# Comodules and contramodules over coalgebras
associated with locally finite categories

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

We explain how to attach a coalgebra $\mathcal{C}$ over a field $k$ to a small
$k$-linear category $\mathsf{E}$ satisfying suitable finiteness conditions. In
this context, we study full-and-faithfulness of the contramodule forgetful
functor, and describe explicitly the categories of locally finite left
$\mathcal{C}$-comodules and left $\mathcal{C}$-contramodules as certain full
subcategories of the category of left $\mathsf{E}$-modules.

###### Contents

  1. 1 Preliminaries on Coalgebras, Comodules, and Contramodules
  2. 2 Locally Finite Categories, Coalgebras, and Comodules
  3. 3 Full-and-Faithfulness of Contramodule Forgetful Functor
  4. 4 Locally Finite Contramodules
  5. 5 Vector Space Duality
  6. 6 Closedness Under Extensions
  7. 7 Upper and Lower Locally Finite Categories

## Introduction

Let $\mathsf{E}$ be a small preadditive category. Then _left
$\mathsf{E}$-modules_ are defined as covariant additive functors
$\mathsf{E}\longrightarrow\mathsf{Ab}$, while _right $\mathsf{E}$-modules_ are
the contravariant additive functors
$\mathsf{E}^{\mathsf{op}}\longrightarrow\mathsf{Ab}$ (where $\mathsf{Ab}$ is
the category of abelian groups). The category of left $\mathsf{E}$-modules
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ is a Grothendieck abelian category
with a set of small projective generators indexed by the objects of the
category $\mathsf{E}$. However, when $\mathsf{E}$ has infinitely many nonzero
objects, the category $\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ does _not_
admit a single small projective generator. So the category
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ is similar to, but different from
the usual categories of modules over (associative, unital) rings. One speaks
of $\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ as the category of modules over
the “ring with several objects” $\mathsf{E}$, or as the category of modules
over a “locally unital” (but nonunital) ring
$R_{\mathsf{E}}=\bigoplus_{x,y\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,y)$
associated with $\mathsf{E}$.

In the language of topological algebra, one can always describe the abelian
category $\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ as the category of
discrete modules, or if one prefers, the category of contramodules over the
topological ring $\mathfrak{R}_{\mathsf{E}}$ associated with $\mathsf{E}$.
There are two versions of $\mathfrak{R}_{\mathsf{E}}$, the left and the right
one. Let us make out choice and put
$\mathfrak{R}_{\mathsf{E}}=\prod_{y\in\mathsf{E}}\bigoplus_{x\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,y)$.
Then $\mathfrak{R}_{\mathsf{E}}$ is the ring of infinite, row-finite matrices
whose entries are morphisms in $\mathsf{E}$. We endow
$\mathfrak{R}_{\mathsf{E}}$ with the obvious topology in which the rows
$\bigoplus_{x\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,y)$ are discrete
abelian groups, while the whole ring $\mathfrak{R}_{\mathsf{E}}$ carries the
product topology of the discrete topologies of the rows.

Then one can easily see that the category of discrete right
$\mathfrak{R}_{\mathsf{E}}$-modules is naturally equivalent to the category of
right $\mathsf{E}$-modules ${\operatorname{\mathsf{Mod--}}}\mathsf{E}$. On the
other hand, applying the result of [14, Corollary 6.3 and Theorem 7.1] to the
projective generator
$\bigoplus_{x\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,{-})$ of the
category of left $\mathsf{E}$-modules
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ shows that
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ is naturally equivalent to the
category of left $\mathfrak{R}_{\mathsf{E}}$-contramodules.

In this paper, we take a different approach and assign (discrete) _coalgebras_
over the field $k$ to some $k$-linear categories. For any coalgebra
$\mathcal{C}$ over $k$, the dual vector space $\mathcal{C}^{*}$ has a natural
structure of topological algebra over $k$; but most topological algebras do
not arise in this way. Furthermore, the dual vector space to an infinite-
dimensional $k$-algebra usually does not have a coalgebra structure. So one
needs to impose some restrictions on a $k$-linear category $\mathsf{E}$ in
order to produce a coalgebra from it. We say that $\mathsf{E}$ is _locally
finite_ if two conditions are satisfied: the $k$-vector space
$\operatorname{Hom}_{\mathsf{E}}(x,y)$ is finite-dimensional for all objects
$x$, $y\in\mathsf{E}$, and the set of all objects $z$ for which there exist
finite chains of composable nonzero morphisms
$x\longrightarrow\dotsb\longrightarrow z$ and
$z\longrightarrow\dotsb\longrightarrow y$ is finite for all $x$,
$y\in\mathsf{E}$. For a locally finite $k$-linear category $\mathsf{E}$, we
construct a coassociative, counital coalgebra $\mathcal{C}_{\mathsf{E}}$. Let
us warn the reader that the dual topological algebra
$\mathcal{C}_{\mathsf{E}}^{*}$ is still _not_ isomorphic to the topological
algebra $\mathfrak{R}_{\mathsf{E}}$ in most cases.

Given a $k$-linear category $\mathsf{E}$, additive functors
$\mathsf{E}\longrightarrow\mathsf{Ab}$ are the same things as $k$-linear
functors $\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--Vect}}}$ (where
$k{\operatorname{\mathsf{--Vect}}}$ is the category of $k$-vector spaces). So
one can interpret left $\mathsf{E}$-modules as covariant $k$-linear functors
$\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--Vect}}}$. We explain that
to every left $\mathcal{C}_{\mathsf{E}}$-comodule, as well as to every left
$\mathcal{C}_{\mathsf{E}}$-contramodule, one can naturally assign a left
$\mathsf{E}$-module. So there are forgetful functors
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
and
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$.
The former functor is always fully faithful; so it is perhaps better to call
it the comodule _inclusion_ functor. Using the technique originally developed
for conilpotent coalgebras in the paper [10, Theorem 2.1], we spell out a
sufficient condition on the category $\mathsf{E}$ (which we call _left strict
local finiteness_) implying that the contramodule forgetful functor is fully
faithful as well. This is the main result of a longish Section 3; see Theorem
3.7.

A left $\mathsf{E}$-module $N$ is said to be _locally finite_ if the
corresponding $k$-linear functor takes values in the category of finite-
dimensional vector spaces $k{\operatorname{\mathsf{--vect}}}$, that is
$N\colon\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--vect}}}$. A left
$\mathcal{C}_{\mathsf{E}}$-comodule or a left
$\mathcal{C}_{\mathsf{E}}$-contramodule is called _locally finite_ if the
corresponding left $\mathsf{E}$-module is locally finite. The main results of
this paper (Theorems 2.7 and 4.2) describe the full subcategories of locally
finite left $\mathcal{C}_{\mathsf{E}}$-comodules and locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodules inside the ambient abelian category
of left $\mathsf{E}$-modules. Specifically, a locally finite left
$\mathsf{E}$-module $M$ arises from a left $\mathcal{C}_{\mathsf{E}}$-comodule
if and only if, for every object $x\in\mathsf{E}$, the set of all objects
$y\in\mathsf{E}$ with nonzero action map
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\longrightarrow M(y)$ is
finite. A locally finite left $\mathsf{E}$-module $P$ arises from a left
$\mathcal{C}_{\mathsf{E}}$-contramodule if and only if, for every object
$y\in\mathsf{E}$, the set of all objects $x\in\mathsf{E}$ with nonzero action
map $\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\longrightarrow P(y)$
is finite. The comodule case is rather easy; the proof of the contramodule
version is much more involved and depends on the assumption of left strict
local finiteness of the $k$-linear category $\mathsf{E}$.

It is clear that, for any $k$-linear category $\mathsf{E}$, the vector space
dualization functor $N\longmapsto P=N^{*}$,
$P(x)=N(x)^{*}=\operatorname{Hom}_{k}(N(x),k)$ for all $x\in\mathsf{E}$,
provides an anti-equivalence between the abelian categories of locally finite
right and left $\mathsf{E}$-modules. It follows from the main results of this
paper that the vector space dualization functor
$\mathcal{N}\longmapsto\mathfrak{P}=\mathcal{N}^{*}=\operatorname{Hom}_{k}(\mathcal{N},k)$
also provides an anti-equivalence between the abelian categories of locally
finite right $\mathcal{C}_{\mathsf{E}}$-comodules and locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodules for a left strictly locally finite
$k$-linear category $\mathsf{E}$. This observation is made in Corollary 5.3.

Given a coalgebra $\mathcal{C}$ over a field $k$, is the essential image of
the comodule inclusion functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$
closed under extensions? More generally, given a dense subring
$R\subset\mathcal{C}^{*}$, is the essential image of the inclusion functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ closed under extensions? There is a vast
body of literature on this question, including such papers as [15, 16, 5, 2,
1, 18, 4]. In particular, for a conilpotent coalgebra $\mathcal{C}$ (called
“pointed irreducible” in the traditional terminology of [17, 6]), the full
subcategory $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ is closed under
extensions in $\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$ if and only if
the full subcategory ${\operatorname{\mathsf{Comod--}}}\mathcal{C}$ is closed
under extensions in ${\operatorname{\mathsf{Mod--}}}\mathcal{C}^{*}$, and if
and only if the coalgebra $\mathcal{C}$ is finitely cogenerated [15, Corollary
2.4 and Section 2.5], [16, Theorem 4.6], [5, Corollary 21], [2, Theorem 2.8],
[1, Proposition 3.13 and Corollary 3.14], [4, Lemma 1.2 and Theorem 4.8]. The
same property characterizes the conilpotent coalgebras $\mathcal{C}$ for which
the contramodule forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$
is fully faithful [10, Theorem 1.2], [13, Example 7.2].

In Section 6 of this paper, we show that the full subcategory
${\operatorname{\mathsf{Comod--}}}\mathcal{C}_{\mathsf{E}}$ is closed under
extensions in ${\operatorname{\mathsf{Mod--}}}\mathsf{E}$ for any left
strictly locally finite $k$-linear category $\mathsf{E}$. This observation is
not really new, but it complements the main results of this paper nicely; so
we include it for the benefit of the reader. Under the same assumptions, the
full subcategory of _locally finite_ left
$\mathcal{C}_{\mathsf{E}}$-contramodules is closed under extensions in the
category of (locally finite or arbitrary) left $\mathsf{E}$-modules. Based on
the discussion of the conilpotent case in [13] (see [13, Theorems 6.2 and 7.1
for $n=2$]) we expect that the left strict local finiteness assumption is
_not_ sufficient to make the whole category of (locally infinite) left
$\mathcal{C}_{\mathsf{E}}$-contramodules closed under extensions in
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Mod}}}$.

In the final Section 7, we discuss _upper finite_ $k$-linear categories
$\mathsf{E}$, for which the abelian category of left
$\mathcal{C}_{\mathsf{E}}$-comodules is always equivalent to the abelian
category of left $\mathsf{E}$-modules. Dual-analogously, for a _lower finite_
$k$-linear category $\mathsf{E}$, the abelian category of left
$\mathcal{C}_{\mathsf{E}}$-contramodules is equivalent to the abelian category
of left $\mathsf{E}$-modules. For right comodules and right contramodules, the
roles of the “upper” and “lower” are reversed. These special cases are much
simpler than those of arbitrary locally finite $k$-linear categories or left
strictly locally finite $k$-linear categories $\mathsf{E}$ considered in the
main results of the paper.

### Acknowledgement

I am grateful to Catharina Stroppel for a kind invitation to Bonn and a
stimulating discussion which prompted me to think about comodules and
contramodules over $k$-linear categories. I also wish to thank Michal Hrbek
and Jan Št’ovíček for very helpful conversations. The author is supported by
the GAČR project 23-05148S and the Czech Academy of Sciences (RVO 67985840).

##  1\. Preliminaries on Coalgebras, Comodules, and Contramodules

All _coalgebras_ in this paper are coassociative and, unless otherwise
mentioned, counital. All coalgebra homomorphisms are presumed to agree with
the counits, and all co/contramodules are co/contraunital by default. The
books [17, 6] are classical reference sources on coalgebras. An introductory
discussion can be found in the present author’s survey [12, Section 3].

A (coassociative, counital) _coalgebra_ $\mathcal{C}$ over a field $k$ is a
$k$-vector space endowed with $k$-linear maps of _comultiplication_
$\mu\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{C}$ and
_counit_ $\epsilon\colon\mathcal{C}\longrightarrow k$ satisfying the usual
coassociativity and counitality axioms. The latter are expressed by the
diagrams

$\mathcal{C}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{C}\rightrightarrows\mathcal{C}\otimes_{k}\mathcal{C}\otimes_{k}\mathcal{C}$

and

$\mathcal{C}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{C}\rightrightarrows\mathcal{C}.$

Here, on the former diagram, the two compositions of the comultiplication map
$\mu\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{C}$ with
the two maps
$\mathcal{C}\otimes_{k}\mathcal{C}\rightrightarrows\mathcal{C}\otimes_{k}\mathcal{C}\otimes_{k}\mathcal{C}$
induced by the comultiplication map must be equal to each other. On the latter
diagram, both the compositions of the comultiplication map with the two maps
$\mathcal{C}\otimes_{k}\mathcal{C}\rightrightarrows\mathcal{C}$ induced by the
counit map $\epsilon\colon\mathcal{C}\longrightarrow k$ must be equal to the
identity map ${\mathrm{id}}_{\mathcal{C}}$.

A _left comodule_ $\mathcal{M}$ over $\mathcal{C}$ is a $k$-vector space
endowed with a $k$-linear map of _left $\mathcal{C}$-coaction_
$\nu\colon\mathcal{M}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{M}$
satisfying the coassociativity and counitality axioms. The latter are
expressed by the diagrams

$\mathcal{M}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{M}\rightrightarrows\mathcal{C}\otimes_{k}\mathcal{C}\otimes_{k}\mathcal{M}$

and

$\mathcal{M}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{M}\longrightarrow\mathcal{M}.$

Here, on the former diagram, the two compositions of the coaction map
$\nu\colon\mathcal{M}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{M}$ with
the two maps
$\mathcal{M}\otimes_{k}\mathcal{C}\rightrightarrows\mathcal{C}\otimes_{k}\mathcal{C}\otimes_{k}\mathcal{M}$
induced by the coaction and comultiplication maps $\nu$ and $\mu$ must be
equal to each other. On the latter diagram, the composition of the coaction
map with the map $\mathcal{C}\otimes_{k}\mathcal{M}\longrightarrow\mathcal{M}$
induced by the counit map must be equal to the identity map
${\mathrm{id}}_{\mathcal{M}}$. A _right comodule_ $\mathcal{N}$ over
$\mathcal{C}$ is a $k$-vector space endowed with a _right coaction_ map
$\nu\colon\mathcal{N}\longrightarrow\mathcal{N}\otimes_{k}\mathcal{C}$
satisfying the similar coassociativity and counitality axioms.

The notion of a contramodule over a coalgebra goes back to Eilenberg and Moore
[3, Section III.5]. For more recent references, see [7, Sections 0.2.4 or
3.1.1], [9, Sections 1.1–1.6], or [12, Section 8].

A _left contramodule_ $\mathfrak{P}$ over $\mathcal{C}$ is a $k$-vector space
endowed with a $k$-linear map of _left $\mathcal{C}$-contraaction_
$\pi\colon\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\longrightarrow\mathfrak{P}$
satisfying the _contraassociativity_ and _contraunitality_ axioms. The latter
are expressed by the diagrams

(1)
$\operatorname{Hom}_{k}(\mathcal{C},\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P}))\simeq\operatorname{Hom}_{k}(\mathcal{C}\otimes_{k}\mathcal{C},\>\mathfrak{P})\,\rightrightarrows\,\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\longrightarrow\mathfrak{P}$

and

(2)
$\mathfrak{P}\longrightarrow\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\longrightarrow\mathfrak{P}.$

Here, on the contraassociativity diagram (1), the map
$\operatorname{Hom}_{k}(\mu,\mathfrak{P})\colon\operatorname{Hom}_{k}(\mathcal{C}\otimes_{k}\mathcal{C},\>\mathfrak{P})\longrightarrow\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})$
is induced by the comultiplication map
$\mu\colon\mathcal{C}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{C}$, while
the map
$\operatorname{Hom}_{k}(\mathcal{C},\pi)\colon\operatorname{Hom}_{k}(\mathcal{C},\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P}))\longrightarrow\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})$
is induced by the contraaction map
$\pi\colon\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\longrightarrow\mathfrak{P}$.
The identification
$\operatorname{Hom}_{k}(\mathcal{C},\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P}))\simeq\operatorname{Hom}_{k}(\mathcal{C}\otimes_{k}\mathcal{C},\>\mathfrak{P})$
is obtained as a particular case of the natural isomorphism
$\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,W))\simeq\operatorname{Hom}_{k}(U\otimes_{k}V,\>W)$,
which holds for all $k$-vector spaces $U$, $V$, and $W$. The two compositions
of the resulting two parallel maps with the contraaction map $\pi$ must be
equal to each other.

On the contraunitality diagram (2), the composition of the map
$\operatorname{Hom}_{k}(\epsilon,\mathfrak{P})\colon\mathfrak{P}\longrightarrow\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})$
induced by the counit map $\epsilon\colon\mathcal{C}\longrightarrow k$ with
the contraaction map
$\pi\colon\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\longrightarrow\mathfrak{P}$
must be equal to the identity map ${\mathrm{id}}_{\mathfrak{P}}$. The
definition of a _right contramodule_ $\mathfrak{Q}$ over $\mathcal{C}$ is
similar, with the only difference that the identification
$\operatorname{Hom}_{k}(\mathcal{C},\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{Q}))\simeq\operatorname{Hom}_{k}(\mathcal{C}\otimes_{k}\mathcal{C},\>\mathfrak{Q})$
arising as a particular case of the natural isomorphism
$\operatorname{Hom}_{k}(V,\operatorname{Hom}_{k}(U,W))\simeq\operatorname{Hom}_{k}(V\otimes_{k}U,\>W)$
is used.

For any right $\mathcal{C}$-comodule $\mathcal{N}$ and any $k$-vector space
$V$, the vector space $\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{N},V)$ has
a natural structure of left $\mathcal{C}$-contramodule. The contraaction map

$\pi_{\mathfrak{P}}=\operatorname{Hom}_{k}(\nu_{\mathcal{N}},V)\colon\\\
\operatorname{Hom}_{k}(\mathcal{C},\operatorname{Hom}_{k}(\mathcal{N},V))\simeq\operatorname{Hom}_{k}(\mathcal{N}\otimes_{k}\mathcal{C},\>V)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{N},V)$

is obtained by applying the contravariant functor
$\operatorname{Hom}_{k}({-},V)$ to the coaction map
$\nu_{\mathcal{N}}\colon\mathcal{N}\longrightarrow\mathcal{N}\otimes_{k}\mathcal{C}$.

The categories $\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ and
${\operatorname{\mathsf{Comod--}}}\mathcal{C}$ of left and right
$\mathcal{C}$-comodules are Grothendieck abelian categories; so they have
enough injective objects. A left $\mathcal{C}$-comodule is injective if and
only if it is a direct summand of a _cofree_ left $\mathcal{C}$-comodule, i.
e., a left $\mathcal{C}$-comodule of the form $\mathcal{C}\otimes_{k}V$, where
$V\in k{\operatorname{\mathsf{--Vect}}}$. For any left $\mathcal{C}$-comodule
$\mathcal{L}$, the vector space of $\mathcal{C}$-comodule morphisms
$\mathcal{L}\longrightarrow\mathcal{C}\otimes_{k}V$ is naturally isomorphic to
the vector space of $k$-linear maps $\mathcal{L}\longrightarrow V$.

The categories $\mathcal{C}{\operatorname{\mathsf{--Contra}}}$ and
${\operatorname{\mathsf{Contra--}}}\mathcal{C}$ of left and right
$\mathcal{C}$-contramodules are locally presentable abelian categories with
enough projective objects. A left $\mathcal{C}$-contramodule is projective if
and only if it is a direct summand of a _free_ left
$\mathcal{C}$-contramodule, i. e., a left $\mathcal{C}$-contramodule of the
form $\operatorname{Hom}_{k}(\mathcal{C},V)$, where $V\in
k{\operatorname{\mathsf{--Vect}}}$. For any left $\mathcal{C}$-contramodule
$\mathfrak{Q}$, the vector space of $\mathcal{C}$-contramodule morphisms
$\operatorname{Hom}_{k}(\mathcal{C},V)\longrightarrow\mathfrak{Q}$ is
naturally isomorphic to the vector space of $k$-linear maps
$V\longrightarrow\mathfrak{Q}$.

The forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ is exact and faithful. It preserves
infinite coproducts, but _not_ infinite products (except when $\mathcal{C}$ is
finite-dimensional). The forgetful functor
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ is exact and faithful as well. It preserves
infinite products, but _not_ infinite coproducts.

Occassionally we will consider coalgebras without counit, defined in the
obvious way (drop the counit map $\epsilon$ and the counitality axiom from the
definitions above). The definitions of comodules and contramodules over a
noncounital coalgebra are obtained by dropping the co/contraunitality axiom in
the definitions of co/contramodules given above.

A noncounital coalgebra $\mathcal{D}$ over a field $k$ is said to be
_conilpotent_ if for every element $d\in D$ there exists an integer $n\geq 1$
such that the iterated comultiplication map
$\mu^{(n)}\colon\mathcal{D}\longrightarrow\mathcal{D}^{\otimes_{k}\,n+1}$
annihilates $d$. The following result is a version of Nakayama lemma for
comodules and contramodules.

######  Lemma 1.1.

Let $\mathcal{D}$ be a conilpotent noncounital coalgebra. Then

(a) for any nonzero left $\mathcal{D}$-comodule $\mathcal{M}$, the coaction
map $\nu\colon\mathcal{M}\longrightarrow\mathcal{D}\otimes_{k}\mathcal{M}$ is
_not_ injective;

(b) for any nonzero left $\mathcal{D}$-contramodule $\mathfrak{P}$, the
contraaction map
$\pi\colon\operatorname{Hom}_{k}(\mathcal{D},\mathfrak{P})\allowbreak\longrightarrow\mathfrak{P}$
is _not_ surjective.

###### Proof.

Part (a) can be found in [13, Lemma 2.1(a)]. For part (b), which is more
complicated, see [7, Lemma A.2.1]; and for a discussion of generalizations,
[12, Lemma 8.1] and [9, Lemmas 2.1 and 3.22]. ∎

The next lemma concerns infinite families of coalgebras. As such, it has no
direct analogue for abstract algebras or rings (but only for _topological_
rings with their topological products).

######  Lemma 1.2.

Let $(\mathcal{C}_{\xi})_{\xi\in\Xi}$ be a family of coalgebras and
$\mathcal{C}=\bigoplus_{\xi\in\Xi}\mathcal{C}_{\xi}$ be their direct sum,
endowed with its natural coalgebra structure.

(a) The category of left $\mathcal{C}$-comodules is equivalent to the
Cartesian product of the categories of left $\mathcal{C}_{\xi}$-comodules over
the indices $\xi\in\Xi$,

$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\,\simeq\,\mathop{\text{\huge$\times$}}\nolimits_{\xi\in\Xi}\,\mathcal{C}_{\xi}{\operatorname{\mathsf{--Comod}}}.$

Any left $\mathcal{C}$-comodule $\mathcal{M}$ decomposes naturally into a
direct sum of left $\mathcal{C}_{\xi}$-comodules $\mathcal{M}_{x}$ over
$\xi\in\Xi$,

$\mathcal{M}\simeq\bigoplus\nolimits_{\xi\in\Xi}\mathcal{M}_{\xi}.$

(b) The category of left $\mathcal{C}$-contramodules is equivalent to the
Cartesian product of the categories of left $\mathcal{C}_{\xi}$-contramodules
over the indices $\xi\in\Xi$,

$\mathcal{C}{\operatorname{\mathsf{--Contra}}}\,\simeq\,\mathop{\text{\huge$\times$}}\nolimits_{\xi\in\Xi}\,\mathcal{C}_{\xi}{\operatorname{\mathsf{--Contra}}}.$

Any left $\mathcal{C}$-contramodule $\mathfrak{P}$ decomposes naturally into a
direct product of left $\mathcal{C}_{\xi}$-contramodules $\mathfrak{P}^{x}$
over $\xi\in\Xi$,

$\mathfrak{P}\simeq\prod\nolimits_{\xi\in\Xi}\mathfrak{P}^{\xi}.$

###### Proof.

Part (a) is easy: the subcomodule $\mathcal{M}_{\xi}\subset\mathcal{M}$ is
recovered as the full preimage of the direct summand
$\mathcal{C}_{\xi}\otimes_{k}\mathcal{M}\subset\mathcal{C}\otimes_{k}\mathcal{M}$
under the coaction map
$\mathcal{M}\longrightarrow\mathcal{C}\otimes_{k}\mathcal{M}$. Alternatively,
part (a) can be obtained as a particular case of [11, Lemma 8.1(a)], where the
applicability of topological algebra language to comodules over coalgebras
over fields is established in [17, Section 2.1] (see the discussion in the
beginning of Section 1.1 below, cf. Remark 3.2). Part (b) is [7, Lemma A.2.2];
cf. [11, Lemma 8.1(b)], which is applicable in view of [9, Section 2.3]. ∎

###  1.1. Change of scalars

We refer to [12, Section 3.1] for a basic introductory discussion of coalgebra
homomorphisms, subcoalgebras and coideals.

The content of this section can be viewed as a particular case of [11, Section
2.9], but then the applicability of the topological algebra exposition in [11]
to comodules and contramodules over coalgebras over a field needs to be
established beforehand. One needs to know that a left $\mathcal{C}$-comodule
is the same thing as a discrete left module over the topological ring
$\mathcal{C}^{*}$, and a left $\mathcal{C}$-contramodule is the same thing as
a left contramodule over the topological ring $\mathcal{C}^{*}$ with its
natural pro-finite-dimensional/linearly compact topology. In this connection,
the relevant references are [17, Section 2.1] for comodules and [9, Section
2.3] for contramodules. The exposition below avoids any references to
topological algebra.

We start with some basic observations about ring homomorphisms, stated in a
slightly unusual notation for the purposes of subsequent dualization to
coalgebras. Let $f\colon C\longrightarrow B$ be a ring homomorphism. Then, via
the restriction of scalars, every left $B$-module acquires a left $C$-module
structure, and every right $B$-module acquires a right $C$-module structure.
So we have the exact, faithful _restriction-of-scalars_ functor $f_{*}\colon
B{\operatorname{\mathsf{--Mod}}}\longrightarrow
C{\operatorname{\mathsf{--Mod}}}$. This construction applies both to the
unital and nonunital ring homomorphisms.

The functor $f_{*}$ has adjoints on both sides. The left adjoint functor
$f^{*}\colon C{\operatorname{\mathsf{--Mod}}}\longrightarrow
B{\operatorname{\mathsf{--Mod}}}$, defined by the rule
$f^{*}(N)=B\otimes_{C}N$, is called the _extension of scalars_. The right
adjoint functor $f^{!}\colon C{\operatorname{\mathsf{--Mod}}}\longrightarrow
B{\operatorname{\mathsf{--Mod}}}$, given by the rule
$f^{!}(N)=\operatorname{Hom}_{C}(B,N)$, is called the _coextension of
scalars_. These formulas presume $f$ to be a unital ring homomorphism.

Now let us assume that the ring homomorphism $f\colon C\longrightarrow B$ is
surjective with the kernel ideal $D\subset C$. Then the functor of restriction
of scalars $f_{*}\colon B{\operatorname{\mathsf{--Mod}}}\longrightarrow
C{\operatorname{\mathsf{--Mod}}}$ is exact and fully faithful. Furthermore,
the ideal $D\subset C$ can be viewed as a nonunital ring; then the inclusion
map $D\longrightarrow C$ is a homomorphism of nonunital rings. The restriction
of scalars with respect to the inclusion map $D\longrightarrow C$ endows every
$C$-module with a $D$-module structure.

For a surjective ring homomorphism $f\colon C\longrightarrow B$, the functors
of (co)extension of scalars $f^{*}$ and $f^{!}$ can be computed as follows.
For any left $C$-module $N$, one has $f^{*}(M)=N/DN$. Here $N/DN$ is the
unique maximal quotient $C$-module of $N$ whose $C$-module structure arises
from a $B$-module structure via the restriction of scalars. For any left
$C$-module $N$, denote by ${}_{D}N\subset N$ the submodule of all elements
annihilated by $D$ in $M$. Then one has $f^{!}(N)={}_{D}N$. The submodule
${}_{D}N\subset N$ is the unique maximal $C$-submodule of $M$ whose $C$-module
structure arises from a $B$-module structure via the restriction of scalars.

Passing from rings to coalgebras, the above picture of a functor with adjoints
on both sides splits in two halves. Let
$g\colon\mathcal{B}\longrightarrow\mathcal{C}$ be a homomorphism of coalgebras
over a field $k$. Then any left $\mathcal{B}$-comodule $\mathcal{M}$ acquires
a natural structure of left $\mathcal{C}$-comodule, provided by the left
$\mathcal{C}$-coaction map

$\mathcal{M}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{B}\otimes_{k}\mathcal{M}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}\otimes_{k}\mathcal{M}$

constructed as the composition of the left $\mathcal{B}$-coaction map with the
map induced by the coalgebra homomorphism $g$. Hence the exact, faithful
functor of _corestriction of scalars_
$g_{\diamond}\colon\mathcal{B}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}{\operatorname{\mathsf{--Comod}}}$.

Dual-analogously, any left $\mathcal{B}$-contramodule $\mathfrak{P}$ acquires
a natural structure of left $\mathcal{C}$-contramodule, provided by the left
$\mathcal{C}$-contraaction map

$\operatorname{Hom}_{k}(\mathcal{C},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{k}(\mathcal{B},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\mathfrak{P}$

constructed as the composition of the map induced by the coalgebra
homomorphism $g$ with the left $\mathcal{B}$-contraaction map. Hence the
exact, faithful functor of _contrarestriction of scalars_
$g_{\sharp}\colon\mathcal{B}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{C}{\operatorname{\mathsf{--Contra}}}$.
These constructions of the functors $g_{\diamond}$ and $g_{\sharp}$ apply both
to the counital and noncounital coalgebra homomorphisms.

The functor of corestriction of scalars $g_{\diamond}$ has a right adjoint
functor
$g^{\diamond}\colon\mathcal{C}{\operatorname{\mathsf{--Comod}}}\allowbreak\longrightarrow\mathcal{B}{\operatorname{\mathsf{--Comod}}}$,
called the _coextension of scalars_. Given a left $\mathcal{C}$-comodule
$\mathcal{N}$, the left $\mathcal{B}$-comodule $g^{\diamond}(\mathcal{N})$ can
be computed explicitly as the cotensor product
$g^{\diamond}(\mathcal{N})=\mathcal{B}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}\mathcal{N}$
(see [8, Sections 2.1 and 4.8] or [7, Sections 0.2.1, 1.2.1, and 7.1.2] for
some details).

Dual-analogously, the functor of contrarestriction of scalars $g_{\sharp}$ has
a left adjoint functor
$g^{\sharp}\colon\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{B}{\operatorname{\mathsf{--Contra}}}$,
called the _contraextension of scalars_. Given a left
$\mathcal{C}$-contramodule $\mathfrak{Q}$, the left $\mathcal{B}$-contramodlue
$g^{\sharp}(\mathfrak{P})$ can be computed explicitly as
$g^{\sharp}(\mathfrak{Q})=\operatorname{Cohom}_{\mathcal{C}}(\mathcal{B},\mathfrak{Q})$
(see [8, Sections 2.2 and 4.8] or [7, Sections 0.2.4, 3.2.1, and 7.1.2] for
some details). These formulas for the functors $g^{\diamond}$ and $g^{\sharp}$
presume $g$ to be a counital coalgebra homomorphism.

Now let us assume that the coalgebra homomorphism
$g\colon\mathcal{B}\longrightarrow\mathcal{C}$ is injective; so $\mathcal{B}$
can be viewed as a subcoalgebra in $\mathcal{C}$. Then the functors of
corestriction and contrarestriction of scalars
$g_{\diamond}\colon\mathcal{B}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}{\operatorname{\mathsf{--Comod}}}$
and
$g_{\sharp}\colon\mathcal{B}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{C}{\operatorname{\mathsf{--Contra}}}$
are exact and fully faithful. Furthermore, the quotient vector space
$\mathcal{D}=\mathcal{C}/g(\mathcal{B})$ acquires a natural structure of
noncounital coalgebra over $k$; this coalgebra structure is uniquely defined
by the condition that the natural surjective map
$\mathcal{C}\longrightarrow\mathcal{D}$ must be a homomorphism of noncounital
coalgebras. The corestriction and contrarestriction of scalars with respect to
the noncounital coalgebra homomorphism $\mathcal{C}\longrightarrow\mathcal{D}$
endow every $\mathcal{C}$-comodule with a $\mathcal{D}$-comodule structure,
and every $\mathcal{C}$-contramodule with a $\mathcal{D}$-contramodule
structure.

For an injective homomorphism of coalgebras
$g\colon\mathcal{B}\longrightarrow\mathcal{C}$, the functors of coextension
and contraextension of scalars $g^{\diamond}$ and $g^{\sharp}$ can be computed
as follows. For any left $\mathcal{C}$-comodule $\mathcal{N}$, denote by
${}_{\mathcal{D}}\mathcal{N}\subset\mathcal{N}$ the kernel of the left
$\mathcal{D}$-coaction map
$\upsilon_{\mathcal{N}}\colon\mathcal{N}\longrightarrow\mathcal{D}\otimes_{k}\mathcal{N}$.
Then one has $g^{\diamond}(\mathcal{N})={}_{\mathcal{D}}\mathcal{N}$. Here the
vector subspace ${}_{\mathcal{D}}\mathcal{N}\subset\mathcal{N}$ is the unique
maximal $\mathcal{C}$-subcomodule of $\mathcal{N}$ whose
$\mathcal{C}$-comodule structure arises from a $\mathcal{B}$-comodule
structure via the corestriction of scalars.

For any left $\mathcal{C}$-contramodule $\mathfrak{Q}$, consider the left
$\mathcal{D}$-contraaction map
$\rho_{\mathfrak{Q}}\colon\operatorname{Hom}_{k}(\mathcal{D},\allowbreak\mathfrak{Q})\longrightarrow\mathfrak{Q}$.
Then one has
$g^{\sharp}(\mathfrak{P})=\operatorname{coker}(\rho_{\mathfrak{Q}})$. Here
$\operatorname{coker}(\rho_{\mathfrak{Q}})=\mathfrak{Q}/\operatorname{im}(\rho_{\mathfrak{Q}})$
is the unique maximal quotient $\mathcal{C}$-contramodule of $\mathfrak{Q}$
whose $\mathcal{C}$-contramodule structure arises from a
$\mathcal{B}$-contramodule structure via the contrarestriction of scalars.

##  2\. Locally Finite Categories, Coalgebras, and Comodules

Let $k$ be a field. A _$k$ -linear category_ $\mathsf{E}$ is a category
enriched in $k$-vector spaces. So, for any two objects $x$, $y\in\mathsf{E}$,
the $k$-vector space of morphisms
$\mathsf{E}_{y,x}=\operatorname{Hom}_{\mathsf{E}}(x,y)$ is given, together
with the multiplication/composition maps

$\mathsf{E}_{z,y}\otimes_{k}\mathsf{E}_{y,x}\longrightarrow\mathsf{E}_{z,x},\text{
\ or equivalently, \
}\operatorname{Hom}_{\mathsf{E}}(y,z)\otimes_{k}\operatorname{Hom}_{\mathsf{E}}(x,y)\longrightarrow\operatorname{Hom}_{\mathsf{E}}(x,z)$

for all $x$, $y$, $z\in\mathsf{E}$, and unit/identity elements
${\mathrm{id}}_{x}\in\mathsf{E}(x,x)$ for all $x\in\mathsf{E}$. The
composition maps must be $k$-linear, and the usual associativity and unitality
axioms are imposed. For simplicity of notation, let us assume that there are
no zero objects in $\mathsf{E}$: so ${\mathrm{id}}_{x}\neq 0$ for every
$x\in\mathsf{E}$.

A _left $\mathsf{E}$-module_ $M$ is a covariant $k$-linear functor
$\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--Vect}}}$. So, for every
object $x\in\mathsf{E}$, a $k$-vector space $M_{x}=M(x)$ is given, together
with the action maps

$\mathsf{E}_{y,x}\otimes_{k}M_{x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM_{y},\quad\text{or
equivalently,}\quad\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$

for all $x$, $y\in\mathsf{E}$. The action maps must be $k$-linear, and the
usual associativity and unitality axioms must be satisfied.

Similarly, a _right $\mathsf{E}$-module_ $N$ is a contravariant $k$-linear
functor $\mathsf{E}^{\mathsf{op}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$. So, for every object $x\in\mathsf{E}$, a
$k$-vector space $N_{x}=N(x)$ is given, together with the action maps

$N_{y}\otimes_{k}\mathsf{E}_{y,x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN_{x},\quad\text{or
equivalently,}\quad\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}N(y)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN(x)$

for all $x$, $y\in\mathsf{E}$.

######  Definition 2.1.

Given a small $k$-linear category $\mathsf{E}$, we define a preorder relation
on the set of all objects of $\mathsf{E}$ by the rule

  * •

$x\preceq y$ if and only if there exists an integer $n\geq 1$ and objects
$x=z_{0}$, $z_{1}$, …, $z_{n-1}$, $z_{n}=y$ in $\mathsf{E}$ such that
$\operatorname{Hom}_{\mathsf{E}}(z_{i-1},z_{i})\neq 0$ for all $1\leq i\leq
n$.

In other words, we write $x\preceq y$ if there exists a finite chain of
composable nonzero morphisms in $\mathsf{E}$ going from $x$ to $y$.

Furthermore, let us write $x\prec y$ whenever $x\preceq y$ and $y\not\preceq
x$. Let us also write $x\sim y$ whenever $x\preceq y$ and $y\preceq x$.

The following piece of terminology is convenient. Let us say that a morphism
$f\in\operatorname{Hom}_{\mathsf{E}}(x,y)$ is _short_ if either $f=0$, or
$x\sim y$; and $f$ is _long_ if either $f=0$, or $x\prec y$. Then short
morphisms form a $k$-linear subcategory in $\mathsf{E}$, while long morphisms
form a two-sided ideal of morphisms in $\mathsf{E}$. Any nonzero morphism in
$\mathsf{E}$ is either short or long, but not both (while the zero morphisms
are both short and long).

######  Definition 2.2.

We will say that a small $k$-linear category $\mathsf{E}$ is _locally finite_
if two conditions are satisfied:

  1. (i)

finite-dimensional vector spaces of morphisms: for any two objects $x$ and
$y\in\mathsf{E}$, the $k$-vector space $\operatorname{Hom}_{\mathsf{E}}(x,y)$
is finite-dimensional;

  2. (ii)

interval finiteness of the preorder on objects: for any two objects $x$ and
$y\in\mathsf{E}$, the set of objects $\\{z\in\mathsf{E}\mid x\preceq z\preceq
y\\}$ is finite.

Notice that our definition of a locally finite $k$-linear category is left-
right symmetric: the passage to the opposite $k$-linear category
$\mathsf{E}^{\mathsf{op}}$ with
$\operatorname{Hom}_{\mathsf{E}^{\mathsf{op}}}(y,x)=\operatorname{Hom}_{\mathsf{E}}(x,y)$
does not affect it. So the results about left $\mathsf{E}$-modules in this
section will be equally applicable to right $\mathsf{E}$-modules.

######  Construction 2.3.

To every locally finite $k$-linear category $\mathsf{E}$, we assign the
related coalgebra $\mathcal{C}_{\mathsf{E}}$ over $k$ by the following rules.
As a $k$-vector space, we put

$\mathcal{C}_{\mathsf{E}}=\bigoplus\nolimits_{x,y\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,y)^{*},$

where $W^{*}=\operatorname{Hom}_{k}(W,k)$ is the notation for the dual vector
space to a $k$-vector space $W$. Put
$\mathcal{C}_{\mathsf{E}}^{x,y}=\operatorname{Hom}_{\mathsf{E}}(x,y)^{*}=\mathsf{E}_{y,x}^{*}$;
so
$\mathcal{C}_{\mathsf{E}}=\bigoplus_{x,y\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,y}$.
The counit map $\epsilon\colon\mathcal{C}_{\mathsf{E}}\longrightarrow k$ is
defined by the rules that the restriction of $\epsilon$ to
$\mathcal{C}_{\mathsf{E}}^{x,y}$ vanishes for $x\neq y$, while the the
restriction of $\epsilon$ to $\mathcal{C}_{\mathsf{E}}^{x,x}$ is the map

$\epsilon_{x}\colon\mathcal{C}_{\mathsf{E}}^{x,x}=\operatorname{Hom}_{\mathsf{E}}(x,x)^{*}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muk$

provided by the pairing with the identity morphism
${\mathrm{id}}_{x}\in\operatorname{Hom}_{\mathsf{E}}(x,x)$. Finally, the
comultiplication map
$\mu\colon\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{C}_{\mathsf{E}}$
is defined by the rule that the restriction of $\mu$ to
$\mathcal{C}_{\mathsf{E}}^{x,y}$ is the map

$\mu_{x,y}\colon\mathcal{C}_{\mathsf{E}}^{x,y}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{x\preceq z\preceq
y}\mathcal{C}_{\mathsf{E}}^{x,z}\otimes_{k}\mathcal{C}_{\mathsf{E}}^{z,y}$

whose components
$\mathcal{C}_{\mathsf{E}}^{x,y}\longrightarrow\mathcal{C}_{\mathsf{E}}^{x,z}\otimes_{k}\mathcal{C}_{\mathsf{E}}^{z,y}$
are the dual maps to the composition/multiplication maps
$\mathsf{E}_{y,z}\otimes_{k}\mathsf{E}_{z,x}\longrightarrow\mathsf{E}_{y,x}$.

Furthermore, consider the cosemisimple coalgebra
$\mathcal{C}^{\mathrm{id}}_{\mathsf{E}}=\bigoplus_{x\in\mathsf{E}}k$, with the
coalgebra structure of the direct sum of the one-dimensional coalgebras $k$
over $k$ (taken over all the objects $x\in\mathsf{E}$). The rule
$\mathsf{E}\longmapsto\mathcal{C}_{\mathsf{E}}$ assigns the coalgebra
$\mathcal{C}^{\mathrm{id}}_{\mathsf{E}}$ to the discrete $k$-linear category
$\mathsf{E}^{\mathrm{id}}$ with the same set of objects as $\mathsf{E}$ (i.
e., the subcategory $\mathsf{E}^{\mathrm{id}}\subset\mathsf{E}$ whose
morphisms are all the scalar multiples of the identity morphisms in
$\mathsf{E}$). Then the inclusion of $k$-linear categories
$\mathsf{E}^{\mathrm{id}}\longrightarrow\mathsf{E}$ induces a surjective
morphism of coalgebras
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{C}^{\mathrm{id}}_{\mathsf{E}}$.
The latter map vanishes on the components
$\mathcal{C}_{\mathsf{E}}^{x,y}\subset\mathcal{C}_{\mathsf{E}}$ with $x\neq
y$, and is given by the maps $\epsilon_{x}$ on the components
$\mathcal{C}_{\mathsf{E}}^{x,x}\subset\mathcal{C}_{\mathsf{E}}$.

######  Construction 2.4.

Let $\mathsf{E}$ be a locally finite $k$-linear category. Then the comodule
inclusion functor

$\Upsilon_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{E}{\operatorname{\mathsf{--Mod}}}$

is constructed as follows. Suppose given a left
$\mathcal{C}_{\mathsf{E}}$-comodule $\mathcal{M}$. Then the corestriction of
scalars with respect to the morphism of coalgebras
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{C}_{\mathsf{E}}^{\mathrm{id}}$
(see Section 1.1) defines a left
$\mathcal{C}_{\mathsf{E}}^{\mathrm{id}}$-comodule structure on $\mathcal{M}$.
By Lemma 1.2(a), the latter structure is equivalent to the datum of a direct
sum decomposition $\mathcal{M}=\bigoplus_{x\in\mathsf{E}}\mathcal{M}_{x}$. The
$k$-vector space $\mathcal{M}_{x}$ is the image of the idempotent linear
operator $\epsilon_{x}\colon\mathcal{M}\longrightarrow\mathcal{M}$ provided by
the action in $\mathcal{M}$ of the idempotent linear function
$\mathcal{C}_{\mathsf{E}}\twoheadrightarrow\mathcal{C}_{\mathsf{E}}^{x,x}\overset{\epsilon_{x}}{\longrightarrow}k$
on $\mathcal{C}_{\mathsf{E}}$.

With respect to this direct sum decomposition of $\mathcal{M}$ and the direct
sum decomposition of $\mathcal{C}_{\mathsf{E}}$ described in Construction 2.3,
the coaction map

$\nu\colon\mathcal{M}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$

has components

$\nu_{x,y}\colon\mathcal{M}_{x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y},\qquad
y\in\mathsf{E},$

while all the other components vanish. More precisely, the map $\nu$ is the
direct sum of maps

(3) $\nu_{x}\colon\mathcal{M}_{x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{y\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y},$

and the maps $\nu_{x,y}$ are the compositions
$\mathcal{M}_{x}\overset{\nu_{x}}{\longrightarrow}\bigoplus\nolimits_{z\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,z}\otimes_{k}\mathcal{M}_{z}\twoheadrightarrow\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$.

Now we put $M(x)=\mathcal{M}_{x}$ for all objects $x\in\mathsf{E}$, and define
the action maps

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$

as the $k$-linear maps corresponding to $\nu_{x,y}$ in view of the
isomorphisms
$\operatorname{Hom}_{\mathsf{E}}(x,y)\simeq(\mathcal{C}_{\mathsf{E}}^{x,y})^{*}$
for finite-dimensional $k$-vector spaces $\mathcal{C}_{\mathsf{E}}^{x,y}$. To
finish the construction, it remains to set
$\Upsilon_{\mathsf{E}}(\mathcal{M})=M$.

######  Proposition 2.5.

For any locally finite $k$-linear category $\mathsf{E}$, the comodule
inclusion functor
$\Upsilon_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
is exact and fully faithful.

###### Proof.

The exactness and faithfulness are obvious from the construction. Let us
briefly explain why the functor is full. Essentially, the point is that the
coaction maps
$\nu_{x,y}\colon\mathcal{M}_{x}\longrightarrow\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$
are uniquely determined by the action maps
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\longrightarrow M(y)$.
Furthermore, the compatibility with the former is equivalent to the
compatibility with the latter, for any pair of left
$\mathcal{C}_{\mathsf{E}}$-comodules $\mathcal{L}$ and $\mathcal{M}$ and a
given $\mathcal{C}_{\mathsf{E}}^{\mathrm{id}}$-comodule map
$\mathcal{L}\longrightarrow\mathcal{M}$. These properties are clear from the
construction as well. It is illuminating to observe that a $k$-linear map
$\nu_{x}\colon\mathcal{M}_{x}\longrightarrow\bigoplus_{y\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$
is determined by its components
$\nu_{x,y}\colon\mathcal{M}_{x}\longrightarrow\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$,
essentially because the direct sum
$\bigoplus_{y\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$
is a vector subspace of the product
$\prod_{y\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$.
∎

######  Definition 2.6.

Let $\mathsf{E}$ be a locally finite $k$-linear category. A left
$\mathsf{E}$-module $M$ is said to be _locally finite_ if, viewed as a functor
$\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--Vect}}}$, it takes values
in the category of finite-dimensional vector spaces
$k{\operatorname{\mathsf{--vect}}}\subset k{\operatorname{\mathsf{--Vect}}}$,
that is $M\colon\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--vect}}}$.
In other words, $M$ is locally finite if and only if the $k$-vector space
$M(x)$ is finite-dimensional for all $x\in\mathsf{E}$. Locally finite right
$\mathsf{E}$-modules are defined similarly.

A left $\mathcal{C}_{\mathsf{E}}$-comodule $\mathcal{M}$ is said to be
_locally finite_ if the corresponding left $\mathsf{E}$-module
$M=\Upsilon_{\mathsf{E}}(\mathcal{M})$ is locally finite. In other words,
$\mathcal{M}$ is locally finite if and only if the vector space
$\mathcal{M}_{x}$ is finite-dimensional for every $x\in\mathsf{E}$. Locally
finite right $\mathcal{C}_{\mathsf{E}}$-comodules are defined similarly.

######  Theorem 2.7.

Let $\mathsf{E}$ be a locally finite $k$-linear category and
$M\colon\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--vect}}}$ be a
locally finite left $\mathsf{E}$-module. Then $M$ arises from some (locally
finite) left $\mathcal{C}_{\mathsf{E}}$-comodule via the comodule inclusion
functor $\Upsilon_{\mathsf{E}}$ if and only if, for every fixed object
$x\in\mathsf{E}$, the set of all objects $y\in\mathsf{E}$ with nonzero action
map

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$

is finite.

###### Proof.

“Only if”: let $\mathcal{M}$ be a locally finite left
$\mathcal{C}_{\mathsf{E}}$-comodule and $x\in\mathsf{E}$ be an object. Then,
for every vector $v\in\mathcal{M}_{x}$, the tensor
$\nu(v)\in\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$ is a finite sum of
decomposable tensors. Hence it must belong to the direct sum $\bigoplus_{y\in
Y_{v}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}\subset\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$,
where $Y_{v}\subset\mathsf{E}$ is some finite set of objects. Since the vector
space $\mathcal{M}_{x}$ is finite-dimensional, it follows that there exists a
finite subset $Y_{x}\subset\mathsf{E}$ such that the subspace
$\nu(\mathcal{M}_{x})\subset\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$ is
contained in the direct sum $\bigoplus_{y\in
Y_{x}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}\subset\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$.
Now Construction 2.4 tells that, for the related left $\mathsf{E}$-module
$M=\Upsilon_{\mathsf{E}}(\mathcal{M})$, the action map
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$ vanishes for
all $y\notin Y_{x}$.

“If”: let $M$ be a (not necessarily locally finite) left $\mathsf{E}$-module
such that for every object $x\in\mathsf{E}$ there is a finite subset of
objects $Y_{x}\subset\mathsf{E}$ with the property that the action map
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$ vanishes for
all $y\notin Y_{x}$. Put $\mathcal{M}_{x}=M(x)$ and
$\mathcal{M}=\bigoplus_{x\in\mathsf{E}}\mathcal{M}_{x}$, and define the
coaction map
$\nu\colon\mathcal{M}\longrightarrow\mathcal{C}_{\mathsf{E}}\otimes_{k}\mathcal{M}$
by the rule that its restriction to $\mathcal{M}_{x}$ is the map

$\nu_{x}\colon\mathcal{M}_{x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{y\in
Y_{x}}\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$

as in (3), with the components
$\nu_{x,y}\colon\mathcal{M}_{x}\longrightarrow\mathcal{C}_{\mathsf{E}}^{x,y}\otimes_{k}\mathcal{M}_{y}$
corresponding to the action maps
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}M(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muM(y)$ in view of the
isomorphisms
$\operatorname{Hom}_{\mathsf{E}}(x,y)\simeq(\mathcal{C}_{\mathsf{E}}^{x,y})^{*}$
for finite-dimensional vector spaces $\mathcal{C}_{\mathsf{E}}^{x,y}$. This
defines a left $\mathcal{C}_{\mathsf{E}}$-comodule structure on $\mathcal{M}$
such that $\Upsilon_{\mathsf{E}}(\mathcal{M})=M$. ∎

##  3\. Full-and-Faithfulness of Contramodule Forgetful Functor

The following construction is the contramodule version of Construction 2.4.

######  Construction 3.1.

Let $\mathsf{E}$ be a locally finite $k$-linear category. Then the
contramodule forgetful functor

$\Theta_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{E}{\operatorname{\mathsf{--Mod}}}$

is defined as follows. Suppose given a left
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$. Then the
contrarestriction of scalars with respect to the morphism of coalgebras
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{C}_{\mathsf{E}}^{\mathrm{id}}$
(see Construction 2.3 and Section 1.1) defines a left
$\mathcal{C}_{\mathsf{E}}^{\mathrm{id}}$-contramodule structure on
$\mathfrak{P}$. By Lemma 1.2(b), the latter structure is equivalent to the
datum of a direct product decomposition
$\mathfrak{P}=\prod_{x\in\mathsf{E}}\mathfrak{P}^{x}$. The $k$-vector space
$\mathfrak{P}^{x}$ is the image of the idempotent linear operator
$\epsilon_{x}\colon\mathfrak{P}\longrightarrow\mathfrak{P}$ provided by the
action in $\mathfrak{P}$ of the idempotent linear function
$\mathcal{C}_{\mathsf{E}}\twoheadrightarrow\mathcal{C}_{\mathsf{E}}^{x,x}\overset{\epsilon_{x}}{\longrightarrow}k$
on $\mathcal{C}_{\mathsf{E}}$.

With respect to this direct product decomposition of $\mathfrak{P}$ and the
direct sum decomposition of $\mathcal{C}_{\mathsf{E}}$ described in
Construction 2.3, the contraaction map

$\pi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\mathfrak{P}$

has components

(4)
$\pi^{y}\colon\prod\nolimits_{x\in\mathsf{E}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathfrak{P}^{y},\qquad y\in\mathsf{E},$

while all the other components vanish. Specifically, this means that the
composition

$\prod\nolimits_{x,y,z}^{(x,y)\neq(z,w)}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{z})\,\rightarrowtail\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\overset{\pi}{\longrightarrow}\mathfrak{P}\,\twoheadrightarrow\,\mathfrak{P}^{w}$

vanishes for all $w\in\mathsf{E}$; or in other words, the map $\pi$ is the
direct product of the maps $\pi^{y}$.

Now we put $P(x)=\mathfrak{P}^{x}$ for all objects $x\in\mathsf{E}$, and
define the action maps

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP(y)$

as the compositions

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\,\simeq\,\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},P(x))\,\rightarrowtail\prod\nolimits_{z\in\mathsf{E}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{z,y},P(z))\overset{\pi^{y}}{\longrightarrow}P(y),$

where the first isomorphism comes from the isomorphism
$\operatorname{Hom}_{\mathsf{E}}(x,y)\simeq(\mathcal{C}_{\mathsf{E}}^{x,y})^{*}$
for a finite-dimensional $k$-vector space $\mathcal{C}_{\mathsf{E}}^{x,y}$,
while the middle morphism is the natural direct summand inclusion. To finish
the construction, it remains to set $\Theta_{\mathsf{E}}(\mathfrak{P})=P$.

######  Remark 3.2.

For any coalgebra $\mathcal{C}$ over a field $k$, the dual $k$-vector space
$\mathcal{C}^{*}$ has a natural structure of a (topological) algebra over $k$.
The construction of the fully faithful comodule inclusion functor
$\Upsilon_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$
goes back, at least, to [17, Section 2.1]; while the construction of the
contramodule forgetful functor
$\Theta_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow
C^{*}{\operatorname{\mathsf{--Mod}}}$ was mentioned in [7, Section A.1.2]. Our
notation comes from the paper [13], where the action of the exact functors
$\Upsilon_{\mathcal{C}}$ and $\Theta_{\mathcal{C}}$ on the Ext spaces in the
respective categories is discussed. In fact, the functor
$\Upsilon_{\mathcal{C}}$ identifies the category of $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$ with the full subcategory in
$\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$ consisting of all the
_discrete_ modules over the topological ring $\mathcal{C}^{*}$ (called
“rational” in [17, 6]).

Let us explain the connection between the functors $\Upsilon_{\mathsf{E}}$ and
$\Theta_{\mathsf{E}}$ defined in Constructions 2.4 and 3.1, on the one hand,
and the functors $\Upsilon_{\mathcal{C}}$ and $\Theta_{\mathcal{C}}$, on the
other hand. For any preadditive category $\mathsf{E}$, the category of left
$\mathsf{E}$-modules $\mathsf{E}{\operatorname{\mathsf{--Mod}}}$ is equivalent
to the category $R_{\mathsf{E}}{\operatorname{\mathsf{--Mod}}}$ of locally
unital left modules over the locally unital ring (ring with enough
idempotents)
$R_{\mathsf{E}}=\bigoplus_{x,y\in\mathsf{E}}\operatorname{Hom}_{\mathsf{E}}(x,y)$.
In the situation at hand with a locally finite $k$-linear category
$\mathsf{E}$, the locally unital $k$-algebra $R_{\mathsf{E}}$ is a dense
subalgebra in the unital topological algebra $\mathcal{C}_{\mathsf{E}}^{*}$
(with its natural pro-finite-dimensional/linearly compact topology).

The functor
$\Upsilon_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
is the composition of the functor
$\Upsilon_{\mathcal{C}_{\mathsf{E}}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}_{\mathsf{E}}^{*}{\operatorname{\mathsf{--Mod}}}$
with the functor of restriction of scalars
$\mathcal{C}_{\mathsf{E}}^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
R_{\mathsf{E}}{\operatorname{\mathsf{--Mod}}}$. Similarly, the functor
$\Theta_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
is the composition of the functor
$\Theta_{\mathcal{C}_{\mathsf{E}}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{C}_{\mathsf{E}}^{*}{\operatorname{\mathsf{--Mod}}}$
with the functor of restriction of scalars
$\mathcal{C}_{\mathsf{E}}^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
R_{\mathsf{E}}{\operatorname{\mathsf{--Mod}}}$. The observations that the
$\mathcal{C}_{\mathsf{E}}$-comodules are the same things as the discrete
$\mathcal{C}_{\mathsf{E}}^{*}$-modules and the subalgebra $R_{\mathsf{E}}$ is
dense in $\mathcal{C}_{\mathsf{E}}^{*}$ explain the fact that the functor
$\Upsilon_{\mathsf{E}}$ is fully faithful, as per Proposition 2.5.

A _coaugmentation_ $\gamma$ of a coalgebra $\mathcal{C}$ over a field $k$ is a
counital coalgebra homomorphism $\gamma\colon k\longrightarrow C$. A
coaugmented coalgebra $(\mathcal{C},\gamma)$ is said to be _conilpotent_ if
the noncounital quotient coalgebra $\mathcal{D}=\mathcal{C}/\gamma(k)$ is
conilpotent in the sense of the definition in Section 1. One can see that any
conilpotent coalgebra $\mathcal{C}$ has a unique coaugmentation [12, Section
3.3], [13, Section 2].

A conilpotent coalgebra $\mathcal{C}$ is said to be _finitely cogenerated_ if
the space $\operatorname{Ext}^{1}_{\mathcal{C}}(k,k)$, computed in the
category of left $\mathcal{C}$-comodules
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}$, with the
$\mathcal{C}$-comodule structure on $k$ defined in terms of the coaugmentation
$\gamma$, is finite-dimensional. According to [10, Theorem 2.1], the
contramodule forgetful functor $\Theta_{\mathcal{C}}$ is fully faithful for
any finitely cogenerated conilpotent coalgebra $\mathcal{C}$. Conversely, [13,
Example 7.2] shows that, for any infinitely cogenerated conilpotent coalgebra
$\mathcal{C}$, the functor $\Theta_{\mathcal{C}}$ is not full.

In fact, [10, Theorem 2.1] tells more. Let $R\subset\mathcal{C}^{*}$ be a
dense subring in the natural (pro-finite-dimensional/linearly compact)
topology on $\mathcal{C}^{*}$. Then the composition of the functor
$\Theta_{\mathcal{C}}\colon\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}$
with the functor of restriction of scalars
$\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ is a fully faithful functor
$\mathcal{C}{\operatorname{\mathsf{--Contra}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$.

The coalgebra $\mathcal{C}_{\mathsf{E}}$ is usually _not_ conilpotent, of
course. The aim of this section is to obtain an analogue of [10, Theorem 2.1]
for the coalgebras $\mathcal{C}_{\mathsf{E}}$.

######  Example 3.3.

It is obvious from the construction that the functor $\Theta_{\mathsf{E}}$ is
exact and faithful for any locally finite $k$-linear category $\mathsf{E}$.
The following counterexample shows, however, that the functor
$\Theta_{\mathsf{E}}$ _need not_ be fully faithful for a locally finite
$k$-linear category $\mathsf{E}$ in general.

Let $\mathsf{E}$ be the $k$-linear category whose set of objects is the set of
all negative integers $\mathbb{Z}_{<0}$. For any integer $n\geq 1$, the
$k$-vector space $\operatorname{Hom}_{\mathsf{E}}(-n,-n)$ is one-dimensional
and spanned by the identity morphism ${\mathrm{id}}_{-n}$. For any integer
$n\geq 2$, the $k$-vector space $\operatorname{Hom}_{\mathsf{E}}(-n,-1)$ is
one-dimensional and spanned by a morphism
$f_{-n,-1}\in\operatorname{Hom}_{\mathsf{E}}(-n,-1)$. There are no other
nonzero morphisms in $\mathsf{E}$.

The category of left $\mathcal{C}_{\mathsf{E}}$-comodules agrees with the
category of left $\mathsf{E}$-modules; in other words, the functor
$\Upsilon_{\mathsf{E}}$ is a category equivalence (cf. Proposition 7.4 below).
The datum of a left $\mathcal{C}_{\mathsf{E}}$-comodule is equivalent to the
datum of a family of $k$-vector spaces $\mathcal{M}_{-n}$, $n\geq 1$, together
with arbitrary $k$-linear maps

$f_{-n,-1}\colon\mathcal{M}_{-n}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathcal{M}_{-1},\qquad n\geq 2.$

The $\mathcal{C}_{\mathsf{E}}$-comodule $\mathcal{M}$ is recovered as
$\mathcal{M}=\bigoplus_{n\geq 1}\mathcal{M}_{-n}$.

The category of right $\mathcal{C}_{\mathsf{E}}$-comodules does _not_ agree
with the category of right $\mathsf{E}$-modules; so the functor
$\Upsilon_{\mathsf{E}^{\mathsf{op}}}$ is _not_ an equivalence. The datum of a
right $\mathcal{C}_{\mathsf{E}}$-comodule is equivalent to the datum of a
family of $k$-vector spaces $\mathcal{N}_{-n}$, $n\geq 1$, together with an
arbitrary $k$-linear map

$G\colon\mathcal{N}_{-1}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{n\geq 2}\mathcal{N}_{-n}.$

The $\mathcal{C}_{\mathsf{E}}$-comodule $\mathcal{N}$ is recovered as
$\mathcal{N}=\bigoplus_{n\geq 1}\mathcal{N}_{-n}$. In particular, if
$\mathcal{N}_{-1}$ is finite-dimensional, then the datum of a map $G$ is a
equivalent to that of a family of linear maps
$g_{-n,-1}\colon\mathcal{N}_{-1}\longrightarrow\mathcal{N}_{-n}$, $n\geq 2$,
such that _only finitely many of the maps $g_{-n,-1}$ are nonzero_ (cf.
Theorem 2.7). On the other hand, the datum of a right $\mathsf{E}$-module is
equivalent to that of a family of $k$-vector spaces $N(-n)$, $n\geq 1$,
together with arbitrary $k$-linear maps

$g_{-n,-1}\colon N(-1)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN(-n),\qquad n\geq
2.$

The datum of a left $\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ is
equivalent to the datum of a family of $k$-vector spaces $\mathfrak{P}^{-n}$,
$n\geq 1$, together with an arbitrary $k$-linear map

$F\colon\prod\nolimits_{n\geq 2}\mathfrak{P}^{-n}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\mathfrak{P}^{-1}.$

The $\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ is recovered as
$\mathfrak{P}=\prod_{n\geq 1}\mathfrak{P}^{-n}$. The functor
$\Theta_{\mathsf{E}}$ assigns to $\mathfrak{P}$ the family of $k$-vector
spaces $P(-n)=\mathfrak{P}^{-n}$ endowed with the family of maps
$f_{-n,-1}\colon P(-n)\longrightarrow P(-1)$ obtained as restrictions of the
map $F$. Essentially, this means restricting the map $F$ to the vector
subspace $\bigoplus_{n\geq 2}\mathfrak{P}^{-n}\subset\prod_{n\geq
2}\mathfrak{P}^{-n}$.

For example, let $\mathfrak{P}^{-n}=k$ be one-dimensional $k$-vector spaces
for all $n\geq 1$. Let $F^{\prime}\colon\prod_{n\geq
2}\mathfrak{P}^{-n}\longrightarrow\mathfrak{P}^{-1}$ be a nonzero linear map
annihilating the subspace $\bigoplus_{n\geq
2}\mathfrak{P}^{-n}\subset\prod_{n\geq 2}\mathfrak{P}^{-n}$, and let
$F^{\prime\prime}\colon\prod_{n\geq
2}\mathfrak{P}^{-n}\longrightarrow\mathfrak{P}^{-1}$ be the zero map. Then the
$\mathcal{C}_{\mathsf{E}}$-contramodules $\mathfrak{P}^{\prime}$ and
$\mathfrak{P}^{\prime\prime}$ defined by the maps $F^{\prime}$ and
$F^{\prime\prime}$ are _not_ isomorphic to each other, but their images under
the functor $\Theta_{\mathsf{E}}$ _are_ isomorphic as left
$\mathsf{E}$-modules. Thus the functor $\Theta_{\mathsf{E}}$ is _not_ fully
faithful.

We start with a discussion of a trivial particular case when it is easy to
show that the functor $\Theta_{\mathsf{E}}$ is not only fully faithful, but
even a category equivalence. Then we will proceed to introduce a much more
general sufficient condition on a locally finite $k$-linear category
$\mathsf{E}$ guaranteeing that the functor $\Theta_{\mathsf{E}}$ is fully
faithful.

######  Remark 3.4.

Let $\mathsf{E}$ be a locally finite $k$-linear category. Denote by
$\mathsf{E}^{\sim}\subset\mathsf{E}$ the $k$-linear subcategory of _short
morphisms_ in $\mathsf{E}$ (in the sense of Definition 2.1). This means that,
given two objects $x$, $y\in\mathsf{E}$, we put
$\operatorname{Hom}_{\mathsf{E}^{\sim}}(x,y)=\operatorname{Hom}_{\mathsf{E}}(x,y)$
if $x\sim y$ and $\operatorname{Hom}_{\mathsf{E}^{\sim}}(x,y)=0$ otherwise. So
we have the $k$-linear inclusion functor
$\mathsf{E}^{\sim}\longrightarrow\mathsf{E}$.

We are more interested, however, in the natural functor in the opposite
direction. The functor $\mathsf{E}\longrightarrow\mathsf{E}^{\sim}$ acts by
the identity map on the objects and by surjective maps on the morphisms, and
represents $\mathsf{E}^{\sim}$ as the quotient category of $\mathsf{E}$ by the
two-sided ideal of _long morphisms_. More precisely, the functor
$\mathsf{E}\longrightarrow\mathsf{E}^{\sim}$ takes every short morphism to
itself and every long morphism to zero.

Consider the vector subspace

$\mathcal{C}_{\mathsf{E}}^{\sim}=\bigoplus\nolimits_{x,y\in\mathsf{E}}^{x\sim
y}\mathcal{C}_{\mathsf{E}}^{x,y}$

in the coalgebra $\mathcal{C}_{\mathsf{E}}$. One can easily see from the
definition of the preorder $\preceq$ on the set of objects of $\mathsf{E}$
that $\mathcal{C}_{\mathsf{E}}^{\sim}$ is a subcoalgebra in
$\mathcal{C}_{\mathsf{E}}$. The coalgebra $\mathcal{C}_{\mathsf{E}}^{\sim}$
arises via Construction 2.3 applied to the locally finite $k$-linear category
$\mathsf{E}^{\sim}$, that is

$\mathcal{C}_{\mathsf{E}}^{\sim}=\mathcal{C}_{\mathsf{E}^{\sim}}.$

The inclusion of coalgebras
$\mathcal{C}_{\mathsf{E}}^{\sim}\longrightarrow\mathcal{C}_{\mathsf{E}}$ is
induced by the surjective $k$-linear functor
$\mathsf{E}\longrightarrow\mathsf{E}^{\sim}$ (while the inclusion of
categories $\mathsf{E}^{\sim}\longrightarrow\mathsf{E}$ induces a surjective
morphism of coalgebras
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{C}_{\mathsf{E}}^{\sim}$,
which we will never use).

The quotient space of any coalgebra by a subcoalgebra has a natural structure
of noncounital coalgebra (see Section 1.1, cf. the discussion in [12, Section
3.1]). So the quotient vector space

$\mathcal{D}_{\mathsf{E}}^{\prec}=\mathcal{C}_{\mathsf{E}}/\mathcal{C}_{\mathsf{E}}^{\sim}=\bigoplus\nolimits_{x,y\in\mathsf{E}}^{x\prec
y}\mathcal{C}_{\mathsf{E}}^{x,y}$

is naturally a noncounital coalgebra over $k$. The natural surjective morphism
(direct summand projection)
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{D}_{\mathsf{E}}^{\prec}$ is a
morphism of noncounital coalgebras.

By the contrarestriction of scalars (see Section 1.1, cf. [12, Section 8.4]),
any left $\mathcal{C}_{\mathsf{E}}$-contramodule acquires a left
$\mathcal{D}_{\mathsf{E}}^{\prec}$-contramodule structure, while any left
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure on a vector space
$\mathfrak{W}$ gives rise to a left $\mathcal{C}_{\mathsf{E}}$-contramodule
structure on $\mathfrak{W}$. The notation in this remark compares with the one
in Section 1.1 by the rules $\mathcal{B}=\mathcal{C}_{\mathsf{E}}^{\sim}$,
$\mathcal{C}=\mathcal{C}_{\mathsf{E}}$, and
$\mathcal{D}=\mathcal{D}_{\mathsf{E}}^{\prec}$.

Our proofs below are based on the following important observation. The
condition of interval finiteness of the preorder $\preceq$ from Definition 2.2
implies that the noncounital coalgebra $\mathcal{D}_{\mathsf{E}}^{\prec}$ is
conilpotent. So Lemma 1.1 is applicable.

######  Lemma 3.5.

For any locally finite $k$-linear category $\mathsf{E}$, the contramodule
forgetful functor
$\Theta_{\mathsf{E}^{\sim}}\colon\mathcal{C}_{\mathsf{E}}^{\sim}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{E}^{\sim}{\operatorname{\mathsf{--Mod}}}$
is an equivalence of categories.

###### Proof.

Let $\Xi$ denote the set of all equivalence classes of objects of the category
$\mathsf{E}$ with respect to the equivalence relation $\sim$. For every
$\xi\in\Xi$, denote by
$\mathsf{E}^{\xi}\subset\mathsf{E}^{\sim}\subset\mathsf{E}$ the full
subcategory on the set of all objects $x\in\xi$. Then the $k$-linear category
$\mathsf{E}^{\sim}$ is the (preadditive) disjoint union of the $k$-linear
categories $\mathsf{E}^{\xi}$ over all $\xi\in\Xi$. Similarly, the coalgebra
$\mathcal{C}_{\mathsf{E}}^{\sim}$ is the direct sum of the coalgebras
$\mathcal{C}_{\mathsf{E}}^{\xi}=\mathcal{C}_{\mathsf{E}^{\xi}}$,

$\mathcal{C}_{\mathsf{E}}^{\sim}=\bigoplus\nolimits_{\xi\in\Xi}\mathcal{C}_{\mathsf{E}}^{\xi}.$

Furthermore, it is clear from Definition 2.2 that the coalgebra
$\mathcal{C}_{\mathsf{E}}^{\xi}$ is finite-dimensional for every $\xi\in\Xi$.
The finite-dimensional, unital algebra
$R_{\mathsf{E}^{\xi}}=\bigoplus_{x,y\in\mathsf{E}^{\xi}}\operatorname{Hom}_{\mathsf{E}^{\xi}}(x,y)$
is naturally isomorphic to the dual algebra
$\bigl{(}\mathcal{C}_{\mathsf{E}}^{\xi}\bigr{)}^{*}$ to the coalgebra
$\mathcal{C}_{\mathsf{E}}^{\xi}$. Hence the functor
$\Theta_{\mathsf{E}^{\xi}}\colon\mathcal{C}_{\mathsf{E}}^{\xi}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{E}^{\xi}{\operatorname{\mathsf{--Mod}}}$
is a category equivalence.

It remains to refer to Lemma 1.2(b) to the effect that any
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule decomposes naturally as an
infinite product of $\mathcal{C}_{\mathsf{E}}^{\xi}$-contramodules over all
$\xi\in\Xi$. So the category
$\mathcal{C}_{\mathsf{E}}^{\sim}{\operatorname{\mathsf{--Contra}}}$ is the
Cartesian product of the categories
$\mathcal{C}_{\mathsf{E}}^{\xi}{\operatorname{\mathsf{--Contra}}}$. Similarly,
the category $\mathsf{E}^{\sim}{\operatorname{\mathsf{--Contra}}}$ is clearly
the Cartesian product of the categories
$\mathsf{E}^{\xi}{\operatorname{\mathsf{--Contra}}}$. The functor
$\Theta_{\mathsf{E}^{\sim}}$ is the Cartesian product of the category
equivalences $\Theta_{\mathsf{E}^{\xi}}$, hence also a category equivalence.

Alternatively, one could observe that the $k$-linear category
$\mathsf{E}^{\sim}$ is lower (and upper) finite in the sense of Definition 7.3
for any locally finite $k$-linear category $\mathsf{E}$. So it suffices to
refer to Proposition 7.5 below. ∎

The following definition spells out a property of locally finite $k$-linear
categories $\mathsf{E}$ implying that the functor $\Theta_{\mathsf{E}}$ is
fully faithful.

######  Definition 3.6.

Let $\mathsf{E}$ be a locally finite $k$-linear category. We will say that
$\mathsf{E}$ is _left strictly locally finite_ if the following condition is
satisfied: For any object $y\in\mathsf{E}$ there exists a finite set of
objects $X_{y}\subset\mathsf{E}$ such that $x\prec y$ (in the sense of
Definition 2.1) for all $x\in X_{y}$ and, for every morphism
$f\in\operatorname{Hom}_{\mathsf{E}}(z,y)$ with $z\prec y$ there exist two
finite families of morphisms $g_{i}\colon z\longrightarrow x_{i}$ and
$h_{i}\colon x_{i}\longrightarrow y$, $1\leq i\leq n$, $n\geq 0$, $x_{i}\in
X_{y}$, such that $f=\sum\nolimits_{i=1}^{n}h_{i}g_{i}$.

For example, the $k$-linear category $\mathsf{E}$ from Example 3.3 is _not_
left strictly locally finite (take $y=-1$).

######  Theorem 3.7.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category. Then
the contramodule forgetful functor
$\Theta_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
is fully faithful.

###### Proof.

Let $\mathfrak{P}$ and $\mathfrak{Q}$ be two left
$\mathcal{C}_{\mathsf{E}}$-contramodules, and let
$\phi\colon\Theta_{\mathsf{E}}(\mathfrak{P})\longrightarrow\Theta_{\mathsf{E}}(\mathfrak{Q})$
be a morphism of the related left $\mathsf{E}$-modules. Following Construction
3.1, both $\mathfrak{P}$ and $\mathfrak{Q}$ decompose naturally as infinite
products $\mathfrak{P}=\prod_{x\in\mathsf{E}}\mathfrak{P}^{x}$ and
$\mathfrak{Q}=\prod_{x\in\mathsf{E}}\mathfrak{Q}^{x}$. The related left
$\mathsf{E}$-modules $P$ and $Q$ are defined so that $P(x)=\mathfrak{P}^{x}$
and $Q(x)=\mathfrak{Q}^{x}$ for all $x\in\mathsf{E}$. The $\mathsf{E}$-module
morphism $\phi$ is a collection of maps $\phi(x)\colon P(x)\longrightarrow
Q(x)$. Consider the product of maps
$\prod_{x\in\mathsf{E}}\phi(x)\colon\mathfrak{P}\longrightarrow\mathfrak{Q}$,
and denote it also by $\phi$. We need to show that $\phi$ is a
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism.

The contraaction map
$\pi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\longrightarrow\mathfrak{P}$
is a surjective morphism of left $\mathcal{C}_{\mathsf{E}}$-contramodules
(where the vector space
$\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})$ is endowed
with the free left $\mathcal{C}_{\mathsf{E}}$-contramodule structure defined
in Section 1). In order to show that
$\phi\colon\mathfrak{P}\longrightarrow\mathfrak{Q}$ is a
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism, is suffices to check that
$\phi\circ\pi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\longrightarrow\mathfrak{Q}$
is a $\mathcal{C}_{\mathsf{E}}$-contramodule morphism. This reduces the
original problem to the case when $\mathfrak{P}$ is a free left
$\mathcal{C}_{\mathsf{E}}$-contramodule. Without loss of generality, we will
therefore assume that
$\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$ is a free
$\mathcal{C}_{\mathsf{E}}$-contramodule generated by some $k$-vector space
$V$.

The counit map $\mathcal{C}_{\mathsf{E}}\longrightarrow k$ induces a natural
inclusion $V\longrightarrow\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$
representing $V$ as space of free generators of
$\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$. The composition
$V\longrightarrow\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)\overset{\phi}{\longrightarrow}\mathfrak{Q}$
defines a $k$-linear map $V\longrightarrow\mathfrak{Q}$, which can be uniquely
extended to a left $\mathcal{C}_{\mathsf{E}}$-contramodule map
$\psi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)\longrightarrow\mathfrak{Q}$.
By construction, the maps $\psi$ and
$\phi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)\longrightarrow\mathfrak{Q}$
agree in restriction to the vector subspace of free
$\mathcal{C}_{\mathsf{E}}$-contramodule generators
$V\subset\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$; we need to show
that they agree on the whole $\mathcal{C}_{\mathsf{E}}$-contramodule
$\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$.

Put $\chi=\phi-\psi$. Then
$\chi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)\longrightarrow\mathfrak{Q}$
is a $k$-linear map about which we know that it corresponds, via the
construction from the first paragraph of this proof, to a left
$\mathsf{E}$-module morphism
$\Theta_{\mathsf{E}}(\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V))\longrightarrow\Theta_{\mathsf{E}}(\mathfrak{Q})$.
We also know that the restriction of $\chi$ to the vector subspace of free
$\mathcal{C}_{\mathsf{E}}$-contramodule generators
$V\subset\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$ vanishes. We need
to show that $\chi$ is a $\mathcal{C}_{\mathsf{E}}$-contramodule morphism, or
equivalently, that $\chi=0$. Replacing if needed the
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{Q}$ by its subcontramodule
generated by the image of $\chi$, we can assume without loss of generality
that no proper $\mathcal{C}_{\mathsf{E}}$-subcontramodule of $\mathfrak{Q}$
contains the image of $\chi$. Then our task is to prove that $\mathfrak{Q}=0$.
It still suffices to show that $\chi$ is a
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism.

Now we are using the notation and discussion of Section 1.1 and Remark 3.4.
Assume for the sake of contradiction that $\mathfrak{Q}\neq 0$. By the
contramodule Nakayama lemma (Lemma 1.1(b)), the composition
$\rho_{\mathfrak{Q}}\colon\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{Q})\longrightarrow\mathfrak{Q}$
of the injective map
$\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{Q})\longrightarrow\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{Q})$
induced by the surjective map
$\mathcal{C}_{\mathsf{E}}\longrightarrow\mathcal{D}_{\mathsf{E}}^{\prec}$ with
the $\mathcal{C}_{\mathsf{E}}$-contraaction map
$\pi_{\mathfrak{Q}}\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{Q})\longrightarrow\mathfrak{Q}$
_cannot_ be surjective. The image of the
$\mathcal{D}_{\mathsf{E}}^{\prec}$-contraaction map $\rho_{\mathfrak{Q}}$ is a
$\mathcal{C}_{\mathsf{E}}$-subcontramodule in $\mathfrak{Q}$; in fact, it is
the maximal $\mathcal{C}_{\mathsf{E}}$-subcontramodule
$\mathfrak{S}\subset\mathfrak{Q}$ with the property that the
$\mathcal{C}_{\mathsf{E}}$-contramodule structure on the quotient contramodule
$\mathfrak{Q}/\mathfrak{S}$ arises from a
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure.

So $\mathfrak{S}=\operatorname{im}(\rho_{\mathfrak{Q}})$ is a proper
$\mathcal{C}_{\mathsf{E}}$-subcontramodule of $\mathfrak{Q}$; and we are
already assuming that no proper $\mathcal{C}_{\mathsf{E}}$-subcontramodule of
$\mathfrak{Q}$ contains the image of $\chi$. Replacing the contramodule
$\mathfrak{Q}$ by the contramodule $\mathfrak{Q}/\mathfrak{S}$ and the map
$\chi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)\longrightarrow\mathfrak{Q}$
by its composition with the natural surjection
$\mathfrak{Q}\longrightarrow\mathfrak{Q}/\mathfrak{S}$, we can assume without
loss of generality that the left $\mathcal{C}_{\mathsf{E}}$-contramodule
structure on $\mathfrak{Q}$ arises from a left
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure. Informally, one can
say that “the contraaction of long morphisms in $\mathfrak{Q}$ vanishes”
(after we’ve killed it by taking the quotient by $\mathfrak{S}$).

At this point, let us return to the notation
$\mathfrak{P}=\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},V)$. Just as the
maps $\phi$ and $\psi$, and all such maps throughout our argument, the map
$\chi$ decomposes as the product

$\chi=\prod\nolimits_{x\in\mathsf{E}}\chi^{x}\colon\prod\nolimits_{x\in\mathsf{E}}\mathfrak{P}^{x}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{x\in\mathsf{E}}\mathfrak{Q}^{x}.$

We have $P=\Theta_{\mathsf{E}}(\mathfrak{P})$ and
$Q=\Theta_{\mathsf{E}}(\mathfrak{Q})$, where $P(x)=\mathfrak{P}^{x}$ and
$Q(x)=\mathfrak{Q}^{x}$ for all $x\in\mathsf{E}$. The collection of maps
$\chi(x)=\chi^{x}\colon P(x)\longrightarrow Q(x)$ is a left
$\mathsf{E}$-module morphism $\chi\colon P\longrightarrow Q$.

Our next aim is to show that the contraaction of long morphisms in
$\mathfrak{P}$ is annihilated by $\chi$. Specifically, the claim is that the
composition

(5)
$\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{P})\overset{\rho_{\mathfrak{P}}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\mathfrak{P}\overset{\chi}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}\mathfrak{Q}$

vanishes. Here $\rho_{\mathfrak{P}}$ is the composition
$\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{P})\,\rightarrowtail\,\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\overset{\pi_{\mathfrak{P}}}{\longrightarrow}\mathfrak{P}$,
similarly to the notation in Section 1.1 and above. The vanishing of (5) would
be obvious if we knew $\chi$ to be a $\mathcal{C}_{\mathsf{E}}$-contramodule
morphism; the point is that we only know that $\chi$ is an $\mathsf{E}$-module
morphism.

With respect to the direct product decomposition over $x\in\mathsf{E}$, the
map (5) consists of the components

(6)
$\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{P})^{y}\,=\,\prod\nolimits_{x\in\mathsf{E}}^{x\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\overset{\rho_{\mathfrak{P}}^{y}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\mathfrak{P}^{y}\overset{\chi^{y}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu}\mathfrak{Q}^{y},\qquad y\in\mathsf{E}.$

In order to prove that the composition (5) vanishes, it suffices to check that
the composition (6) vanishes for every $y\in\mathsf{E}$.

So let us fix an object $y\in\mathsf{E}$. We still have not used the
assumption that $\mathsf{E}$ is left strictly locally finite so far in this
proof. Now we are using it. Consider the finite set of objects
$X_{y}\subset\mathsf{E}$ corresponding to the object $y$ as per Definition
3.6. Put

$U_{y}=\bigoplus\nolimits_{x\in X_{y}}\mathcal{C}_{\mathsf{E}}^{x,y}.$

So $U_{y}$ is a finite-dimensional $k$-vector space. Denote by
$q_{y}\colon\mathcal{C}_{\mathsf{E}}\longrightarrow U_{y}$ the direct summand
projection. Fix an object $z\in\mathsf{E}$ such that $z\prec y$, and consider
the composition

(7) $\mathcal{C}_{\mathsf{E}}^{z,y}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{x\in\mathsf{E}}\mathcal{C}_{\mathsf{E}}^{z,x}\otimes_{k}\mathcal{C}_{\mathsf{E}}^{x,y}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{x\in
X_{y}}\mathcal{C}_{\mathsf{E}}^{z,x}\otimes_{k}\mathcal{C}_{\mathsf{E}}^{x,y}.$

Here the leftmost map is the $(z,y)$-component of the comultiplication in
$\mathcal{C}_{\mathsf{E}}$, while the rightmost map is induced by the direct
summand projection $q_{y}$.

The key observation is that the condition in Definition 3.6 implies
injectivity of the composition (7). Indeed, the domain of the composition (7)
is the dual vector space to $\operatorname{Hom}_{\mathsf{E}}(z,y)$. The
condition in Definition 3.6 tells precisely that the
composition/multiplication map in the category $\mathsf{E}$,

$\bigoplus\nolimits_{x\in
X_{y}}\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}\operatorname{Hom}_{\mathsf{E}}(z,x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{\mathsf{E}}(x,z),$

which is $k$-vector space dual to the composition in (7), is surjective. Hence
the composition in (7) is injective.

Now injectivity of the map (7) implies surjectivity of the map obtained by
applying the contravariant functor
$\operatorname{Hom}_{k}({-},\mathfrak{P}^{z})$ to (7). Thus the map

(8) $\prod\nolimits_{z\in\mathsf{E},\,x\in X_{y}}^{z\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{z,x},\mathfrak{P}^{z}))\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{z\in\mathsf{E}}^{z\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{z,y},\mathfrak{P}^{z})$

induced by the comultiplication in $\mathcal{C}_{\mathsf{E}}$ is surjective.
The contraassociativity axiom for the $\mathcal{C}_{\mathsf{E}}$-contramodule
$\mathfrak{P}$ describes the composition of the map (8) with the contraaction
map

(9) $\prod\nolimits_{z\in\mathsf{E}}^{z\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{z,y},\mathfrak{P}^{z})\overset{\rho_{\mathfrak{P}}^{y}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}\mathfrak{P}^{y}$

as the composition of two maps induced by the contraaction map
$\pi_{\mathfrak{P}}$. We have shown that the image of the contraaction map (9)
is contained in the image of the composition

(10) $\prod\nolimits_{x\in
X_{y}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{x\in\mathsf{E}}^{x\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\overset{\rho_{\mathfrak{P}}^{y}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}\mathfrak{P}^{y}.$

Here the leftmost map in (10) is the direct summand inclusion induced by the
direct summand projection $q_{y}$.

Informally speaking, we have shown that _the image of the contraaction of the
long morphisms in $\mathfrak{P}$, projected to $\mathfrak{P}^{y}$, is
contained in the image of the contraaction of the morphisms from $U_{y}$,
projected to $\mathfrak{P}^{y}$_. But $X_{y}$ is only a finite set of objects,
and $U_{y}$ is only a finite-dimensional vector space. For this reason, the
“contraaction of the morphisms from $U_{y}$” in the
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ is a part of the action
of the category $\mathsf{E}$ in the $\mathsf{E}$-module $P$. And we know that
the map $\chi\colon P\longrightarrow Q$ preserves the $\mathsf{E}$-module
structure. We also know that the contraaction of long morphisms in
$\mathfrak{Q}$ vanishes; so in particular the action of long morphisms in $Q$
vanishes. And all the morphisms from the objects $x\in X_{y}$ to the object
$y$ are long. These arguments explain why the composition (6) has to vanish.

More precisely, the composition in (10) is isomorphic to the
$\mathsf{E}$-action map

(11) $\bigoplus\nolimits_{x\in
X_{y}}\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP(y).$

Since the image of (9) is contained in (actually, coincides with) the image of
(10), and since $\chi\colon P\longrightarrow Q$ is a left $\mathsf{E}$-module
morphism, it follows that the image of the composition (6) is contained in the
image of the composition

(12) $\bigoplus\nolimits_{x\in
X_{y}}\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muP(y)\overset{\chi(y)}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}Q(y),$

which is contained in the image of the $\mathsf{E}$-action map

(13) $\bigoplus\nolimits_{x\in
X_{y}}\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}Q(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muQ(y).$

Finally, the image of (13) is contained in the image of the
$\mathcal{D}_{\mathsf{E}}^{\prec}$-contraaction map

(14) $\prod\nolimits_{x\in\mathsf{E}}^{x\prec
y}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{Q}^{x})\overset{\rho_{\mathfrak{Q}}^{y}}{\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu}\mathfrak{Q}^{y},$

which we know to be zero by the preceding arguments in this proof.

We have shown that the composition (6) vanishes, and consequently so does the
composition (5), as claimed. Let $\mathfrak{T}\subset\mathfrak{P}$ be the
image of the $\mathcal{D}_{\mathsf{E}}^{\prec}$-contraaction map
$\rho_{\mathfrak{P}}\colon\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\mathfrak{P})\longrightarrow\mathfrak{P}$.
So $\mathfrak{T}$ is the maximal $\mathcal{C}_{\mathsf{E}}$-subcontramodule in
$\mathfrak{P}$ with the property that the
$\mathcal{C}_{\mathsf{E}}$-contramodule structure on
$\mathfrak{P}/\mathfrak{T}$ arises from a
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure. We have shown that
the $k$-linear map $\chi\colon\mathfrak{P}\longrightarrow\mathfrak{Q}$
factorizes as
$\mathfrak{P}\longrightarrow\mathfrak{P}/\mathfrak{T}\longrightarrow\mathfrak{Q}$.
By construction, $\mathfrak{P}\longrightarrow\mathfrak{P}/\mathfrak{T}$ is a
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism. It remains to prove that
$\mathfrak{P}/\mathfrak{T}\longrightarrow\mathfrak{Q}$ is a
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism.

Put $T=\Theta_{\mathsf{E}}(\mathfrak{T})$; so $T$ is a left
$\mathsf{E}$-module. Applying the functor $\Theta_{\mathsf{E}}$ to the
surjective $\mathcal{C}_{\mathsf{E}}$-contramodule morphism
$\mathfrak{P}\longrightarrow\mathfrak{P}/\mathfrak{T}$, we obtain a surjective
$\mathsf{E}$-module morphism $P\longrightarrow P/T$. The $\mathsf{E}$-module
morphism $\chi\colon P\longrightarrow Q$ factorizes as $P\longrightarrow
P/T\longrightarrow Q$, where $P/T\longrightarrow Q$ is an $\mathsf{E}$-module
morphism. Now the $\mathsf{E}$-module structure on
$P/T=\Theta_{\mathsf{E}}(\mathfrak{P}/\mathfrak{T})$ arises from a
$\mathsf{E}^{\sim}$-module structure (in other words, the action of the ideal
of long morphisms vanishes in $P/T$), since the
$\mathcal{C}_{\mathsf{E}}$-contramodule structure on
$\mathfrak{P}/\mathfrak{T}$ arises from a
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure. Similarly, the
$\mathsf{E}$-module structure on $Q$ arises from a $\mathsf{E}^{\sim}$-module
structure (i. e., once again, the action of the ideal of long morphisms
vanishes in $Q$). So $P/T\longrightarrow Q$ is a morphism of left
$\mathsf{E}^{\sim}$-modules.

At last, by Lemma 3.5, the functor $\Theta_{\mathsf{E}^{\sim}}$ is an
equivalence of categories; in particular, it is fully faithful. Since
$P/T\longrightarrow Q$ is a morphism of $\mathsf{E}^{\sim}$-modules, it
follows that $\mathfrak{P}/\mathfrak{T}\longrightarrow\mathfrak{Q}$ is a
morphism of $\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodules. Thus
$\mathfrak{P}/\mathfrak{T}\longrightarrow\mathfrak{Q}$ is a morphism of
$\mathcal{C}_{\mathsf{E}}$-contramodules as well. We can conclude that the
composition of two $\mathcal{C}_{\mathsf{E}}$-contramodule morphisms
$\mathfrak{P}\longrightarrow\mathfrak{P}/\mathfrak{T}\longrightarrow\mathfrak{Q}$
is also a $\mathcal{C}_{\mathsf{E}}$-contramodule morphism, finishing the
proof of the theorem. ∎

##  4\. Locally Finite Contramodules

The following definition is the contramodule version of Definition 2.6.

######  Definition 4.1.

Let $\mathsf{E}$ be a locally finite $k$-linear category. A left
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ is said to be _locally
finite_ if the corresponding left $\mathsf{E}$-module
$P=\Theta_{\mathsf{E}}(\mathfrak{P})$ (defined in Construction 3.1) is locally
finite. In other words, $\mathfrak{P}$ is locally finite if and only if the
vector space $\mathfrak{P}^{x}$ is finite-dimensional for every object
$x\in\mathsf{E}$. Locally finite right
$\mathcal{C}_{\mathsf{E}}$-contramodules are defined similarly.

The aim of this section is to prove the following contramodule version of
Theorem 2.7.

######  Theorem 4.2.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category and
$P\colon\mathsf{E}\longrightarrow k{\operatorname{\mathsf{--vect}}}$ be a
locally finite left $\mathsf{E}$-module. Then $P$ arises from some (locally
finite) left $\mathcal{C}_{\mathsf{E}}$-contramodule via the contramodule
forgetful functor $\Theta_{\mathsf{E}}$ if and only if, for every fixed object
$y\in\mathsf{E}$, the set of all objects $x\in\mathsf{E}$ with nonzero action
map

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP(y)$

is finite.

Notice that the functor $\Theta_{\mathsf{E}}$ is fully faithful in the
assumptions of Theorem 4.2, by Theorem 3.7. So the
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ such that
$\Theta_{\mathsf{E}}(\mathfrak{P})\simeq P$ is unique if it exists. Let us
also emphasize that the left strict local finiteness assumption on the
$k$-linear category $\mathsf{E}$ _cannot_ be simply dropped or replaced by the
local finiteness in Theorem 4.2. Indeed, _every_ left module $P$ over the
$k$-linear category $\mathsf{E}$ from Example 3.3 belongs to the essential
image of the functor $\Theta_{\mathsf{E}}$, but $P$ need not satisfy the
condition of Theorem 4.2 for the object $y=-1$.

###### Proof of the “if” implication in Theorem 4.2.

This implication requires _neither_ the left strict local finiteness
assumption on $\mathsf{E}$, _nor_ the local finiteness assumption on $P$. Let
$P$ be a (not necessarily locally finite) left $\mathsf{E}$-module such that
for every object $y\in\mathsf{E}$ there is a finite subset of objects
$Z_{y}\subset\mathsf{E}$ with the property that the action map
$\operatorname{Hom}_{\mathsf{E}}(z,y)\otimes_{k}P(z)\longrightarrow P(y)$
vanishes for all $z\in\mathsf{E}$, $z\notin Z_{y}$. Put
$\mathfrak{P}^{x}=P(x)$ and
$\mathfrak{P}=\prod_{x\in\mathsf{E}}\mathfrak{P}^{x}$, and define the
contraaction map
$\pi\colon\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}},\mathfrak{P})\longrightarrow\mathfrak{P}$
by the following rule. The map $\pi$ consists of the components

$\pi^{y}\colon\prod\nolimits_{x\in\mathsf{E}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathfrak{P}^{y},\qquad y\in\mathsf{E}$

as in (4). The map $\pi^{y}$ is the composition

(15)
$\prod\nolimits_{x\in\mathsf{E}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{x,y},\mathfrak{P}^{x})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\prod\nolimits_{z\in
Z_{y}}\operatorname{Hom}_{k}(\mathcal{C}_{\mathsf{E}}^{z,y},\mathfrak{P}^{z})\\\
\,\simeq\,\bigoplus\nolimits_{z\in
Z_{y}}\operatorname{Hom}_{\mathsf{E}}(z,y)\otimes_{k}P(z)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muP(y).$

Here the leftmost map in (15) is the direct summand projection from the
product of the components indexed by all the objects of $\mathsf{E}$ to the
product of the components indexed by the subset $Z_{y}\subset\mathsf{E}$. The
rightmost map in (15) is provided by the action of $\mathsf{E}$ in $P$. The
middle isomorphism holds since
$\operatorname{Hom}_{\mathsf{E}}(z,y)\simeq\bigl{(}\mathcal{C}_{\mathsf{E}}^{z,y}\bigr{)}^{*}$
is a finite-dimensional vector space and the set $Z_{y}$ is finite. This
defines a left $\mathcal{C}_{\mathsf{E}}$-contramodule structure on
$\mathfrak{P}$ such that $\Theta_{\mathsf{E}}(\mathfrak{P})=P$. ∎

Before proceeding to prove the “only if” implication, let us introduce a
further series of combinatorial definitions and formulate some lemmas.

######  Definition 4.3.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $x$,
$y\in\mathsf{E}$ be a pair of objects such that $x\preceq y$. The _distance_
$d(x,y)$ from $x$ to $y$ is defined as the supremum of the set of all integers
$n\geq 0$ for which there exists a sequence of objects $z_{1}$, …,
$z_{n-1}\in\mathsf{E}$ such that $x\prec z_{1}\prec z_{2}\prec\dotsb\prec
z_{n-1}\prec y$. In particular, $d(x,y)\geq 1$ whenever $x\prec y$. We put
$d(x,y)=0$ if $x\sim y$.

The point of this definition is that $d(x,y)$ is a finite integer for every
pair of objects $x\preceq y$, because the set of all objects $\\{z\mid
x\preceq z\preceq y\\}$ is finite by Definition 2.2(ii).

######  Definition 4.4.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $y\in\mathsf{E}$
be an object. Let $X\subset\mathsf{E}$ be a finite subset in $\mathsf{E}$ such
that $x\prec y$ for all $x\in X$. We will say that $X$ is a _frontier_ for $y$
if there is a finite subset of objects $W\subset\mathsf{E}$ such that, for for
every morphism $f\in\operatorname{Hom}_{\mathsf{E}}(z,y)$ with $z\prec y$ and
$z\notin W$, there exist two finite families of morphisms $g_{i}\colon
z\longrightarrow x_{i}$ and $h_{i}\colon x_{i}\longrightarrow y$, $1\leq i\leq
n$, $n\geq 0$, $x_{i}\in X$, for which $f=\sum\nolimits_{i=1}^{n}h_{i}g_{i}$.
The finite subset of objects $W\subset\mathsf{E}$ will be called the _frontier
exception_ for a frontier $X$ of an object $y$.

In particular, Definition 3.6 can be restated by saying that a locally finite
$k$-linear category $\mathsf{E}$ is left strictly locally finite if and only
every object $y\in\mathsf{E}$ admits a frontier. To be more precise,
Definition 3.6 requires existence of a frontier with an empty frontier
exception. An empty exception can be always achieved by expanding the
frontier. For any frontier $X$ of an object $y$ with a frontier exception $W$,
the union $X\cup(W\cap\\{z\in\mathsf{E}\mid z\prec y\\})$ is a frontier for
$y$ with an empty frontier exception.

######  Definition 4.5.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category.
Suppose chosen a finite subset of objects $X_{y}\subset\mathsf{E}$ for every
object $y\in\mathsf{E}$ as per Defintion 3.6. Let us call $X_{y}$ the
_standard frontier_ of $y$.

Put $X_{y}^{(1)}=X_{y}$. Proceeding by induction on $n\geq 1$, set
$X_{y}^{(n+1)}=\bigcup_{z\in X_{y}^{(n)}}X_{z}$. So $X_{y}^{(n)}$ is a finite
set of objects in $\mathsf{E}$ for every object $y\in\mathsf{E}$. Let us call
$X_{y}^{(n)}$ the _degree $n$ standard frontier_ of $y$.

######  Lemma 4.6.

Our terminology is consistent: for any object $y\in\mathsf{E}$ and any integer
$n\geq 1$, the degree $n$ standard frontier $X_{y}^{(n)}$ is a frontier for
$y$.

###### Proof.

Quite generally, if $X$ is a frontier for $y$ with a frontier exception $W$,
then $\bigcup_{z\in X}X_{z}$ is also a frontier for $y$ with the frontier
exception $W\cup X^{\sim}$, where $X^{\sim}$ is the set of all objects of
$\mathsf{E}$ that are $\sim$-equivalent to some objects from $X$. Since the
$\sim$-equivalence classes are finite by Defintion 3.6(ii), the set $X^{\sim}$
is finite. Hence $X_{y}^{(n)}$ is a frontier for $y$ with a frontier exception
$\bigcup_{i=1}^{n-1}X_{y}^{(n)\,\sim}$. ∎

The following definition assigns a name to the class of left
$\mathsf{E}$-modules we are interested in.

######  Definition 4.7.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $P$ be a left
$\mathsf{E}$-module. We will say that $P$ is _contrafinite_ if it satisfies
the condition of Theorem 4.2. To repeat, this means that, for every object
$y\in\mathsf{E}$, there exists a finite subset of objects
$A_{P,y}\subset\mathsf{E}$ such that, for all objects $x\in\mathsf{E}$,
$x\notin A_{P,y}$, the action map
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\longrightarrow P(y)$
vanishes.

It is clear that the class of all contrafinite left $\mathsf{E}$-modules is
closed under submodules, quotients, and finite direct sums. The next lemma
plays a key role.

######  Lemma 4.8.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category. Then
the class of all contrafinite left $\mathsf{E}$-modules is closed under
extensions in $\mathsf{E}{\operatorname{\mathsf{--Mod}}}$.

###### Proof.

Let $0\longrightarrow P\longrightarrow T\longrightarrow Q\longrightarrow 0$ be
a short exact sequence of left $\mathsf{E}$-modules. Assume that the
$\mathsf{E}$-modules $P$ and $Q$ are contrafinite. We have to prove that the
$\mathsf{E}$-module $T$ is contrafinite as well. Given an object
$y\in\mathsf{E}$, we need to establish existence of a finite subset of objects
$A_{T,y}\subset\mathsf{E}$.

Consider the degree $n$ standard frontier $X_{y}^{(n)}\subset\mathsf{E}$, and
let $W_{y}^{(n)}\subset\mathsf{E}$ be its frontier exception constructed in
the proof of Lemma 4.6. For any object $x\in X_{y}^{(n)}$, the distance
$d(x,y)$ is greater than or equal to $n$. As the distance between a pair of
objects $u$ and $v$ with $u\preceq v$ is always finite, we can choose $n$
large enough so that _no object of $A_{P,y}\cup A_{Q,y}$ belongs to
$X_{y}^{(n)}$_. Then, for any object $x\in X_{y}^{(n)}$, both the maps
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\longrightarrow P(y)$ and
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}Q(x)\longrightarrow Q(y)$
vanish. It follows that for any object $z\prec y$ with $z\notin W_{y}^{(n)}$,
both the maps
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}P(x)\longrightarrow P(y)$ and
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}Q(x)\longrightarrow Q(y)$
vanish; but we will not need to use this fact.

Now consider the degree $m$ standard frontier $X_{y}^{(m)}\subset\mathsf{E}$
with $m>n$. The set $B=\bigcup_{x\in X_{y}^{(n)}}(A_{P,x}\cup A_{Q,x})$ is
finite. For the same reason as above, we can choose $m$ large enough so that
_no object of $B$ belongs to $X_{y}^{(m)}$_. Then, for any objects $x\in
X_{y}^{(n)}$ and $z\in X_{y}^{(m)}$, both the maps
$\operatorname{Hom}_{\mathsf{E}}(z,x)\otimes_{k}P(z)\longrightarrow P(x)$ and
$\operatorname{Hom}_{\mathsf{E}}(z,x)\otimes_{k}Q(z)\longrightarrow Q(x)$
vanish. It follows that for any objects $x\in X_{y}^{(n)}$ and $u\prec x$ with
$u\notin W_{x}^{(m-n)}$, both the maps
$\operatorname{Hom}_{\mathsf{E}}(u,x)\otimes_{k}P(u)\longrightarrow P(x)$ and
$\operatorname{Hom}_{\mathsf{E}}(u,x)\otimes_{k}Q(u)\longrightarrow Q(x)$
vanish; now this fact is relevant for us.

Put $A_{T,y}=\\{y\\}^{\sim}\cup W_{y}^{(m)}$, where
$\\{y\\}^{\sim}\subset\mathsf{E}$ is the finite set of all objects
$\sim$-equivalent to $y$. Given an object $u\notin A_{T,y}$ and a nonzero
morphism $f\in\operatorname{Hom}_{\mathsf{E}}(u,y)$, we have to show that the
map $T(f)\colon T(u)\longrightarrow T(y)$ vanishes. Notice that we necessarily
have $u\prec y$ and $u\notin W_{y}^{(n)}$. Since $X_{y}^{(n)}$ is a frontier
for $y$ with a frontier exception $W_{y}^{(n)}$ by Lemma 4.6, the map $f$ can
be represented as

$f=\sum\nolimits_{i=1}^{r}h_{i}g_{i},$

where $g_{i}\in\operatorname{Hom}_{\mathsf{E}}(u,x_{i})$ and
$h_{x}\in\operatorname{Hom}_{\mathsf{E}}(x_{i},y)$, $1\leq i\leq r$, $r\geq
1$, and $x_{i}\in X_{y}^{(n)}$. Without loss of generality we can assume that
$g_{i}\neq 0$ for every $1\leq i\leq r$.

For a given index $i$, $1\leq i\leq r$, put $g=g_{i}$, $h=h_{i}$ and
$x=x_{i}$. The existence of a nonzero morphism $g\colon u\longrightarrow x$
and the condition that $u\notin W_{y}^{(m)}\supset X_{y}^{(n)\,\sim}$ imply
that $u\prec x$. Since $W_{x}^{(m-n)}\subset W_{y}^{(m)}$, we also have
$u\notin W_{x}^{(m-n)}$. Therefore, both the maps $P(g)\colon
P(u)\longrightarrow P(x)$ and $Q(g)\colon Q(u)\longrightarrow Q(x)$ vanish. So
do both the maps $P(h)\colon P(x)\longrightarrow P(y)$ and $Q(h)\colon
Q(x)\longrightarrow Q(y)$, as we have seen.

Finally, we can conclude that the map $T(hg)\colon T(u)\longrightarrow T(y)$
vanishes, as the product of any two strictly uppertriangular $2\times 2$ block
matrices is zero. Thus $T(f)=\sum_{i=1}^{r}T(h_{i}g_{i})=0$, as desired. ∎

Now the following sequence of lemmas leads to the proof of the remaining
(main) implication in Theorem 4.2. To begin with, here is another piece of
auxiliary terminology. Given a left $\mathsf{E}$-module $P$, we will say that
an $\mathsf{E}$-submodule $Q\subset P$ is _big_ if the quotient
$\mathsf{E}$-module $P/Q$ is contrafinite.

######  Lemma 4.9.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $P$ be a left
$\mathsf{E}$-module. Let $Q^{\prime}$ and $Q^{\prime\prime}\subset P$ be two
big $\mathsf{E}$-submodules in $P$. Then the intersection $Q^{\prime}\cap
Q^{\prime\prime}$ is also a big $\mathsf{E}$-submodule in $P$.

###### Proof.

The quotient module $P/(Q^{\prime}\cap Q^{\prime\prime})$ is a submodule of
the direct sum $P/Q^{\prime}\oplus P/Q^{\prime\prime}$. So the assertion
follows from the facts that the class of all contrafinite left
$\mathsf{E}$-modules is closed under submodules and finite direct sums in
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$. ∎

######  Lemma 4.10.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $P$ be a locally
finite left $\mathsf{E}$-module. Let $(Q_{\xi}\subset P)_{\xi\in\Xi}$ be a
family of big $\mathsf{E}$-submodules in $P$. Then the intersection
$\bigcap_{\xi\in\Xi}Q_{\xi}$ is also a big $\mathsf{E}$-submodule in $P$.

###### Proof.

Given a left $\mathsf{E}$-module $T$ and an object $y\in\mathsf{E}$, let us
say that the $\mathsf{E}$-module $T$ is _$y$ -contrafinite_ if a finite subset
of objects $A_{T,y}\subset\mathsf{E}$ satisfying the condition of Definition
4.7 exists (for one particular object $y$). Given a left $\mathsf{E}$-module
$P$, let us say that an $\mathsf{E}$-submodule $K\subset P$ is _$y$ -big_ if
the quotient $\mathsf{E}$-module $P/K$ is $y$-contrafinite. The key
observation is that, for a fixed $\mathsf{E}$-module $P$ and a fixed object
$y\in\mathsf{E}$, the property of an $\mathsf{E}$-submodule $K\subset P$ to be
$y$-big depends only on the vector subspace $K(y)\subset P(y)$ (and _not_ on
the vector subspaces $K(x)\subset P(x)$ for all the other objects $x\neq y$ in
$\mathsf{E}$).

Returning to the situation at hand, for every given object $y\in\mathsf{E}$,
the $k$-vector space $P(y)$ is finite-dimensional by assumption. Hence there
exists a finite subset of indices $\Upsilon_{y}\subset\Xi$ such that
$\bigcap_{\xi\in\Xi}Q_{\xi}(y)=\bigcap_{\upsilon\in\Upsilon_{y}}Q_{\upsilon}(y)$.
Now the $\mathsf{E}$-submodule
$\bigcap_{\upsilon\in\Upsilon_{y}}Q_{\upsilon}\subset P$ is big, and in
particular $y$-big, by Lemma 4.9. It follows that the $\mathsf{E}$-submodule
$\bigcap_{\xi\in\Xi}Q_{\xi}\subset P$ is $y$-big. As this holds for all
$y\in\mathsf{E}$, we can conclude that the $\mathsf{E}$-submodule
$\bigcap_{\xi\in\Xi}Q_{\xi}$ is big in $P$. ∎

######  Lemma 4.11.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $\mathfrak{P}$ be
a nonzero left $\mathcal{C}_{\mathsf{E}}$-contramodule. Then the left
$\mathsf{E}$-module $\Theta_{\mathsf{E}}(\mathfrak{P})$ has a proper big
$\mathsf{E}$-submodule.

###### Proof.

The argument is based on the discussion in Section 1.1 and Remark 3.4,
together with the contramodule Nakayama lemma (Lemma 1.1(b)). Let
$\rho_{\mathfrak{P}}\colon\operatorname{Hom}_{k}(\mathcal{D}_{\mathsf{E}}^{\prec},\allowbreak\mathfrak{P})\longrightarrow\mathfrak{P}$
denote the $\mathcal{D}_{\mathsf{E}}^{\prec}$-contraaction map. Following
Section 1.1, the vector subspace
$\mathfrak{Q}=\operatorname{im}(\rho_{\mathfrak{P}})\subset\mathfrak{P}$ is a
$\mathcal{C}_{\mathsf{E}}$-subcontramodule. Furthermore, the
$\mathcal{C}_{\mathsf{E}}$-contramodule structure on
$\mathfrak{P}/\mathfrak{Q}$ arises from a
$\mathcal{C}_{\mathsf{E}}^{\sim}$-contramodule structure. By Lemma 1.1(b),
$\mathfrak{Q}$ is a proper subcontramodule of $\mathfrak{P}$.

It remains to put $Q=\Theta_{\mathsf{E}}(\mathfrak{Q})$ and notice that $Q$ is
a proper $\mathsf{E}$-submodule of $P$ such that the quotient
$\mathsf{E}$-module $P/Q$ arises from an $\mathsf{E}^{\sim}$-module. It is
clear from the interval finiteness condition in Definition 2.2 that any left
$\mathsf{E}^{\sim}$-module is contrafinite (also as an $\mathsf{E}$-module).
So $P/Q$ is a nonzero contrafinite left $\mathsf{E}$-module and $Q$ is a
proper big $\mathsf{E}$-submodule in $P$. ∎

###### Proof of the “only if” implication in Theorem 4.2.

Let $\mathfrak{P}$ be a locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodule. In the terminology of Definition 4.7,
we need to show that the left $\mathsf{E}$-module
$P=\Theta_{\mathsf{E}}(\mathfrak{P})$ is contrafinite.

By Lemma 4.10, the intersection of all big $\mathsf{E}$-submodules in $P$ is a
big $\mathsf{E}$-submodule $K\subset P$. So $K$ is the unique minimal big
$\mathsf{E}$-submodule in $P$. The quotient $\mathsf{E}$-module $Q=P/K$ is
locally finite and contrafinite. In order to prove that $P$ is contrafinite,
we will show that $K=0$.

By the “if” implication of the theorem (which we have proved already in the
beginning of this section), the left $\mathsf{E}$-module $Q$ arises from some
left $\mathsf{E}$-contramodule $\mathfrak{Q}$; so
$Q=\Theta_{\mathsf{E}}(\mathfrak{Q})$. As the functor $\Theta_{\mathsf{E}}$ is
(exact and) fully faithful by Theorem 3.7, the surjective $\mathsf{E}$-module
morphism $P\longrightarrow Q$ arises from a surjective
$\mathcal{C}_{\mathsf{E}}$-contramodule morphism
$\mathfrak{P}\longrightarrow\mathfrak{Q}$. Put
$\mathfrak{K}=\ker(\mathfrak{P}\to\mathfrak{Q})$. Then
$K=\Theta_{\mathsf{E}}(\mathfrak{K})$; so the $\mathsf{E}$-submodule $K\subset
P$ also arises from a $\mathcal{C}_{\mathsf{E}}$-subcontramodule
$\mathfrak{K}\subset\mathfrak{P}$.

Assume for the sake of contradiction that $K\neq 0$. Then Lemma 4.11 tells
that the left $\mathsf{E}$-module $K$ has a proper big $\mathsf{E}$-submodule
$L\subset K$. Now we have a short exact sequence of left $\mathsf{E}$-modules
$0\longrightarrow K/L\longrightarrow P/L\longrightarrow P/K=Q\longrightarrow
0$. The $\mathsf{E}$-submodule $L$ is big in $K$, and the
$\mathsf{E}$-submodule $K$ is big in $P$; so both the $\mathsf{E}$-modules
$K/L$ and $Q$ are contrafinite. By Lemma 4.8, it follows that the
$\mathsf{E}$-module $P/L$ is contrafinite, too. So $L$ is a big
$\mathsf{E}$-submodule in $P$.

But $L$ is a proper $\mathsf{E}$-submodule of $K$, and $K$ is the (unique)
minimal big $\mathsf{E}$-submodule in $P$ by construction. The contradiction
proves that $K=0$ and the left $\mathsf{E}$-module $P$ is contrafinite. ∎

######  Question 4.12.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category and
$\mathfrak{P}$ be a left $\mathcal{C}_{\mathsf{E}}$-contramodule. Fix an
object $y\in\mathsf{E}$ and assume that the $k$-vector space
$\mathfrak{P}^{y}$ (for one particular object $y$) is finite-dimensional. Put
$P=\Theta_{\mathsf{E}}(\mathfrak{P})$. Does it follow that there exists a
finite set of objects $A_{P,y}\subset\mathsf{E}$ satisfying the condition of
Definition 4.7 (for the particular object $y$)? In other words, is the left
$\mathsf{E}$-module $P$ necessarily $y$-contrafinite in the sense of the
terminology in the proof of Lemma 4.10 ? Our proof of the “only if”
implication in Theorem 4.2 does not seem to answer this question.

##  5\. Vector Space Duality

Let $\mathsf{E}$ be a $k$-linear category and
$N\colon\mathsf{E}^{\mathsf{op}}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$ be a right $\mathsf{E}$-module. Then the
rule $P(x)=N(x)^{*}=\operatorname{Hom}_{k}(N(x),k)$ defines a left
$\mathsf{E}$-module $P=N^{*}\colon\mathsf{E}\longrightarrow
k{\operatorname{\mathsf{--Vect}}}$.

On the other hand, let $\mathcal{C}$ be a coalgebra over $k$ and $\mathcal{N}$
be a right $\mathcal{C}$-comodule. Then the construction from Section 1
defines a left $\mathcal{C}$-contramodule structure on the $k$-vector space
$\mathfrak{P}=\mathcal{N}^{*}=\operatorname{Hom}_{k}(\mathcal{N},k)$.

Now let $\mathsf{E}$ be a locally finite $k$-linear category, and let
$\mathcal{C}_{\mathsf{E}}$ be the related coalgebra as per Construction 2.3.
Then we have a commutative square of exact, faithful functors

(16) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
34.38895pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{({\operatorname{\mathsf{Comod--}}}\mathcal{C}_{\mathsf{E}})^{\mathsf{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.13895pt\raise 5.85695pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.85695pt\hbox{$\scriptstyle{\mathcal{N}\mapsto\mathcal{N}^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 88.38895pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
115.39175pt\raise-20.60277pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.70833pt\hbox{$\scriptstyle{\Theta_{\mathsf{E}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
115.39175pt\raise-31.2611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{({\operatorname{\mathsf{Mod--}}}\mathsf{E})^{\mathsf{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.73062pt\raise-35.34859pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.85695pt\hbox{$\scriptstyle{N\mapsto N^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
95.65561pt\raise-41.20554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.38895pt\raise-41.20554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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95.65561pt\raise-41.20554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{E}{\operatorname{\mathsf{--Mod}}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

Here the functor $\Upsilon_{\mathsf{E}^{\mathsf{op}}}$ is the right (co)module
version of the functor $\Upsilon_{\mathsf{E}}$ from Construction 2.4, while
the functor $\Theta_{\mathsf{E}}$ was defined in Construction 3.1.

Notice that a $\mathcal{C}_{\mathsf{E}}$-comodule $\mathcal{N}$ is a direct
sum of its components,
$\mathcal{N}=\bigoplus_{x\in\mathsf{E}}\mathcal{N}_{x}$, while a
$\mathcal{C}_{\mathsf{E}}$-contramodule $\mathfrak{P}$ is a direct product of
its components, $\mathfrak{P}=\prod_{x\in\mathsf{E}}\mathfrak{P}^{x}$. The
diagram (16) is commutative due to the fact that the vector space dualization
functor $\mathcal{N}\longmapsto\mathcal{N}^{*}$ takes the direct sums to the
direct products.

Let us introduce some further notation. Denote the full subcategories of
locally finite $\mathsf{E}$-modules (see Definition 2.6) by
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}_{\mathsf{lf}}\subset\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
and
${\operatorname{\mathsf{Mod_{\mathsf{lf}}--}}}\mathsf{E}\subset{\operatorname{\mathsf{Mod--}}}\mathsf{E}$.
It is clear that the functor $N\longmapsto P=N^{*}$ restricts to a anti-
equivalence of categories

(17)
$({\operatorname{\mathsf{Mod_{\mathsf{lf}}--}}}\mathsf{E})^{\mathsf{op}}\xrightarrow[\,\,N\mapsto
N^{*}\,]{\simeq}\mathsf{E}{\operatorname{\mathsf{--Mod}}}_{\mathsf{lf}}.$

This follows simply from the fact that the dual vector space functor is an
anti-equivalence on the category of finite-dimensional vector spaces,
$(k{\operatorname{\mathsf{--vect}}})^{\mathsf{op}}\simeq
k{\operatorname{\mathsf{--vect}}}$.

Furthermore, denote the full subcategories of locally finite
$\mathcal{C}_{\mathsf{E}}$-comodules by
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}_{\mathsf{lf}}\subset\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}$
and
${\operatorname{\mathsf{Comod_{\mathsf{lf}}--}}}\mathcal{C}_{\mathsf{E}}\subset{\operatorname{\mathsf{Comod--}}}\mathcal{C}_{\mathsf{E}}$.
Similarly, let the full subcategory of locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodules (see Definition 4.1) be denoted by
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}_{\mathsf{lf}}\subset\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}$.
It is clear that the functor
$\mathcal{N}\longmapsto\mathfrak{P}=\mathcal{N}^{*}$ restricts to an exact,
faithful functor

(18)
$({\operatorname{\mathsf{Comod_{\mathsf{lf}}--}}}\mathcal{C}_{\mathsf{E}})^{\mathsf{op}}\xrightarrow{\,\,\mathcal{N}\mapsto\mathcal{N}^{*}\,}\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}_{\mathsf{lf}}.$

In fact, one can say more.

######  Proposition 5.1.

For any locally finite $k$-linear category $\mathsf{E}$, the contravariant
functor
$({\operatorname{\mathsf{Comod_{\mathsf{lf}}--}}}\mathcal{C}_{\mathsf{E}})^{\mathsf{op}}\longrightarrow\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}_{\mathsf{lf}}$
(18) is fully faithful.

###### Proof.

For any coalgebra $\mathcal{C}$ over $k$, a $k$-vector space map
$f\colon\mathcal{M}\longrightarrow\mathcal{N}$ between two right comodules
$\mathcal{M}$, $\mathcal{N}$ over $\mathcal{C}$ is a $\mathcal{C}$-comodule
morphism if and only if the dual map
$f^{*}\colon\mathcal{N}^{*}\longrightarrow\mathcal{M}^{*}$ is a left
$\mathcal{C}$-contramodule morphism. Not all $\mathcal{C}$-contramodule
morphisms $g\colon\mathcal{N}^{*}\longrightarrow\mathcal{M}^{*}$, viewed as
$k$-vector space maps, arise from $k$-vector space maps
$\mathcal{M}\longrightarrow\mathcal{N}$ in general. But in the situation at
hand they do, because all morphisms
$g\colon\mathfrak{P}\longrightarrow\mathfrak{Q}$ in
$\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}$ respect the
decomposition into the product of the components,
$\mathfrak{P}=\prod_{x\in\mathsf{E}}\mathfrak{P}^{x}$ and
$\mathfrak{Q}=\prod_{x\in\mathsf{E}}\mathfrak{Q}^{x}$, and the components
$\mathcal{M}_{x}$ and $\mathcal{N}_{x}$ of all comodules $\mathcal{M}$,
$\mathcal{N}\in{\operatorname{\mathsf{Comod_{\mathsf{lf}}--}}}\mathcal{C}_{\mathsf{E}}$
are finite-dimensional by the definition. ∎

Before stating a corollary (of Theorems 2.7 and 4.2), let us formulate
explicitly the right (co)module version of Theorem 2.7.

######  Proposition 5.2.

Let $\mathsf{E}$ be a locally finite $k$-linear category and
$N\colon\mathsf{E}^{\mathsf{op}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ be a locally finite right
$\mathsf{E}$-module. Then $N$ arises from some (locally finite) right
$\mathcal{C}_{\mathsf{E}}$-comodule via the comodule inclusion functor
$\Upsilon_{\mathsf{E}^{\mathsf{op}}}$ if and only if, for every fixed object
$y\in\mathsf{E}$, the set of all objects $x\in\mathsf{E}$ with nonzero action
map

$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}N(y)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5muN(x)$

is finite.

###### Proof.

Apply Theorem 2.7 to the locally finite $k$-linear category
$\mathsf{E}^{\mathsf{op}}$ in place of $\mathsf{E}$. ∎

The following corollary is the main result of this section.

######  Corollary 5.3.

For any left strictly locally finite $k$-linear category $\mathsf{E}$, the
vector space dualization functor (18) is anti-equivalence between the abelian
category of locally finite right $\mathcal{C}_{\mathsf{E}}$-comodules and the
abelian category of locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodules,

$({\operatorname{\mathsf{Comod_{\mathsf{lf}}--}}}\mathcal{C}_{\mathsf{E}})^{\mathsf{op}}\xrightarrow[\,\,\mathcal{N}\mapsto\mathcal{N}^{*}\,]{\simeq}\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Contra}}}_{\mathsf{lf}}.$

###### Proof.

Compare the descriptions of locally finite right
$\mathcal{C}_{\mathsf{E}}$-comodules and locally finite left
$\mathcal{C}_{\mathsf{E}}$-contramodules provided by Proposition 5.2 and
Theorem 4.2, and take into account the anti-equivalence of module categories
(17) together with commutativity of the dualization diagram (16). ∎

######  Remark 5.4.

The theory of finite-dimensional comodules and contramodules over arbitrary
coalgebras $\mathcal{C}$ over $k$ is quite different from the theory of
locally finite comodules and contramodules over the coalgebras
$\mathcal{C}_{\mathsf{E}}$.

Let
$\mathcal{C}{\operatorname{\mathsf{--comod}}}\subset\mathcal{C}{\operatorname{\mathsf{--Comod}}}$,
${\operatorname{\mathsf{comod--}}}\mathcal{C}\subset{\operatorname{\mathsf{Comod--}}}\mathcal{C}$,
and
$\mathcal{C}{\operatorname{\mathsf{--contra}}}\subset\mathcal{C}{\operatorname{\mathsf{--Contra}}}$
denote the full subcategories of finite-dimensional comodules and
contramodules. Then the vector space dualization functor
$\mathcal{N}\longmapsto\mathcal{M}=\mathcal{N}^{*}$ provides an anti-
equivalence between the categories of finite-dimensional left and right
$\mathcal{C}$-comodules,

(19)
$({\operatorname{\mathsf{comod--}}}\mathcal{C})^{\mathsf{op}}\xrightarrow[\,\,\mathcal{N}\mapsto\mathcal{N}^{*}\,]{\simeq}\mathcal{C}{\operatorname{\mathsf{--comod}}}.$

There is also the exact, fully faithful dualization functor

(20)
$({\operatorname{\mathsf{comod--}}}\mathcal{C})^{\mathsf{op}}\xrightarrow{\,\,\mathcal{N}\mapsto\mathcal{N}^{*}\,}\mathcal{C}{\operatorname{\mathsf{--contra}}}.$

Consequently, any finite-dimensional left $\mathcal{C}$-comodule acquires a
left $\mathcal{C}$-contramodule structure, so there is an exact, fully
faithful functor $\mathcal{M}\longmapsto\mathfrak{P}=\mathcal{M}$ defined on
finite-dimensional comodules [7, Section A.1.2], [12, Section 8.5],

(21)
$\mathcal{C}{\operatorname{\mathsf{--comod}}}\xrightarrow{\,\,\mathcal{M}\mapsto\mathcal{M}\,}\mathcal{C}{\operatorname{\mathsf{--contra}}}.$

The functors (20) and (21) are _not_ essentially surjective for an infinite-
dimensional coalgebra $\mathcal{C}$ in general [7, Section A.1.2]. Comparing
the discussion of irreducible comodules and contramodules in [7, Section
A.1.2], [12, Section 8.5] with the discussion of the Ext spaces for finite-
dimensional comodules and contramodules in [7, Section A.1.2], [13, proof of
Proposition 3.3], one can see that, for a conilpotent coalgebra $\mathcal{C}$,
the functors (20–21) are essentially surjective if and only if the coalgebra
$\mathcal{C}$ is finitely cogenerated (cf. Remark 3.2).

For comparison, _no functors similar to ( 19) or (21) exist_ for locally
finite comodules and contramodules over the coalgebra
$\mathcal{C}_{\mathsf{E}}$ corresponding to a left/right strictly locally
finite $k$-linear category $\mathsf{E}$ in general, as one can easily observe
in the case of the category $\mathsf{E}$ from Example 7.1 and Remark 7.2
below. On the other hand, Example 3.3 shows that the left strict local
finiteness condition on the category $\mathsf{E}$ _cannot_ be replaced by the
weaker condition of local finiteness of $\mathsf{E}$ in Corollary 5.3.

##  6\. Closedness Under Extensions

We start with an obvious corollary of the arguments in Section 4, particularly
of Theorem 4.2 and Lemma 4.8.

######  Corollary 6.1.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category. Then
the class of all left $\mathsf{E}$-modules of the form
$\Theta_{\mathsf{E}}(\mathfrak{P})$, where $\mathfrak{P}$ ranges over the
locally finite left $\mathcal{C}_{\mathsf{E}}$-contramodules, is closed under
extensions in the abelian category of (locally finite) left
$\mathsf{E}$-modules.

###### Proof.

It is obvious that any extension of two locally finite $\mathsf{E}$-modules is
locally finite. Theorem 4.2 tells that a locally finite left
$\mathsf{E}$-module $P$ has the form $\Theta_{\mathsf{E}}(\mathfrak{P})$ for
some (locally finite) left $\mathsf{E}$-contramodule $\mathfrak{P}$ if and
only if $P$ is contrafinite in the sense of Definition 4.7. By Lemma 4.8, any
extension of two contrafinite left $\mathsf{E}$-modules is contrafinite. ∎

The aim of this section is to prove a comodule version of Corollary 6.1, which
does not require the local finiteness assumption on the (co)modules. The
following definition is a “comodule” version of Definition 4.7.

######  Definition 6.2.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $N$ be a right
$\mathsf{E}$-module. We will say that $N$ is _cofinite_ if it satisfies the
condition of Proposition 5.2. To repeat, this means that, for every object
$y\in\mathsf{E}$, there is a finite subset of objects
$A_{N,y}\subset\mathsf{E}$ such that, for all objects $x\in\mathsf{E}$,
$x\notin A_{N,y}$, the action map
$\operatorname{Hom}_{\mathsf{E}}(x,y)\otimes_{k}N(y)\longrightarrow N(x)$
vanishes.

######  Lemma 6.3.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category. Then
the class of all cofinite right $\mathsf{E}$-modules is closed under
extensions in ${\operatorname{\mathsf{Mod--}}}\mathsf{E}$.

###### Proof.

This is the dual version of Lemma 4.8, provable by exactly the same argument
with the action maps going in the opposite direction. Alternatively, one can
reduce the problem at hand to the assertion of Lemma 4.8 using the vector
space dualization functor $N\longmapsto N^{*}$ of Section 5. One easily
observes that a right $\mathsf{E}$-module $N$ is cofinite if and only if the
left $\mathsf{E}$-module $P=N^{*}$ is contrafinite. Given a short exact
sequence $0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow
0$ of right $\mathsf{E}$-modules, consider the short exact sequence of left
$\mathsf{E}$-modules $0\longrightarrow N^{*}\longrightarrow
M^{*}\longrightarrow L^{*}\longrightarrow 0$. If the right
$\mathsf{E}$-modules $L$ and $N$ are cofinite, the left $\mathsf{E}$-modules
$N^{*}$ and $L^{*}$ are contrafinite, and Lemma 4.8 tells that the left
$\mathsf{E}$-module $M^{*}$ is contrafinite. Hence the right
$\mathsf{E}$-module $M$ is cofinite. ∎

######  Corollary 6.4.

Let $\mathsf{E}$ be a left strictly locally finite $k$-linear category. Then
the class of all right $\mathsf{E}$-modules of the form
$\Upsilon_{\mathsf{E}^{\mathsf{op}}}(\mathcal{N})$, where $\mathcal{N}$ ranges
over the locally finite right $\mathcal{C}_{\mathsf{E}}$-comodules, is closed
under extensions in the abelian category of (locally finite) right
$\mathsf{E}$-modules.

###### Proof.

Similar to the proof of Corollary 6.4, and based on Proposition 5.2 and Lemma
6.3. ∎

######  Remark 6.5.

Let $\mathsf{E}$ be a locally finite $k$-linear category and $M$ be a left
$\mathsf{E}$-module. Then $M$ arises from a left
$\mathcal{C}_{\mathsf{E}}$-comodule via the comodule inclusion functor
$\Upsilon_{\mathsf{E}}$ if and only if, for every object $x\in\mathsf{E}$, the
action map

$M(x)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{y\in\mathsf{E}}\operatorname{Hom}_{k}(\operatorname{Hom}_{\mathsf{E}}(x,y),M(y))$

factorizes through the vector subspace

$\bigoplus\nolimits_{y\in\mathsf{E}}\operatorname{Hom}_{k}(\operatorname{Hom}_{\mathsf{E}}(x,y),M(y))\,\subset\,\prod\nolimits_{y\in\mathsf{E}}\operatorname{Hom}_{k}(\operatorname{Hom}_{\mathsf{E}}(x,y),M(y)),$

that is

$M(x)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus\nolimits_{y\in\mathsf{E}}\operatorname{Hom}_{k}(\operatorname{Hom}_{\mathsf{E}}(x,y),M(y)),$

as in (3). It follows that the essential image of the functor
$\Upsilon_{\mathsf{E}}\colon\mathcal{C}_{\mathsf{E}}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathsf{E}{\operatorname{\mathsf{--Mod}}}$
is closed under subobjects, quotients, direct sums, and direct limits in
$\mathsf{E}{\operatorname{\mathsf{--Mod}}}$. Similarly, the essential image of
the functor
$\Upsilon_{\mathsf{E}^{\mathsf{op}}}\colon{\operatorname{\mathsf{Comod--}}}\mathcal{C}_{\mathsf{E}}\longrightarrow{\operatorname{\mathsf{Mod--}}}\mathsf{E}$
is closed under subobjects, quotients, direct sums, and direct limits in
${\operatorname{\mathsf{Mod--}}}\mathsf{E}$.

More generally, for any coalgebra $\mathcal{C}$ over $k$ and any dense subring
$R\subset\mathcal{C}^{*}$, the essential image of the fully faithful
composition of inclusion/forgetful functors
$\mathcal{C}{\operatorname{\mathsf{--Comod}}}\longrightarrow\mathcal{C}^{*}{\operatorname{\mathsf{--Mod}}}\longrightarrow
R{\operatorname{\mathsf{--Mod}}}$ is closed under submodules, quotients,
direct sums, and direct limits in $R{\operatorname{\mathsf{--Mod}}}$.

######  Lemma 6.6.

Let $\mathsf{E}$ be a locally finite $k$-linear category. Then every
$\mathsf{E}$-module is the direct limit of its locally finite
$\mathsf{E}$-submodules.

###### Proof.

Given an object $y\in\mathsf{E}$, the _free right $\mathsf{E}$-module with one
generator sitting at $y$_ is defined as the functor
$\mathsf{E}^{\mathsf{op}}\longrightarrow k{\operatorname{\mathsf{--Vect}}}$
represented by the object $y$. In other words, the free right
$\mathsf{E}$-module $F_{y}$ has the components
$F_{y}(x)=\operatorname{Hom}_{\mathsf{E}}(x,y)$ and the action maps induced by
the multiplication/composition of morphisms in $\mathsf{E}$. A right
$\mathsf{E}$-module is said to be _finitely generated_ if it is a quotient of
a finitely generated free right $\mathsf{E}$-module, i. e., a quotient of a
finite direct sum of right $\mathsf{E}$-modules of the form $F_{y}$,
$y\in\mathsf{E}$. The point is that every $\mathsf{E}$-module is the direct
limit of its finitely generated $\mathsf{E}$-submodules. This holds over any
$k$-linear category $\mathsf{E}$.

When $\mathsf{E}$ is locally finite, all finitely generated