T_{\bullet}(A,M)\to Q^{A}_{\bullet}(A,M)$ of right
$H$-comodules in $\mathcal{M}^{H}$.

###### Proof.

It follows from the universality of right Kan extension. ∎

Note that the $H$-coinvariant part of $T_{\bullet}(A,M)$ is an analogy of
Connes and Moscovich’s Hopf-cyclic homology. If we endow $M$ the trivial
$A$-action and the trivial $A$-coaction, from the equations in the proof of
Proposition 4.9, we have the non-Hopf cyclic structure. Therefore, we can
compare Hopf-cyclic modules with non-Hopf cyclic modules as an analogue of the
characteristic map in the paper of Connes and Moscovici as follows.

######  Corollary 4.13.

There is a map $Q^{A}_{\bullet}(A,k)\to Q^{A}_{\bullet}(A,A)$ of right
$H$-comodules in $\mathcal{M}^{H}$ induced from the unit map.

∎

######  Remark 4.14.

We can define a cyclic $k$-module $Q_{\bullet}(A,M)$ by taking $H$-coinvariant
submodule $Q_{\bullet}^{A}(A,M)^{coH}$ of $Q_{\bullet}^{A}(A,M)$. On the other
hand, only object $A^{\otimes n}\otimes M$ in each degree in
$Q^{A}_{\bullet}(A,M)$ admits the diagonal $A$-action. As an expansion of an
idea of taking $k$ in Corollary 4.13, we will take an $A$-coinvariant part of
$M$. Assume that $M$ is a Hopf $A$-module in $\mathcal{M}^{H}$. If we take
$A$-coinvariant part of $M$, then we have $A\otimes M^{coA}\cong M$ by the
fundamental theorem. The isomorphism is $H$-colinear left $A$-module map.
Therefore, we have $A^{\otimes n}\otimes M^{coA}\cong A^{\otimes(n-1)}\otimes
M$ in each degree, and the degree shift of $Q_{\bullet}^{A}(A,M)$ appeared.

If $H$ is a finite dimensional commutative Hopf algebra, we have the
composition of functors

$\textstyle{\mathrm{LMod}_{A}(\mathcal{M}^{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{\mathcal{M}^{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{M}_{H^{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{Ch(H),}$

where $\mathcal{M}_{H^{\ast}}$ is the category of $H^{\ast}$-modules, the
forgetful functor $U$ is the right adjoint and the inclusion $i$ is the
Quillen functor and the fonctor $F=(-)\otimes Tk$ given by tensoring a Tate
resolution of the base field is the left adjoint.

We send the (co)cyclic $H$-comodule under the functor $F$. By [Kay], the
approximation theorem of cocyclic comodule theory can be regarded as a (local)
left Kan extension and the coapproximation as right Kan extension. Therefore,
Hopf-(co)cyclic homology in $\mathcal{M}^{H}$ can be calculated via the
associated Hopf-(co)cyclic theory in $Ch(H)$.

## References


# PTMTorrent: A Dataset for Mining Open-source Pre-trained Model Packages

Wenxin Jiang 1§${}^{\textsuperscript{\href
https://orcid.org/0000-0003-2608-8576}}$ Nicholas
Synovic2§${}^{\textsuperscript{\href https://orcid.org/0000-0003-0413-4594}}$
Purvish Jajal1${}^{\textsuperscript{\href
https://orcid.org/0000-0002-1199-6363}}$ Taylor R.
Schorlemmer1${}^{\textsuperscript{\href
https://orcid.org/0000-0003-2181-5527}}$ Arav
Tewari1${}^{\textsuperscript{\href https://orcid.org/0000-0002-1512-858X}}$
Bhavesh Pareek1${}^{\textsuperscript{\href
https://orcid.org/0000-0002-6885-9810}}$ George K.
Thiruvathukal2${}^{\textsuperscript{\href
https://orcid.org/0000-0002-0452-5571}}$ James C.
Davis1${}^{\textsuperscript{\href https://orcid.org/0000-0003-2495-686X}}$
1Purdue University and 2Loyola University Chicago

###### Abstract

Due to the cost of developing and training deep learning models from scratch,
machine learning engineers have begun to reuse pre-trained models (PTMs) and
fine-tune them for downstream tasks. PTM registries known as “model hubs”
support engineers in distributing and reusing deep learning models. PTM
packages include pre-trained weights, documentation, model architectures,
datasets, and metadata. Mining the information in PTM packages will enable the
discovery of engineering phenomena and tools to support software engineers.
However, accessing this information is difficult — there are many PTM
registries, and both the registries and the individual packages may have rate
limiting for accessing the data.

We present an open-source dataset, PTMTorrent, to facilitate the evaluation
and understanding of PTM packages. This paper describes the creation,
structure, usage, and limitations of the dataset. The dataset includes a
snapshot of 5 model hubs and a total of 15,913 PTM packages. These packages
are represented in a uniform data schema for cross-hub mining. We describe
prior uses of this data and suggest research opportunities for mining using
our dataset.

The PTMTorrent dataset (v1) is available at: https://app.globus.org/file-
manager?origin_id=55e17a6e-9d8f-11ed-a2a2-8383522b48d9&origin_path=%2F%7E%2F.

Our dataset generation tools are available on GitHub:
https://doi.org/10.5281/zenodo.7570357

###### Index Terms:

Open-Source Software, Data Mining, Machine learning, Empirical software
engineering

§§footnotetext: Authors contributed equally.

##  I Introduction

Modern software systems reuse Deep Neural Networks (DNNs) to build intelligent
and adaptive systems [1, 2]. Engineering a DNN from scratch is challenging for
many reasons, including the variation in deep learning libraries [3, 4] and
the high expense of training models [5]. Organizations and developers can
address some of these challenges and reduce the cost and effort associated
with DNN development by reusing _pre-trained DNN models_ (PTMs) [6, 7]. PTMs
are shared via _deep learning model registries_ , which are modeled on
traditional software package registries such as NPM [8]. These PTM packages
include reusable components, such as model architectures, weights, licenses,
and other metadata. Deep learning model registries enable engineers to develop
their models with re-usability in mind [9, 10]. Although PTM reuse is still in
its early stages, the most popular PTMs are downloaded millions of times each
month [11, 12].

As PTM reuse becomes more widespread, the engineering community will benefit
from research into PTM reuse practices, challenges, and tools [12, 11]. By
analogy to traditional software, mining PTM software repositories can help us
understand development trends [13, 14, 15] and usage patterns [16, 17].
However, mining the software repositories associated with PTM packages is
difficult for three reasons related to _data availability_. First, researchers
must look in many places — PTM packages are distributed across many competing
PTM registries [11]. Second, researchers must access the packages — PTMs
include complex DNN models and weights with sizes over 1 TB, and access to
these packages may be hindered by throttling or rate limiting [18]. Third, for
scientific replicability, this large-scale data needs to be hosted long-term.

To enable mining of PTM packages, we share _PTMTorrent_ , the first many-hub
dataset of PTM packages. PTMTorrent contains 15,913 PTMs from 5 different PTM
registries identified in our prior work [11]: Hugging Face [19], Model Zoo
[20], PyTorch Hub [21], ONNX Model Zoo [22], and Modelhub [23]. Our dataset is
hosted on a high-performance storage system (HPSS) maintained by Purdue
University’s Research Computing center. The dataset includes the metadata of
each PTM and the package histories for each GitHub repository. These packages
are represented in a uniform data schema for cross-hub mining. Out dataset
supports many directions for further research, including studies of the PTM
supply chain, PTM package evolution, PTM mining tools, and DNN architectural
trends.

Figure 1:  Data collection and preprocessing workflow for PTMTorrent. We
standardize the PTM metadata by using a data schema, collecting it from PTM
packages and the corresponding GitHub repository.

##  II The PTMTorrent Dataset

###  II-A Data Source

In prior work we mapped the major model hubs and indicated that there exist
open, gated, and commercial hubs [11]. Open and gated hubs tend to be larger
and more widely used because they accept contributions from anyone, and can be
accessed by anyone. Commercial hubs are offered by individual companies to
share vetted models with their clients. Due to the limited access to
commercial model hubs, we only provide a snapshot of the open hubs (Hugging
Face) and some of the gated hubs (Model Zoo, PyTorch Hub, Modelhub, and ONNX
Model Zoo).

The PTMTorrent dataset contains the repository histories of 15,913 PTM
packages available as of January 2023. They are provided as complete git
clones, resulting in a compressed footprint of ~61TB. Each PTM package was
cloned at its most recent version, including the model card, architecture,
weights, and other information provided by the maintainers (e.g., training
configuration, hyper-parameters).

Figure 1 indicates the collection and preprocessing approaches of our dataset.

We collected PTM packages from all open and gated model hubs per Jiang et al.
[11], excluding TensorFlow Hub because it does not support version control
features. We downloaded all PTM packages from Model Zoo, PyTorch Hub, ONNX
Model Zoo, and Modelhub. Due to the size of Hugging Face, we downloaded only
the top 10% most-downloaded PTMs.111Although we collected a small amount of
the full Hugging Face registry, this “top 10%” snapshot includes all Hugging
Face PTMs with over 30 downloads. Overall, our dataset contains 15,913
packages from 5 PTM registries, distributed as described in Table I.

TABLE I:  Details about the PTMTorrent content for each of the 5 model registries we collected.  Name | # Models | Data Size
---|---|---
Hugging Face [24] | 12,401 | 61TB
Model Zoo [20] | 3,245 | 115GB
PyTorch Hub [21] | 49 | 1.5GB
ONNX Model Zoo [22] | 185 | 441MB
Modelhub [23] | 33 | 721MB
PTMTorrent | 15,913 | ~61TB

###  II-B Data Schema

Figure 2 shows the overview of the data schema we used to standardize the
dataset. We extracted common entities into a general PTM schema. Each PTM
registry has some custom features, so we customized the schema slightly for
each model registry. The full data schema is encoded following the JSON Schema
format,222See https://json-schema.org/draft/2020-12/json-schema-core.html and
is available in the GitHub repository associated with this project.

###  II-C Data Storage

As shown in Table I, the entire PTMTorrent dataset (v1) needs ~61TB of storage
space. A cost-effective storage system is required to serve this dataset.
Commercial services are cost-prohibitive at this scale, e.g., we estimated a
monthly cost of over $1000 to store and serve this dataset from Amazon Web
Services. We opted instead for an internal resource available at Purdue
University: the Purdue Fortress tape-based hierarchical storage system.333For
more information about Fortress, see
https://www.rcac.purdue.edu/knowledge/fortress/overview. Our GitHub repository
includes a guide on how to access data stored in Globus. To facilitate
external distribution of our dataset, we offer a Globus share [25] named
PTMTorrent.

###  II-D Maintainability and Extensibility

The sizes of PTM registries are increasing rapidly. For example, Hugging Face
provided 63,182 public PTM packages on August 2022, and now it provides
124,427 packages. We believe the number of open-source PTM packages will
increase in the foreseeable future. Therefore, maintainability and
extensibility are two important properties of PTMTorrent.

The PTMTorrent dataset is designed to be maintainable by re-running our
scripts to gather any additional changes that may have been made to the PTM
registry since its last collection. Expect a biannual update.

For extensibility, new model hubs can be incorporated into the dataset. We
follow an open-source model and will review Issue and Pull Request
contributions on GitHub. The PTMTorrent data schema captures most elements of
a PTM package, though some specialization is needed. The downloaders for a new
model hub can be developed based on the examples of the already-supported
model hubs in our open-source data collection tools. An extender must provide
2-4 scripts following the pattern we used on the other hubs.

Figure 2:  An overview of PTMTorrent’s data schema. Each model hub shares a
general schema (grey boxes), with hub-specific data stored in customized
schema (colored boxes). The full schema is available in JSON in the dataset
generation repository.

##  III Originality and Relevance

Prior works have extracted information from open-source projects to a dataset
and provide it for future analysis, such as GHTorrent [26], TravisTorrent
[27], and RTPTorrent [28]. These datasets can be used for further mining
software repository researches and help the community better understand open-
source software projects [29, 14, 15, 30].

Similarly, our dataset captures the open-source PTM packages from many model
hubs. The structure of our dataset imitates prior datasets that were focused
on traditional open-source software [26, 28]. Compared to prior work,
PTMTorrent focuses on PTM packages, including the metadata, architecture,
dataset, and performance metrics. Our dataset provides a way for users to
efficiently download and access large amount of data on PTM packages and
relevant repositories.

##  IV Usage Examples

###  IV-A Prior Usage in the Literature

In prior work, we used a part of PTMTorrent (the Hugging Face part) to measure
potential risks in the Hugging Face model registry [12]. We measured the
dependencies of model architecture and datasets, PTM documentation, and GPG
commit signing in Hugging Face PTMs. Our analysis identified potential
software supply chain concerns facing PTM reusers, including spoofing,
tampering, and repudiation.

In prior work, we also used metadata from Hugging Face to measure model
discrepancies and maintainers’ reach [11]. Our analysis showed that existing
defenses appear insufficient for ensuring the security of PTMs.

The PTMTorrent dataset provides more opportunities for mining PTM data by
covering more PTM registries and providing greater structure. We believe that
these large amount of PTM packages can be analyzed in similar ways as
traditional packages [31, 32, 33].

###  IV-B Applying an Existing MSR Tool

Since PTMTorrent consists of git repositories, it is possible to use existing
software repository mining tools on the PTM packages. Our GitHub repository
includes a demonstration of this. We used our PRIME tool [34] to analyze
software process metrics on a subset of the dataset.

##  V Limitations

PTMTorrent is incomplete. It is biased towards the top 10% most-downloaded
PTMs in HuggingFace (though this is almost all PTMs with any downloads, cf.
section II-A). There are other model hubs, such as Papers With Code [35],
PINTO Model Zoo [36], and Jetson Zoo [37]. Beyond these, there are other deep
learning-specific registries that lack versioning or packaging features. The
initial PTMTorrent release provides PTMs from model hubs that are similar to
traditional software packages, as defined by Jiang et al. [11]. We leave their
capture for future work.

Another limitation of our data is the non-standardized granularity. The
current version of PTMTorrent lacks detailed metadata and does not provide
uniform information, e.g., datasets, model architectures. During the data
collection, we notice that the provided information from PTM registries can be
quite different and we use customized data schemas for each PTM registry. As a
result, it is difficult to analyze all the PTM packages under the same
umbrella when using our dataset.

For example, Hugging Face provides detailed documentation and structured
metadata, as well as relevant configuration files for each PTM, while ONNX
Model Zoo provides PTM metadata through unstructured Markdown files. Thereby
making metadata extraction challenging. To mitigate this problem, we have a
parent data schema for all the PTM registries and child schemas for each
specific registry that represents their custom data.

##  VI Future Work

In addition to the risk measurements presented by Jiang et al. [12, 11], the
PTMTorrent dataset can be used in different ways. We suggest three research
directions: PTM supply chain analysis, tools for PTM reuse, and mining tool
development.

###  VI-A Supporting Future PTM Supply Chain Analysis

Prior work has focused on understanding the characteristics of package
registries and their supply chains. Zimmermann et al. analyzed the metadata of
NPM packages and identified the potential threats on downstream users [31].
Ladisa et al. proposed an attack taxonomy on open-source supply chains,
including code contributions to package distribution [38]. Similar studies are
also important in PTM supply chain alongside studies focused on PTM-specific
aspects. We propose that future studies can analyze PTMTorrent dataset to
understand the characteristics of the PTM supply chain, including the
dependency analysis [31], vulnerabilities [39], and code knowledge transfer
[40]. Recent advances in AI, such as ChatGPT [41], that clearly build upon
composing various PTMs strongly suggest that being able to study how PTMs and
are composed to build more complex systems (a trait shared with traditional
software) will become more important. We hope our dataset will aid in
performing such analyses.

###  VI-B Expanding PTM Model Registry Analysis

Researchers can extract more information from these model registries by
reusing or developing software metrics for PTM packages, including provenance,
reproducibility, and portability [12]. PTM registries can help us develop
comprehensive attributes and provide these details in the PTM dashboard,
similar to the measured attributes from NPM [42] and PyPi [43].

Our prior study has indicated that engineers can have trouble finding the best
PTM that matches their requirements, and it can therefore be hard to identify
the portability and reproducibility of the open-source PTMs [12]. Montes et
al. shows that there exist notable discrepancies among different model zoos
[44]. With more detailed and comprehensive metadata provided for each PTM and
the corresponding usage patterns on downstream tasks, it will be possible to
develop a recommender system to help engineers find the right set of PTMs for
a given application and requirements [45]. PTM Registry contributors can
develop sophisticated visualization tools—with the aid of our dataset—that
help PTM users understand the strengths and limitations of each model.

###  VI-C Furthering the State of Mining Tool Development

Given the lack of standardization among different PTM registries (section V),
it was challenging to standardize all the metadata. PTMTorrent may not have
everything needed for every type of analysis. Researchers can augment the
dataset during the data collection and processing stage for other subsequent
mining needs. We have included the relevant GitHub pages of each PTM in out
dataset, and therefore the extraction can be done either based on the provided
documentation from PTM registries [46] or source code from the underlying
repositories [47].

##  VII Acknowledgements

This work was supported by gifts from Google and Cisco and by NSF awards
#2107230, #2229703, #2107020, and #2104319.

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