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\title{Notes on Tannakian duality}

%\author{Yifei Zhao}
\date{\today}
%\email{yifei@math.harvard.edu}

\begin{document}
\maketitle

%\begin{abstract}
%\end{abstract}

\setcounter{tocdepth}{2}
\tableofcontents


\newpage

\subsubsection*{Convention} For a fixed (commutative) ring $k$, we use the term \emph{tensor category} to refer to a $k$-linear symmetric monoidal category $(\cal C, \otimes, \mathbf 1)$. A $k$-linear symmetric monoidal functor between such will be called a \emph{tensor functor}. A mapping between tensor functors is a natural transformation that is compatible with the $k$-linear symmetric monoidal structures.

\subsubsection*{Summary}
These notes concern two kinds of objects: affine category schemes $\cal X$ (= category objects in affine schemes) and triples $(\cal C, R, F)$, where:
\begin{enumerate}
	\item $\cal C$ is a (small) tensor category;
	\item $R$ is a (commutative) $k$-algebra;
	\item $F$ is a tensor functor from $\cal C$ to the tensor category $R\Mod^{\Box}$ of dualizable $R$-modules.
\end{enumerate}
We do not define a ``Tannakian category.'' Whatever this notion is, it should be a triple $(\cal C, R, F)$ as above satisfying certain conditions.

We explain two constructions. Given an affine category scheme $\cal X$, we associate to it a triple where $\cal C$ is the category of ``finite-rank representations'' of $\cal X$, and $F$ is the forgetful functor of ``taking underlying vector space.'' Conversely, given a triple $(\cal C, R, F)$, we associate to it an affine category scheme $\cal X$.

Tannakian reconstruction poses the following question: give necessary and sufficient conditions on affine category schemes $\cal X$ and triples $(\cal C, R, F)$ for the above constructions to be inverses of each other. These notes do not contain a satisfactory answer to this question. Instead, we give:
\begin{enumerate}
	\item a sufficient condition on affine category schemes to be reconstructed. This includes transitive affine groupoids (as studied in [Deligne, Cat\'egories Tannakiennes]), affine monoids over a field, and some non-transitive groupoids (whose representation category is not abelian).
	\item a sufficient condition on $(\cal C, R, F)$ for the \emph{large category} $\tn{Ind}(\cal C)$ to be reconstructed. This condition involves abelian-ness of $\cal C$ and seems rather restrictive.
\end{enumerate}
\noindent
We have not yet investigated in any detail the reconstruction of the small category $\cal C$ itself.




%%

%%


\newpage

%%

%%

\section{Affine category schemes}

We fix a ground ring $k$. In this section, we define coalgebroids and affine category schemes, their geometric counterparts. Of particular importance is the category of finite-rank representations $\cal X\Rep^f$ of an affine category schemes $\cal X$.

The main result of this section is that an affine category scheme $\cal X$, whose regular representation is a filtered colimit of finite-rank ones, can be reconstructed from $\cal X\Rep^f$.

\subsection{Coalgebroids}

\subsubsection{} A \emph{coalgebroid} (over $k$) consists of a commutative $k$-algebra $R^0$, a commutative $(R^0\otimes R^0)$-algebra $R^1$ where we view the two $R^0$-structures as multiplications on the left and right, together with the following additional data:
\begin{enumerate}
	\item an $(R^0\otimes R^0)$-algebra map $\Delta : R^1 \rightarrow R^1 \underset{R^0}{\otimes} R^1$, called \emph{comultiplication};
	\item an $(R^0\otimes R^0)$-algebra map $\varepsilon : R^1 \rightarrow R^0$, called \emph{counit},
\end{enumerate}
such that the associative and unital conditions are satisfied.

%%

\subsubsection{}
\label{sec-category-scheme}
A coalgebroid is the dual notion of an affine category scheme (i.e., a category object in affine schemes).

Indeed, writing $X^0 := \Spec(R^0)$ and $X^1 := \Spec(R^1)$, we obtain a diagram:
$$
\xysmall{
	X^1 \ar@<0.5ex>[r]^-{\tn{pr}_1} \ar@<-0.5ex>[r]_-{\tn{pr}_2} & X^0.
}
$$
Here, the projection $\tn{pr}_1$ (resp.~$\tn{pr}_2$) corresponds to the left (resp.~right) $R^0$-algebra structure on $R^1$. Furthermore, we obtain a composition law $m$ and a unit $e$, satisfying associative and unital conditions. The composition law can be conveniently expressed by the following diagram:
$$
\xymatrix@!=1em{
	& & X^1 \underset{X^0}{\times} X^1 \ar[dl]_{\pi_1}\ar[dr]^{\pi_2} \ar[rr]^-{m} &  & X^1 \ar[ddllll]_{\tn{pr}_1}\ar[dd]^{\tn{pr}_2} \\
	& X^1 \ar[dl]_{\tn{pr}_1} \ar[dr]^{\tn{pr}_2} & \Box & X^1 \ar[dl]_{\tn{pr}_1} \ar[dr]^{\tn{pr}_2} \\
	X^0 & & X^0 & & X^0
}
$$
where the middle square is Cartesian. The unit can be expressed by the following diagram:
$$
\xysmall@!=2em{
	X^0 \ar[r]^e\ar[d]_{\tn{id}}\ar[drr]_{\tn{id}} & X^1 \ar[dl]_{\tn{pr}_1}\ar[dr]^{\tn{pr}_2} \\
	X^0 & & X^0
}
$$

\begin{rem}
Informally, we view a point of $X^0$ as an object $x$ and a point of $X^1$ as an arrow $f : x \rightarrow x'$ where $x = \tn{pr}_1(f)$ and $x' = \tn{pr}_2(f)$.
\end{rem}

%%

\subsubsection{} Given a coalgebroid $R^{\bullet} = (R^0, R^1, \Delta, \varepsilon)$, an \emph{$\cal R$-comodule} consists of:
\begin{enumerate}
	\item an $R^0$-module $\cal M$;
	\item an $R^0$-module map $\rho : \cal M \rightarrow R^1 \underset{R^0}{\otimes} \cal M$ (for the second object, $R^0$ acts on $R^1$ on the left), called \emph{coaction},
\end{enumerate}
such that the associative and unital conditions are satisfied; namely, the following two compositions are equal:
$$
\xymatrix@C=3em{
	\cal M \ar[r]^-{\rho} & R^1 \underset{R^0}{\otimes} \cal M \ar@<0.5ex>[r]^-{\Delta\otimes\tn{id}_{\cal M}} \ar@<-0.5ex>[r]_-{\tn{id}_{R^1}\otimes\rho} & R^1 \underset{R^0}{\otimes} R^1  \underset{R^0}{\otimes}\cal M,
}
$$
and the following composition is $\tn{id}_{\cal M}$:
$$
\xymatrix@C=2.5em{
\cal M \ar[r]^-{\rho} & R^1 \underset{R^0}{\otimes} \cal M \ar[r]^-{\varepsilon \otimes \tn{id}_{\cal M}} & \cal M.
}
$$
A morphism of $\cal R$-comodules is a map $\varphi : \cal M \rightarrow \cal M'$ of $R^0$-modules making the following diagram commute:
$$
\xymatrix@R=1.5em{
	\cal M \ar[r]^-{\varphi}\ar[d]^{\rho_{\cal M}} & \cal M' \ar[d]^{\rho_{\cal M'}} \\
	R^1  \underset{R^0}{\otimes}\cal M \ar[r]^-{\tn{id}_{R^1}\otimes\varphi} & R^1  \underset{R^0}{\otimes}\cal M'
}
$$
Thus $\cal R$-comodules form a category, denoted by $\cal R\Comod$.

In terms of the geometric description in \S\ref{sec-category-scheme}, the datum $\rho$ is equivalent to a morphism:
$$
\rho : \cal M \rightarrow (\tn{pr}_1)_*\tn{pr}_2^*\cal M,
$$
which is compatible with $m$ and $e$ in the obvious sense. One can alternatively view $\rho$ as a morphism $\tn{pr}_1^*\cal M \rightarrow \tn{pr}_2^*\cal M$ by adjunction.

\begin{rem}
Informally, the datum of an $\cal R$-comodule consists a module $\cal M_x$ for any point $x$ of $X^0$ and a morphism $f_* : \cal M_{x} \rightarrow \cal M_{x'}$ for every point $f : x \rightarrow x'$ of $X^1$.
\end{rem}

%%

\subsubsection{}
\label{sec-induced-rep}
The comultiplication $\Delta$ equips the $R^0$-module $R^1$ (via left multiplication) the structure of an $\cal R$-comodule. We refer to it as the \emph{regular representation}.

More generally, given any $R^0$-module $\cal N$, the $R^0$-module $R^1\underset{R^0}{\otimes} \cal N$ together with comultiplication $\Delta \otimes \tn{id}_{\cal N}$ forms an $\cal R$-comodule. We call this the coinduced comodule of $\cal N$ and denote it by $\tn{Coind}(\cal N)$.

\begin{lem}
\label{lem-comodule-adjunction}
The following pair of functors forms an adjunction:
$$
\xysmall{
	\tn{oblv} : \cal R\Comod \ar@<0.5ex>[r] & R^0\Mod : \tn{Coind}, \ar@<0.5ex>[l]
}
$$
where $\tn{oblv}$ is conservative and $\tn{Coind}$ commutes with all colimits.
\end{lem}
\begin{proof}
Given an $\cal R$-comodule $\cal M$ and an $R^0$-module $\cal N$, we construct a natural bijection:
$$
\tn{Hom}_{R^0\Mod}(\cal M, \cal N) \xrightarrow{\sim} \tn{Hom}_{\cal R\Comod}(\cal M, R^1 \underset{R^0}{\otimes} \cal N)
$$
as follows. It associates to an $R^0$-module map $\varphi : \cal M \rightarrow \cal N$ the morphism:
$$
\cal M \xrightarrow{\rho_{\cal M}} R^1 \underset{R^0}{\otimes} \cal M \xrightarrow{\tn{id}_{R^1}\otimes\varphi} R^1 \underset{R^0}{\otimes} \cal M,
$$
where $\rho_{\cal M}$ is the $\cal R$-comodule structure on $\cal M$. One checks that this is a map of $\cal R$-comodules using the compatibility of $\rho_{\cal M}$ with comultiplication $\Delta$. The inverse of this construction associates to every $\cal R$-comodule map $\cal M \rightarrow R^1\underset{R^0}{\otimes} \cal N$ its post-composition with $\varepsilon\otimes\tn{id}_{\cal N} : R^1\underset{R^0}{\otimes} \cal N \rightarrow \cal N$.

The fact that $\tn{oblv}$ is conservative is obvious from the definition. Thus, in order to check that $\tn{Coind}$ preserves all colimits, it suffices to check that $\tn{oblv}\circ\tn{Coind}$ preserves all colimits. The latter follows from the fact that the tensor operation $R^1\underset{R^0}{\otimes}- : R^0\Mod \rightarrow R^0\Mod$ commutes with all colimits.
\end{proof}

%%

\subsubsection{}
We will now characterize the compact objects in $\cal R\Comod$.

\begin{lem}
\label{lem-compact-comodules}
Let $\cal R = (R^0, R^1, \Delta, \varepsilon)$ be a coalgebroid. The following are equivalent for any object $\cal M \in \cal R\Comod$:
\begin{enumerate}
	\item $\cal M$ is compact;
	\item $\tn{oblv}(\cal M) \in R^0\Mod$ is compact.
\end{enumerate}
\end{lem}
\noindent
The second condition means concretely that $\tn{oblv}(\cal M)$ is a finitely presented $R^0$-module.
\begin{proof}[(1) $\implies$ (2)]
By Lemma \ref{lem-comodule-adjunction}, $\tn{oblv}$ admits a right adjoint sending $\cal N$ to the $\cal R$-comodule $R^1\underset{R^0}{\otimes} \cal N$. Since this functor commutes with filtered colimits, the left adjoint $\tn{oblv}$ preserves compact objects.
\end{proof}

\begin{proof}[(2) $\implies$ (1)]
Suppose $\tn{oblv}(\cal M) \in \cal R^0\Mod$ is compact. Let $\cal M'_i$ ($i\in\cal I$) be a filtered system in $\cal R\Comod$ with colimit $\cal M'$. We have a commutative diagram:
\begin{equation}
\label{eq-compact-comodule}
\xysmall{
	\underset{i\in\cal I}{\tn{colim}}\,\tn{Hom}_{\cal R\Comod}(\cal M, \cal M_i') \ar[r]^-{\alpha}\ar[d] & \tn{Hom}_{\cal R\Comod}(\cal M, \cal M') \ar[d] \\
	\underset{i\in\cal I}{\tn{colim}}\,\tn{Hom}_{R^0\Mod}(\cal M, \cal M_i') \ar[r]^-{\sim} & \tn{Hom}_{R^0\Mod}(\cal M, \cal M')
}
\end{equation}
where the lower horizontal arrow is bijective because $\tn{oblv}(\cal M)$ is compact and $\tn{oblv}$ commutes with filtered colimits (being a left adjoint). On the other hand, the vertical arrows are injective because $\tn{oblv}$ is faithful. In particular, $\alpha$ is injective.

In order to show that $\alpha$ is surjective, we need show that any morphism $\varphi : \cal M \rightarrow \cal M'$ of $\cal R$-comodules comes from a morphism $\varphi_i : \cal M \rightarrow \cal M_i'$ of $\cal R$-comodules for some $i \in \cal I$. The commutative diagram \eqref{eq-compact-comodule} shows that $\tn{oblv}(\varphi)$ comes from a morphism $\nu : \cal M \rightarrow \cal M_j'$ \emph{of $R^0$-modules}. In the following diagram, the outer rectangle and the right square commute:
$$
\xymatrix@R=1.5em{
	\cal M \ar[r]^-{\nu}\ar[d]^{\rho_{\cal M}}\ar@{}[dr]|-{(?)} & \cal M'_j \ar[r]\ar[d]^{\rho_{\cal M'_j}} & \cal M' \ar[d]^{\rho_{\cal M'}} \\
	R^1 \underset{R^0}{\otimes} \cal M \ar[r]^{\tn{id}_{R^1}\otimes\nu} & R^1 \underset{R^0}{\otimes} \cal M'_j \ar[r] & R^1 \underset{R^0}{\otimes} \cal M'
}
$$
Thus, the two circuits of the left square $\rho_{\cal M'_j} \circ\nu$ and $(\tn{id}_{R^1}\otimes\nu) \circ \rho_{\cal M}$ are equalized after further mapping to $R^1\underset{R^0}{\otimes} \cal M'$. Since the following canonical map of $R^0$-modules is a bijective ($\cal I_{j/} \rightarrow \cal I$ is cofinal):
$$
\underset{i \in \cal I_{j/}}{\tn{colim}}\,R^1 \underset{R^0}{\otimes} \cal M_i' \xrightarrow{\sim} R^1 \underset{R^0}{\otimes} \cal M',
$$
we see that $\rho_{\cal M'_j} \circ\nu$ and $(\tn{id}_{R^1}\otimes\nu) \circ \rho_{\cal M}$ are equalized after mapping to $R^1 \underset{R^0}{\otimes} \cal M'_i$ along some arrow $j \rightarrow i$ in $\cal I$. Thus the composition $\cal M \xrightarrow{\nu} \cal M_j' \rightarrow \cal M_i'$ is a map of $\cal R$-comodules which gives rise to $\varphi$.
\end{proof}

%%

%%

\subsection{Representations ``of finite rank''}

\subsubsection{} Let us now consider the analogue of finite-dimensional representations for affine category schemes.

For any $k$-algebra $R$, let $R\Mod^{\Box}$ denote the full subcategory of $R\Mod$ consisting of dualizable objects. One knows that $R\Mod^{\Box}$ consists of finite projective $R$-modules. It has the structure of an exact category, when equipped with the tautological exact structure from $R\Mod$.

%%

\subsubsection{}
Suppose we are given a coalgebroid $\cal R = (R^0, R^1, \Delta, \varepsilon)$. Let $\cal X = (X^0, X^1, m, e)$ be the corresponding affine category scheme. Define:
$$
\cal X\Rep^f := R^1\Comod(R^0\Mod^{\Box}).
$$
Write $F : \cal X\Rep^f \rightarrow R^0\Mod^{\Box}$ for the forgetful functor. In other words, $\cal X\Rep^f$ is the full subcategory of $\cal R\Comod = R^1\Comod(R^0\Mod)$ whose underlying $R^0$-module is dualizable.

\begin{lem}
\label{lem-ind-rep-inside-comod}
The embedding $\cal X\Rep^f \hookrightarrow \cal R\Comod$ extends to a fully faithful embdding:
\begin{equation}
\label{eq-ind-rep-inside-comod}
\tn{Ind}(\cal X\Rep^f) \hookrightarrow \cal R\Comod.
\end{equation}
\end{lem}
\begin{proof}
One first notes that $\cal R\Comod$ contains all colimits, in particular filtered ones. Thus $\cal X\Rep^f \hookrightarrow \cal R\Comod$ extends to a functor $\tn{Ind}(\cal X\Rep^f) \rightarrow \cal R\Comod$. On the other hand, Lemma \ref{lem-compact-comodules} shows that $\cal X\Rep^f$ belongs to the full subcategory of compact objects inside $\cal R\Comod$. Hence the extended functor remains fully faithful.
\end{proof}

%%

\subsubsection{}
The tensor product of $R^1$-comodules and the trivial $R^1$-comodule $\mathbf 1:= R^0$ equip the category $\cal X\Rep^f$ with the structure of a tensor category. With respect to the tautological tensor structure on $R^0\Mod^{\Box}$, the forgetful functor $F$ has the structure of a tensor functor.

Note that the embedding \eqref{eq-ind-rep-inside-comod} is also a tensor functor, when $\tn{Ind}(\cal X\Rep^f)$ is equipped with the ind-extended tensor structure.

%%

\subsubsection{} Let us consider the functor:
$$
\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F) : (R^0\otimes R^0)\ComAlg \rightarrow \tn{Set}
$$
which sends a commutative $(R^0\otimes R^0)$-algebra $R'$ to the set of morphisms of tensor functors $\tn{Hom}^{\otimes}(F\underset{R^0}{\otimes}{}_1 R', F\underset{R^0}{\otimes}{}_2 R')$, where the subscripts $i=1,2$ refer to the two $R^0$-structures of $R'$.

There is a canonical map of presheaves over $X^0\times X^0$:
\begin{equation}
\label{eq-category-scheme-reconstruction}
X^1 \rightarrow \mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F),
\end{equation}
defined as follows. For a test affine scheme $S \rightarrow X^1$, we shall specify a map of tensor functors $F \underset{R^0}{\otimes} {}_1 \cal O_S \rightarrow F\underset{R^0}{\otimes} {}_2\cal O_S$. For any $\cal V\in\cal X\Rep^f$, denote by $\underline{\cal V} := F(\cal V)$ its underlying object in $R^0\Mod^{\Box}$. The $\cal R$-comodule structure on $\underline{\cal V}$ corresponds to a map:
$$
\underline{\cal V} \underset{R^0}{\otimes} {}_1R^1 \rightarrow \underline{\cal V} \underset{R^0}{\otimes} {}_2R^1.
$$
Base change along $R^1 \rightarrow \cal O_S$ yields the desired map, obviously functorial in $S$.

%%

%%

\subsection{Reconstruction of affine category schemes}

\subsubsection{} Let $\cal X = (X^0, X^1, m, e)$ be an affine category scheme. The goal of this subsection is to give a sufficient condition for the map \eqref{eq-category-scheme-reconstruction} to be an isomorphism.

%%

\subsubsection{}
Recall the regular representation of \S\ref{sec-induced-rep}, as an object $\tn{reg}\in\cal R\Comod$, whose underlying $R^0$-module is $R^1$ with respect to the left (i.e., $\tn{pr}_1$) $R^0$-structure. Its comodule structure is given by the comultiplication map $\Delta : R^1 \rightarrow R^1 \underset{R^0}{\otimes} R^1$. 

On the other hand, Lemma \ref{lem-ind-rep-inside-comod} tells us that it makes sense to say whether an object of $\cal R\Comod$ belongs to $\tn{Ind}(\cal X\Rep^f)$.

\begin{lem}
\label{lem-reconstruct-affine-cat-scheme}
Suppose $\tn{reg} \in \cal R\Comod$ belongs to $\tn{Ind}(\cal X\Rep^f)$. Then the canonical map of presheaves \eqref{eq-category-scheme-reconstruction} is an isomorphism:
$$
X^1 \xrightarrow{\sim} \mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F).
$$
\end{lem}
\begin{proof}
We explicitly construct the inverse of \eqref{eq-category-scheme-reconstruction} in \S\ref{lem-reconstruct-affine-cat-scheme-obs}--\ref{lem-reconstruct-affine-cat-scheme-final} below.
\end{proof}

%%

\subsubsection{Observation}
\label{lem-reconstruct-affine-cat-scheme-obs}
\label{} We note that the hypothesis $\tn{reg} \in \tn{Ind}(\cal X\Rep^f)$ implies that $\tn{Coind}(\cal M) \in \tn{Ind}(\cal X\Rep^f)$ for any flat $R^0$-module $\cal M$. Indeed, any such $\cal M$ is a filtered colimit $\underset{\alpha}{\tn{colim}}(\cal M_{\alpha})$ where $\cal M_{\alpha}$ is a free $R^0$-module of finite rank (Lazard's theorem). It suffices to verify the claim when $\cal M$ is free of finite rank. In this situation, $\tn{Coind}(\cal M)$ is a finite direct sum of copies of $\tn{reg}$, which belongs to $\tn{Ind}(\cal X\Rep^f)$.

The hypothesis also implies that the underlying $R^0$-module $\underline{\tn{reg}}$ is flat, being a filtered colimit of finite projective $R^0$-modules.

%%

\subsubsection{} Fix a test affine $S \rightarrow X^0\times X^0$ together with a map of tensor functors from $\cal X\Rep^f$ to $\cal O_S\Mod^{\Box}$:
\begin{equation}
\label{eq-map-of-tensor-functors}
\alpha : F \underset{R^0}{\otimes} {}_1\cal O_S \rightarrow F \underset{R^0}{\otimes} {}_2\cal O_S.
\end{equation}
Regarding $F \underset{R^0}{\otimes} {}_1\cal O_S$ as a tensor functor $\cal X\Rep^f \rightarrow \cal O_S\Mod$, one obtains an ind-extension
$$
\tn{Ind}(\cal X\Rep^f) \rightarrow \cal O_S\Mod,\quad \cal V \leadsto \underline{\cal V} \underset{R^0}{\otimes} {}_1\cal O_S,
$$
where $\underline{\cal V}$ is the underlying $R^0$-module of $\cal V$, seen as an object of $\cal R\Comod$ under the embedding of Lemma \ref{lem-ind-rep-inside-comod}. Indeed, this expression of the ind-extended functor follows from the fact that both functors below preserve filtered colimits:
$$
\cal R\Comod \xrightarrow{\tn{oblv}} R^0\Mod \xrightarrow{-\underset{R^0}{\otimes}{}_1\cal O_S} \cal O_S\Mod.
$$
It is clear that an analogous statement holds for $F \underset{R^0}{\otimes} {}_2\cal O_S$.

By the functoriality of ind-extension, \eqref{eq-map-of-tensor-functors} induces a map between two tensor functors from $\tn{Ind}(\cal X\Rep^f)$ to $\cal O_S\Mod$, again denoted by $\alpha$. The hypothesis says that the regular representation belongs to $\tn{Ind}(\cal X\Rep^f)$, so we may evaluate $\alpha$ on it to obtain a map of $\cal O_S$-modules:
\begin{equation}
\label{eq-map-of-tensor-functors-reg}
\alpha_{\tn{reg}} : \cal O_S \underset{{}_1, R^0}{\otimes} R^1 \rightarrow \cal O_S \underset{{}_2, R^0}{\otimes} R^1.
\end{equation}
(We exchange the positions to emphasize that the tensor uses the left $R^0$-structure of $R^1$.) One may also view $\alpha_{\tn{reg}}$ as a map of $R^0$-modules $R^1 \rightarrow \cal O_S \underset{{}_2, R^0}{\otimes}R^1$, where the $R^0$-module structure on $\cal O_S \underset{{}_2, R^0}{\otimes}R^1$ comes from the first $R^0$-structure on $\cal O_S$.

%%

\subsubsection{} We claim that $\alpha_{\tn{reg}}$ \eqref{eq-map-of-tensor-functors-reg} canonically upgrades to a map of (commutative) $\cal O_S$-algebras. Since $\alpha_{\tn{reg}}$ is a map between tensor functors, it suffices to show that $\tn{reg} \in \tn{Ind}(\cal X\Rep^f)$ is canonically a (commutative) algebra object. Indeed, the structure maps:
$$
\tn{reg}\otimes\tn{reg} \rightarrow \tn{reg},\quad \mathbf 1 \rightarrow \tn{reg}
$$
are the ones induced from $R^1\underset{{}_1, R^0, {}_1}{\otimes} R^1 \rightarrow R^1$ (multiplication) and $R^0 \rightarrow R^1$ (left $R^0$-structure), respectively. We omit checking that these maps are maps of comodules, where the coaction on $R^1\underset{{}_1, R^0, {}_1}{\otimes} R^1$ is the diagonal one.

%%

\subsubsection{} Having proved that \eqref{eq-map-of-tensor-functors-reg} is a map of $\cal O_S$-algebras, we view it as a map between affine schemes:
\begin{equation}
\label{eq-map-of-tensor-functors-reg-geom}
\alpha_{\tn{reg}} : S \underset{{}_2, X^0}{\times} X^1 \rightarrow  X^1.
\end{equation}
We claim that $\alpha_{\tn{reg}}$ is \emph{$\cal X$-equivariant}, in the sense that the following diagram commutes:
\begin{equation}
\label{eq-equivariance}
\xymatrix@R=1.5em@C=3.5em{
	S \underset{{}_2, X^0}{\times} X^1 \underset{X^0}{\times} X^1 \ar[r]^-{\alpha_{\tn{reg}}\times\tn{id}_{X^1}}\ar[d]^{\tn{id}_S\times m} &  X^1 \underset{X^0}{\times} X^1 \ar[d]^{\tn{id}_S\times m} \\
	S \underset{{}_2, X^0}{\times} X^1 \ar[r]^-{\alpha_{\tn{reg}}} &  X^1
}
\end{equation}
where $m$ denotes the composition law of the category scheme $\cal X$. To prove this statement, we first use the fact that the morphism in $\tn{Ind}(\cal X\Rep^f)$:
$$
\tn{reg} \rightarrow \tn{Coind}(\underline{\tn{reg}}),
$$
given by $\Delta : R^1 \rightarrow R^1 {}_2\underset{R^0}{\otimes}{}_1 R^1$, is a map of algebra objects; then we appeal to the following result to identify $\alpha_{\tn{Coind}(\underline{\tn{reg}})}$ with $\alpha_{\tn{reg}} \times\tn{id}_{X^1}$ (applied to $\cal N := \underline{\tn{reg}}$).

\begin{claim}
\label{claim-alpha-of-coinduced}
For any flat $R^0$-module $\cal N$, there holds:
$$
\alpha_{\tn{Coind}(\cal N)} = \alpha_{\tn{reg}} \otimes \tn{id}_{\cal N} : R^1\underset{R^0}{\otimes}\cal N \rightarrow \cal O_S  \underset{{}_2, R^0}{\otimes}R^1\underset{R^0}{\otimes}\cal N
$$
\end{claim}
\begin{proof}
Since $\cal N$ is a filtered colimit of free $R^0$-modules of finite rank and $\alpha_{\cal V_1 \oplus \cal V_2} = \alpha_{\cal V_1} \oplus \alpha_{\cal V_2}$ for $\cal V_1, \cal V_2\in \cal X\Rep^f$, the question reduces to the case $\cal N = R^0$, where it is obvious.
\end{proof}

%%

\subsubsection{} Next, we establish two statements.
\begin{enumerate}
	\item $\alpha_{\tn{reg}}$ \eqref{eq-map-of-tensor-functors-reg-geom} is the unique $\cal X$-equivariant extension of its restriction to identity:
	$$
	e^*\alpha_{\tn{reg}} : S \xrightarrow{\tn{id}\times e} S \underset{{}_2, X^0}{\times} X^1 \xrightarrow{\alpha_{\tn{reg}}} X^1.
	$$
	Namely, $\alpha_{\tn{reg}}$ identifies with the composition:
	$$
	S \underset{{}_2, X^0}{\times} X^1 \xrightarrow{e^*\alpha_{\tn{reg}} \times\tn{id}_{X^1}} X^1 \underset{X^0}{\times} X^1 \xrightarrow{m} X^1.
	$$
	
	\item for any $\cal V\in \cal X\Rep^f$, the map $\alpha_{\cal V}$ of $\cal O_S$-modules is the unique one making the following diagram commutes:
	$$
	\xymatrix@C=3.5em@R=1.5em{
		 \underline{\cal V}  \ar[r]^-{\alpha_{\cal V}} \ar[d]^{\rho_{\cal V}} & \cal O_S \underset{{}_2, R^0}{\otimes} \underline{\cal V} \ar[d]^{\tn{id}_S\otimes \rho_{\cal V}} \\
		R^1\underset{R^0}{\otimes}\underline{\cal V} \ar[r]^-{\alpha_{\tn{reg}}\otimes\tn{id}_{\underline{\cal V}}} \ar[r] &  \cal O_S \underset{{}_2, R^0}{\otimes}  R^1\underset{R^0}{\otimes}\underline{\cal V}
	}
	$$
	where $\rho_{\cal V}$ denotes the coaction map.
\end{enumerate}
\noindent
The first statement follows from restricting the commutative diagram \eqref{eq-equivariance} along the map:
$$
S \underset{{}_2, X^0}{\times} X^0 \underset{X^0}{\times} X^1 \xrightarrow{\tn{id}_S\times e\times\tn{id}_{X^1}} S \underset{{}_2, X^0}{\times} X^1 \underset{X^0}{\times} X^1.
$$
For the second statement, $\alpha_{\cal V}$ makes the diagram commute because $\rho_{\cal V} : \cal V \rightarrow \tn{Coind}(\underline{\cal V})$ is a morphism in $\tn{Ind}(\cal X\Rep^f)$ and $\alpha_{\tn{Coind}(\underline{\cal V})} = \alpha_{\tn{reg}} \otimes \tn{id}_{\underline{\cal V}}$ by Claim \ref{claim-alpha-of-coinduced}. The map $\alpha_{\cal V}$ is the unique one with this property since both vertical arrows are injective, as demonstrated by the fact that the composition:
$$
\underline{\cal V} \xrightarrow{\rho_{\cal V}}  R^1\underset{R^0}{\otimes}\underline{\cal V} \xrightarrow{\varepsilon\cdot\tn{id}_{\underline{\cal V}}} \underline{\cal V}
$$
is the identity on $\underline{\cal V}$.

%%

\subsubsection{}
\label{lem-reconstruct-affine-cat-scheme-final}We define the map:
$$
\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F) \rightarrow X^1
$$
of presheaves by sending an $S$-point $\alpha$ of $\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F)$ to the $S$-point $e^*\alpha_{\tn{reg}}$.

It is clear that if $\alpha$ is the mapping of functors associated to some $g \in \tn{Maps}(S, X^1)$, then $e^*\alpha_{\tn{reg}}$ recovers $g$. Conversely, the map between tensor functors determined by $e^*\alpha_{\tn{reg}} \in \tn{Maps}(S, X^1)$ agrees with $\alpha$, by the two statements established above. This completes the proof of Lemma \ref{lem-reconstruct-affine-cat-scheme}.

%%

%%



\newpage


\section{Tannakian categories}

%%

We continue to fix the ground ring $k$. In this section, we discuss some constructions associated to the set-up of a tensor category $\cal C$, a commutative $k$-algebra $R$, and a tensor functor $F$ valued in dualizable $R$-modules:
$$
F : \cal C \rightarrow R\Mod^{\Box}.
$$
Write $X^0 := \Spec(R)$. Our first goal is to show that a certain mapping space of pullbacks of $F$ is representable by an affine category scheme $\cal X = (X^0, X^1, m, e)$.

In the case where $\cal C$ is abelian and $F$ is exact and faithful, we will identify the large category $\tn{Ind}(\cal C)$ as comodules over $\cal X$.

\subsection{Representabiliy of mapping spaces}

\subsubsection{} Let $R$ be a commutative $k$-algebra. For any functor $F : \cal C \rightarrow R\Mod$ and any commutative $R$-algebra $R'$, we denote by:
$$
F\otimes R' : \cal C\rightarrow R'\Mod
$$
the functor which attaches to each $c\in \cal C$ the $R'$-module $F(c)\underset{R}{\otimes} R'$. Clearly, if $F$ lands in $R\Mod^{\Box}$, then $F\otimes R'$ lands in $R'\Mod^{\Box}$.

When $\cal C$ has a symmetric monoidal structure and $F$ is a symmetric monoidal functor, $F\otimes R'$ inherits a symmetric monoidal structure.

%%

\subsubsection{} The following lemma is the basis of our representability statements.

\begin{lem}
\label{lem-representable}
Suppose $(\cal C, \otimes, \mathbf 1)$ is a tensor category and $R$ is a commutative $k$-algebra. Let
$$
F_1, F_2 : \cal C \rightarrow R\Mod^{\Box}
$$
be tensor functors. Then the presheaf:
$$
\mathbf{Hom}^{\otimes}(F_1, F_2) : R\ComAlg \rightarrow \tn{Set}
$$
which sends every commutative $R$-algebra $R'$ to the set of morphisms of tensor functors $\tn{Hom}^{\otimes}(F_1 \otimes R', F_2\otimes R')$ is representable by an affine scheme over $\Spec(R)$.
\end{lem}
\begin{proof}
The proof is constructive and proceeds in several step.

\noindent
\emph{Step 1}. Consider the functor:
\begin{equation}
\label{eq-coend-functor}
H : \tn{Tw}(\cal C)\rightarrow R\Mod
\end{equation}
defined as follows:
\begin{enumerate}
	\item to $c \xrightarrow{\varphi} d$, it associates the $R$-module $F_1(c) \otimes F_2(d)^{\vee}$;
	\item to a morphism $\varphi \rightarrow \varphi'$ in $\tn{Tw}(\cal C)$:
	$$
	\xysmall{
		c \ar[r]^-{\varphi} \ar[d] & d \\
		c' \ar[r]^-{\varphi'} & d' \ar[u]
	}
	$$
	it associates the natural map:
	$$
	F_1(c)\otimes F_2(d)^{\vee} \rightarrow F_1(c')\otimes F_2(d')^{\vee}.
	$$
\end{enumerate}
Since $R\Mod$ contains all colimits, we may set
$$
\cal A := \underset{\tn{Tw}(\cal C)}{\tn{colim}} \, H \in R\Mod.
$$

%%

\noindent
\emph{Step 2}. We will equip $\cal A$ with the structure of a commutative $R$-algebra. By Lemma \ref{lem-colimit-commutative-algebra}, it suffices to upgrade \eqref{eq-coend-functor} to a symmetric monoidal functor, where the symmetric monoidal structure on $\tn{Tw}(\cal C)$ is the one from \S\ref{sec-twisted-arrow-tensor}.

The required data are specified by:
$$
H(\mathbf 1\xrightarrow{\varphi} \mathbf 1) = F_1(\mathbf 1)\otimes F_2(\mathbf 1)^{\vee} \xrightarrow{\sim} R,
$$
and the commutative diagram:
$$
\xysmall{
	H(c\xrightarrow{\varphi} d) \otimes H(c'\xrightarrow{\varphi'} d') \ar[r]\ar@{=}[d] & H(c\otimes c' \xrightarrow{\varphi\otimes\varphi'} d\otimes d') \ar@{=}[d] \\
	(F_1(c)\otimes F_2(d)^{\vee})\otimes (F_1(c')\otimes F_2(d')^{\vee}) \ar[r]^-{\sim} & F_1(c\otimes c') \otimes F_2(d\otimes d')^{\vee}
}
$$

%%

\noindent
\emph{Step 3}. For any commutative $R$-algebra $R'$, we shall construct an isomorphism:
\begin{equation}
\label{eq-module-equivalence}
\tn{Hom}_{R\Mod}(\cal A, R') \xrightarrow{\sim} \tn{Hom}(F_1\otimes R', F_2\otimes R')
\end{equation}
functorial in $R'$, where $\tn{Hom}$ is calculated as plain (i.e., not symmetric monoidal) functors from $\cal C$ to $R'\Mod$.

Indeed, there are equivalences:
\begin{align*}
	\tn{Hom}_{R\Mod}(\cal A, R') & \xrightarrow{\sim} \underset{\tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}}\, \tn{Hom}_{R\Mod}(F_1(c)\otimes F_2(d)^{\vee}, R') \\
	& \xrightarrow{\sim} \underset{\tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}}\, \tn{Hom}_{R'\Mod}(F_1(c)\otimes R', F_2(d) \otimes R'),
\end{align*}
and the latter identifies with $\tn{Hom}(F_1\otimes R', F_2\otimes R')$ by Lemma \ref{lem-twisted-arrow}.

%%

\noindent
\emph{Step 4}. We show that an $R$-module map $f : \cal A \rightarrow R'$ is a map of $R$-algebras if and only if its image $\alpha$ under \eqref{eq-module-equivalence} is compatible with the symmetric monoidal structures on $F_1\otimes R'$ and $F_2\otimes R'$. For any object $c\in\cal C$, we let $\alpha_c$ denote the component:
$$
\alpha_c : F_1(c) \rightarrow F_2(c)\otimes R'.
$$
We consider the unit and product structures seperately.
\begin{enumerate}
	\item The statement that $f$ restricts to the given map $R\rightarrow R'$ is equivalent to the commutativity of the the diagram:
	$$
	\xysmall{
		R \ar[dr]\ar[r]^-{\sim} & F_1(\mathbf 1) \ar[d]^-{\alpha_{\mathbf 1}} \\
		 & F_2(\mathbf 1)\otimes R'
	}
	$$
	which is in turn equivalent to the compatibility between $\alpha$ and the unital structures on $F_1\otimes R'$ and $F_2\otimes R'$.
	
	\item The statement that $f$ preserves products is equivalent to the commutativity of the diagram for all $\varphi : c \rightarrow d$ and $\varphi' : c'\rightarrow d'$:
	$$
	\xysmall{
		(F_1(c) \otimes F_2(d)^{\vee}) \otimes (F_1(c')\otimes F_2(d')^{\vee}) \ar[r]^-{\sim}\ar[d]^{f\otimes f} & F_1(c\otimes c') \otimes F_2(d\otimes d')^{\vee} \ar[d]^f \\
		R' \otimes R' \ar[r] & R'
	}
	$$
	where we slightly abused the notation $f$ for its restriction to each term of the form $F_1(c)\otimes F_2(d)^{\vee}$. The functoriality of the upper horizontal isomorphism in $\varphi$, $\varphi'$ allows to reduce this statement to the special case $\varphi = \tn{id}_c$ and $\varphi' = \tn{id}_{c'}$. This latter statement is equivalent to the commutativity of the square:
	$$
	\xysmall{
		F_1(c) \otimes F_1(c') \ar[d]^{\alpha_c\otimes\alpha_{c'}} \ar[r] & F_1(c\otimes c') \ar[d]^{\alpha_{c\otimes c'}} \\
		(F_2(c)\otimes R') \otimes (F_2(c')\otimes R') \ar[r] & F_2(c\otimes c')\otimes R'
	}
	$$
	which expresses the compatibility of $\alpha$ with monoidal products on $F_1\otimes R'$ and $F_2\otimes R'$.
\end{enumerate}
Therefore, the isomorphism \eqref{eq-module-equivalence} restricts to an isomorphism:
\begin{equation}
\label{eq-algebra-equivalence}
\tn{Hom}_{R\ComAlg}(\cal A, R') \xrightarrow{\sim} \tn{Hom}^{\otimes}(F_1\otimes R', F_2\otimes R'),
\end{equation}
which is functorial in $R'$.
\end{proof}


%%

%%

\subsection{Obtaining affine category schemes}
\label{sec-comonad-alg}

\subsubsection{} In this subsection, we will obtain from a tensor category $\cal C$ and a tensor functor $F : \cal C \rightarrow R\Mod^{\Box}$ an affine category scheme $\cal X$. We begin with the following representability result of the scheme of ``arrows.''

\begin{lem}
\label{lem-groupoid-representable}
Suppose $(\cal C, \otimes, \mathbf 1)$ is a tensor category and $R$ is a commutative $k$-algebra. Let:
$$
F : \cal C \rightarrow R\Mod^{\Box}
$$
be a tensor functor. Then the presheaf:
$$
\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F) : (R\otimes R)\ComAlg \rightarrow \tn{Set}
$$
which sends a commutative $(R\otimes R)$-algebra $R'$ to the set of morphisms of tensor functors $\tn{Hom}^{\otimes}(F \underset{R}{\otimes}{}_1R', F \underset{R}{\otimes}{}_2 R')$, where the subscripts $i=1,2$ refer to the two $R$-structures of $R'$, is representable by an affine scheme over $\Spec(R\otimes R)$.
\end{lem}
\begin{proof}
We write $F_1 : \cal C \rightarrow(R\otimes R)\Mod^{\tn{dualizable}}$ for the functor:
$$
\tn{pr}_1^*F : c \leadsto F(c) \underset{R}{\otimes}{}_1 (R \otimes R) \cong F(c) \underset{k}{\otimes} R,
$$
and $F_2 :  \cal C \rightarrow(R\otimes R)\Mod^{\tn{dualizable}}$ for the functor:
$$
\tn{pr}_2^*F : c \leadsto F(c) \underset{R}{\otimes}{}_2 (R \otimes R) \cong  R \underset{k}{\otimes} F(c) .
$$
Then we apply Lemma \ref{lem-representable} to $F_1$ and $F_2$.
\end{proof}

In particular, the output of the construction of Lemma \ref{lem-representable} gives the representing commutative $(R\otimes R)$-algebra:
\begin{equation}
\label{eq-coend-expression}
\cal A \cong \underset{(c\rightarrow d) \in \tn{Tw}(\cal C)}{\tn{colim}} F(c) \underset{k}{\otimes} F(d)^{\vee},
\end{equation}
where the dual is taken in $R\Mod^{\Box}$.

%%

\subsubsection{} Let us remain in the set-up of Lemma \ref{lem-groupoid-representable}. Denote by $X^0 := \Spec(R)$ and $X^1:= \Spec(\cal A)$ where $\cal A$ is the representing commutative $(R\otimes R)$-algebra. We denote the two projection maps, corresponding to the two $R$-structures of $\cal A$, by:
$$
\xysmall{
	X^1 \ar@<0.5ex>[r]^-{\tn{pr}_1} \ar@<-0.5ex>[r]_-{\tn{pr}_2} & X^0.
}
$$
Our current goal is to equip this diagram with the structure of a category scheme. This involves specifying the additional data of composition and identity:
\begin{align}
	X^1 {}_{\tn{pr}_2}\underset{X^0}{\times} {}_{\tn{pr}_1} X^1 &\rightarrow X^1 \label{eq-composition} \\
	X^0 &\rightarrow X^1 \label{eq-identity}
\end{align}
More precisely, \eqref{eq-composition} arises from the map of functors $(R\otimes R\otimes R)\ComAlg \rightarrow\tn{Set}$:
$$
\circ : \mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F) \times \mathbf{Hom}^{\otimes}(\tn{pr}_2^*F, \tn{pr}_3^*F) \rightarrow \mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_3^*F),
$$
and \eqref{eq-identity} arises from the map $\cal A\rightarrow R$ of commutative $(R\otimes R)$-algebras, where $R$ is given the diagonal $(R\otimes R)$-action, represented by:
$$
\tn{id} \in \tn{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F),\quad \tn{pr}_1^*F \cong \tn{pr}_2^*F.
$$

%%

\subsubsection{} In algebraic terms, the two maps above give rise to $k$-algebra morphisms
\begin{align}
	\cal A &\rightarrow \cal A {}_2\underset{R}{\otimes}{}_1\cal A  \label{eq-composition-alg}  \\
	\cal A &\rightarrow R \label{eq-identity-alg}
\end{align}
The first map is $(R\otimes R)$-linear, where the two $R$'s act on $\cal A {}_2\underset{R}{\otimes}{}_1\cal A$ by outer multiplications. (We think of $1$ as the left $R$-structure and $2$ as the right $R$-structure.) The second map is also $(R\otimes R)$-linear, where $R$ is equipped with the diagonal action.

Regarding $\cal A$ as an $(R, R)$-bimodule, the endomorphism:
\begin{equation}
\label{eq-comonad-alg}
\cal A\underset{R}{\otimes}(-): R\Mod \rightarrow R\Mod
\end{equation}
upgrades to a comonad acting on $R\Mod$, with comultiplication given by \eqref{eq-composition-alg} and counit given by \eqref{eq-identity-alg}. This is the comonad associated to the forgetful--coinduction adjunction (see \S\ref{sec-induced-rep}).

%%

%%


\subsection{The ind-category as comodules}

\subsubsection{Set-up}
\label{set-up-monadicity}
Let $(\cal C, \otimes, \mathbf 1)$ be a tensor category \emph{containing finite colimits}. Suppose $R$ is a commutative $k$-algebra and
$$
F : \cal C \rightarrow R\Mod
$$
is a tensor functor preserving finite colimits.

Thus, $\tn{Ind}(\cal C)$ contains all colimits and the ind-extension $\tn{Ind}(F) : \tn{Ind}(\cal C) \rightarrow R\Mod$ preserves all colimits (Lemma \ref{lem-ind-all-colimits}). By the adjoint functor theorem, it admits a right adjoint $G$:
$$
\xysmall{
	\tn{Ind}(F) : \tn{Ind}(\cal C) \ar@<0.5ex>[r] & R\Mod : G. \ar@<0.5ex>[l]
}
$$
We let $T$ denote the comonad $\tn{Ind}(F) \circ G$ acting on $R\Mod$. In particular, $\tn{Ind}(F)$ canonically factors as follows, where $\tn{oblv}$ denotes the forgetful functor.
$$
\xysmall{
	\tn{Ind}(\cal C) \ar[r]^-{\widetilde F}\ar[dr]_{\tn{Ind}(F)} & T\Comod(R\Mod) \ar[d]^{\tn{oblv}} \\
	& R\Mod
}
$$

%%

\subsubsection{} The following is a consequence of the Barr--Beck theorem.

\begin{lem}
\label{lem-large-category}
In set-up \S\ref{set-up-monadicity}, if $\cal C$ is abelian and $F : \cal C \rightarrow R\Mod$ is exact and faithful. Then the functor $\widetilde F$ is an equivalence:
$$
\widetilde F : \tn{Ind}(\cal C) \xrightarrow{\sim} T\Comod(R\Mod).
$$
\end{lem}
\begin{proof}
Since filtered colimits in $R\Mod$ are exact, the ind-extension:
\begin{equation}
\label{eq-ind-extension-forgetful}
F : \tn{Ind}(\cal C) \rightarrow R\Mod
\end{equation}
is still an exact and faithful functor between abelian categories. We observe:
\begin{enumerate}
	\item The functor \eqref{eq-ind-extension-forgetful} is conservative. Indeed, if a morphism $\varphi : c \rightarrow d$ in $\tn{Ind}(\cal C)$ becomes an isomorphism after applying $F$, the exactness of $F$ implies that $F(\tn{Ker}(\varphi)) = 0$ and $F(\tn{Coker}(\varphi)) = 0$. The faithfulness of $F$ then implies that $\tn{Ker}(\varphi)$ and $\tn{Coker}(\varphi)$ are zero, so $\varphi$ is an isomorphism.
	\item $\tn{Ind}(\cal C)$ admits cosplit equalizers, and the functor \eqref{eq-ind-extension-forgetful} preserves them. Indeed, any abelian category contains finite limits, and the functor \eqref{eq-ind-extension-forgetful} preserves them because it is exact.
\end{enumerate}
The result thus follows from the Barr--Beck theorem (Lemma \ref{lem-barr-beck}).
\end{proof}

%%

%%

\subsection{Identification of the comonad}

\subsubsection{} Suppose $(\cal C, \otimes, \mathbf 1)$ is a tensor category and $R$ is a commutative $k$-algebra. Let:
$$
F : \cal C \rightarrow R\Mod^{\Box}
$$
be a tensor functor. The materials from \S\ref{sec-comonad-alg} yield an affine scheme $\Spec(\cal A)$ representing $\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F)$, and moreover a comonad $\cal A\underset{R}{\otimes} (-)$ \eqref{eq-comonad-alg} acting on $R\Mod$.

On the other hand, if $\cal C$ contains finite colimits and the induced functor $\cal C \rightarrow R\Mod$ (denoted by the same letter $F$) preserves them, then the ind-extension together with its right adjoint define a comonad $T := \tn{Ind}(F)\circ G$ acting on $R\Mod$ as in \S\ref{set-up-monadicity}.

%%

\subsubsection{} Note that there is a morphism of endo-functors on $R\Mod$:
\begin{equation}
\label{eq-comonad-identification}
\tn{Ind}(F)\circ G \rightarrow \cal A\underset{R}{\otimes}(-).
\end{equation}
Indeed, given any $R$-module $\cal N$. The unit $R \rightarrow \cal A$ of the right $R$-module structure gives rise to a map $\cal N \rightarrow \cal A\underset{R}{\otimes}\cal N$. Upon applying $G$, we obtain a map $G(\cal N) \rightarrow G(\cal A\underset{R}{\otimes} \cal N)$ in $\tn{Ind}(\cal C)$, and finally \eqref{eq-comonad-identification} comes from adjunction.

\begin{lem}
\label{lem-comonad-identification}
The map \eqref{eq-comonad-identification} is an isomorphism of comonads.
\end{lem}
\begin{proof}
Let us first prove that for $R$-modules $\cal N, \cal N'$, the canonical map defined above:
\begin{equation}
\label{eq-comonad-identification-evaluate}
\tn{Hom}_{R\Mod}(\cal A\underset{R}{\otimes}\cal N, \cal N') \rightarrow \tn{Hom}_{R\Mod}(\tn{Ind}(F) \circ G(\cal N), \cal N')
\end{equation}
is an isomorphism. The right-hand-side identifies with $\tn{Hom}_{\tn{Ind}(\cal C)}(G(\cal N), G(\cal N'))$ by adjunction. To identify the left-hand-side, we recall the structure of $\cal A$ as an $(R,R)$-bimodule \eqref{eq-coend-expression}, which yields:
$$
\tn{Hom}_{R\Mod}(\cal A\underset{R}{\otimes}\cal N, \cal N') \cong \underset{(c\rightarrow d)\in\tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}} \tn{Hom}_{R\Mod}((F(c) \underset{k}{\otimes} F(d)^{\vee}) \underset{R}{\otimes} \cal N, \cal N').
$$
(To interpret the right-hand-side: the tensor with $\cal N$ is formed with respect to the $R$-module structure on $F(c)\underset{k}{\otimes} F(d)^{\vee}$ acting on $F(d)^{\vee}$. The remaining $R$-module structure on:
$$
(F(c) \underset{k}{\otimes} F(d)^{\vee}) \underset{R}{\otimes} \cal N \cong F(c)\underset{k}{\otimes} (F(d)^{\vee}\underset{R}{\otimes} \cal N)
$$
acts only on the factor $F(c)$, which we then use to form the hom-space with $\cal N'$.) Consequently, we have
\begin{align}
& \underset{(c\rightarrow d)\in\tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}} \tn{Hom}_{R\Mod}((F(c) \underset{k}{\otimes} F(d)^{\vee}) \underset{R}{\otimes} \cal N, \cal N') \notag  \\
\xrightarrow{\sim} & \underset{(c\rightarrow d)\in\tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}} \tn{Hom}_{k\Mod}( F(d)^{\vee} \underset{R}{\otimes} \cal N, F(c)^{\vee}\underset{R}{\otimes}\cal N') \notag \\
\xrightarrow{\sim} & \tn{Hom}_{\tn{Fun}(\cal C^{\tn{op}}, k\Mod)}(F(-)^{\vee}\underset{R}{\otimes}\cal N, F(-)^{\vee}\underset{R}{\otimes} \cal N'). \label{eq-functor-hom-space}
\end{align}
where we applied Lemma \ref{lem-twisted-arrow} in the final step. On the other hand, the functor:
$$
F(-)^{\vee}\underset{R}{\otimes} \cal N : \cal C^{\tn{op}} \rightarrow k\Mod
$$
is ind-represented by $G(\cal N)$. Namely, there is a canonical isomorphism of $k$-modules, functorial in $c\in\cal C$:
$$
\tn{Hom}_{\tn{Ind}(\cal C)}(c, G(\cal N)) \xrightarrow{\sim} \tn{Hom}_{R\Mod}(F(c), \cal N) \xrightarrow{\sim} F(c)^{\vee}\underset{R}{\otimes} \cal N
$$
Therefore, the hom-space of functors \eqref{eq-functor-hom-space} is equivalent to $\tn{Hom}_{\tn{Ind}(\cal C)}(G(\cal N), G(\cal N'))$. This shows that \eqref{eq-comonad-identification-evaluate} is an isomorphism.

It remains to prove that this morphism is compatible with the monad structures. Namely, the following diagrams commute:
$$
\xysmall{
	\tn{Ind}(F)\circ G \ar[d]^{\tn{id}\cdot \tn{unit}\cdot\tn{id}}\ar[r]^-{\sim} & \cal A\underset{R}{\otimes}(-) \ar[d]^{\Delta\otimes\tn{id}} \\
	\tn{Ind}(F)\circ G \circ \tn{Ind}(F)\circ G \ar[r]^-{\sim} & \cal A\underset{R}{\otimes}\cal A\underset{R}{\otimes}(-)
}
\quad
\xysmall{
	\tn{Ind}(F)\circ G \ar[d]^{\tn{counit}}\ar[r]^-{\sim} & \cal A\underset{R}{\otimes}(-) \ar[d]^{\varepsilon} \\
	\tn{id} \ar[r]^-{\sim} & \tn{id}
}
$$
We leave these verifications to the interested reader.
\end{proof}

%%

\subsection{Reconstruction of the ind-category}

\subsubsection{} We will now put together the results from this section to prove a reconstruction statement of the ind-category. Let us assume:
\begin{enumerate}
	\item $\cal C$ is an \emph{abelian} tensor category;
	\item $R$ is a commutative $k$-algebra;
	\item $F : \cal C \rightarrow R\Mod^{\Box}$ is an \emph{exact}, \emph{faithful} tensor functor.
\end{enumerate}
Let $\cal X = (R, \cal A, \Delta, \varepsilon)$ be the coalgebroid representing $\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F)$, constructed in \S\ref{sec-comonad-alg}. Let $\cal A\Comod$ denote the tensor category of $\cal A$-comodules (in $R\Mod$).

\begin{lem}
With the above notations, there is a canonical equivalence:
$$
\widetilde F : \tn{Ind}(\cal C) \xrightarrow{\sim} \cal A\Comod,
$$
making the following diagram commute:
$$
\xysmall{
	\tn{Ind}(\cal C) \ar[r]^-{\widetilde F}\ar[dr]_{\tn{Ind}\,F} & \cal A\Comod \ar[d]^{\tn{oblv}} \\
	& R\Mod
}
$$
\end{lem}
\begin{proof}
We first apply Lemma \ref{lem-large-category} to obtain an equivalence:
$$
\tn{Ind}(\cal C) \xrightarrow{\sim} T\Comod(R\Mod),
$$
where $T$ is the comonad $\tn{Ind}(F)\circ G$ (see \S\ref{set-up-monadicity}). By Lemma \ref{lem-comonad-identification}, the comonad $T$ identifes with tensoring with the co-algebra $\cal A$ representing $\mathbf{Hom}^{\otimes}(\tn{pr}_1^*F, \tn{pr}_2^*F)$.
\end{proof}

%%

%%


%%





\newpage


\section{Recognizing small Tannakian categories}

\subsection{Ramblings}

This section is not yet written. In the previous section, we showed that for a tensor category $\cal C$ with a tensor functor:
$$
F : \cal C \rightarrow R\Mod^{\Box}
$$
satisfying additional conditions, such as $\cal C$ being abelian and $F$ exact and faithful, one can recover $\tn{Ind}(\cal C)$ as $\cal A$-comodules in $R\Mod$. Here, $X^0 = \Spec(R)$ and $X^1 = \Spec(\cal A)$ is the representing affine category scheme $\cal X$.

Given the additional assumption that $\cal C$ is rigid, one can show that $\cal X$ is a groupoid scheme. Assuming further that $k$ is a field, one can recover the small category $\cal C$ as $\cal A$-comodules in $R\Mod^{\Box}$ using the following approach:
\begin{enumerate}
	\item prove that $X^1 \rightarrow X^0\times X^0$ is faithfully flat, i.e., the groupoid is transitive. In [Deligne, Cat\'egories Tannakiennes], this is done using a construction of tensor products of Tannakian categories. Namely, one regards $\cal A = \tn{colim}\,F(c)\otimes F(d)^{\vee}$ as the image of an \emph{ind-object} under the fiber functor:
	$$
	F\otimes F : \cal C \otimes\cal C \rightarrow (R\otimes R)\Mod^{\Box}
	$$
	This ind-object of $\cal C\otimes\cal C$, which is \emph{a priori} $\tn{colim}\,c \boxtimes d^{\vee}$, can be expressed as a filtered colimit of quotients of $\bigoplus_{c\in C_i}c\boxtimes c^{\vee}$ as one expresses $\cal C$ as a filtered colimit of its finite full subcategories $\cal C_i$.
	
	\item one uses the trick that representations of a transitive groupoid (over a field) are equivalent to representations of the stabilzer group of a field-valued point. Thus, one obtains the desired equivalence by taking compact objects on both sides of:
	$$
	\tn{Ind}(\cal C) \cong \cal A\Comod.
	$$
\end{enumerate}

I am not yet sure that this is the generality that makes most sense. In particular, it assumes $\cal C$ to be abelian (so $\tn{Vect}(\mathbb A^1/\mathbb G_m)$ is excluded) and rigid (so representations of monoids are excluded), and the assumption that $k$ is a field appears \emph{ad hoc}. (On the other hand, in order to find a more general statement, it seems that one should first think about reconstruction of large categories with the abelian-ness assumption removed, so I am delaying writing up this section.)


%%

%%






\newpage

\appendix

\section{Category theory lemmas}

\subsection{Recovering a category from its ind-completion}

\subsubsection{} The following discussion applies to categories in general as well as $k$-linear categories for a fixed ring $k$. Let $\cal C$ be a category (resp.~a $k$-linear category).

An object $c\in \cal C$ is called \emph{compact} if for every filtered diagram $d_i \in \cal C$ ($i\in \cal I$) whose colimit exists, the natural map below is an isomorphism:
$$
\underset{i\in \cal I}{\tn{colim}} \, \tn{Hom}(c, d_i) \rightarrow \tn{Hom}(c, \underset{i\in \cal I}{\tn{colim}} \, d_i).
$$
Let $\cal C^{\tn{cpt}}$ denote the full subcategory of $\cal C$ spanned by compact objects.

%%

\subsubsection{} Let $c\in \cal C$ be any object. A morphism $e : c \rightarrow c$ is called \emph{idempotent} if $e^2 = e$. A \emph{splitting} of an idempotent $e : c \rightarrow c$ is the datum $(d, \iota, r)$ where $d\in \cal C$ is an object and $\iota, r$ are morphisms:
$$
c \xrightarrow{r} d \xrightarrow{\iota} c,\quad \text{such that $r\circ \iota = \tn{id}_d$ and $\iota\circ r = e$}.
$$
A splitting of $e$, if exists, is unique up to a unique isomorphism. We call the resulting object $d$ the \emph{image} of $e$.

\begin{lem}
\label{lem-splitting-idempotent}
Let $e : c \rightarrow c$ be an idempotent. The following are equivalent:
\begin{enumerate}
	\item $e$ splits;
	\item the diagram $c\xrightarrow{e} c$ admits a limit;
	\item the diagram $c\xrightarrow{e} c$ admits a colimit.
\end{enumerate}
\noindent
When these conditions hold, the image of $e$ identifies with both the limit and colimit of the diagram $c\xrightarrow{e} c$. \qed
\end{lem}

%%

\subsubsection{} We call $\cal C$ \emph{idempotent-complete} (or \emph{Karoubian}) if every idempotent in $\cal C$ splits.\footnote{This notion is discussed in [09SF] in the context of pre-additive categories.} By Lemma \ref{lem-splitting-idempotent}, a category $\cal C$ which admits finite limits \emph{or} finite colimits is idempotent-complete. This yields:

\begin{lem}
\label{lem-abelian-karoubian}
An abelian category $\cal C$ is idempotent-complete.\qed
\end{lem}

%%

\subsubsection{} Let $\tn{Ind}(\cal C)$ denote the ind-completion of $\cal C$. We view $\cal C$ as a full subcategory of $\tn{Ind}(\cal C)$ via the tautological embedding $\cal C\hookrightarrow\tn{Ind}(\cal C)$. By definition of $\tn{Ind}(\cal C)$, this embedding factors through:
$$
\cal C \hookrightarrow\tn{Ind}(\cal C)^{\tn{cpt}} \hookrightarrow\tn{Ind}(\cal C).
$$

\begin{lem}
\label{lem-compacts-in-ind}
Suppose $d\in \tn{Ind}(\cal C)$. The following are equivalent.
\begin{enumerate}
	\item $d \in \tn{Ind}(\cal C)^{\tn{cpt}}$;
	\item there is an idempotent $e : c\rightarrow c$ in $\cal C$ such that $d$ is isomorphic to the image of $e$.
\end{enumerate}
\end{lem}
\begin{proof}[(2) $\implies$ (1)]
Suppose $e : c \rightarrow c$ is an idempotent in $\cal C$ with image $d$. For any filtered diagram $b_i$ in $\tn{Ind}(\cal C)$ ($i\in\cal I$) with colimit $b$, the following diagram commutes:
$$
\xysmall{
	\underset{i \in \cal I}{\tn{colim}}\,\tn{Hom}(d. b_i) \ar[r]^-{\alpha_d} \ar[d] & \tn{Hom}(d, b) \ar[d] \\
	\underset{i \in \cal I}{\tn{colim}}\,\tn{Hom}(c. b_i) \ar[r]^-{\alpha_c} \ar[d] & \tn{Hom}(c, b) \ar[d] \\
	\underset{i \in \cal I}{\tn{colim}}\,\tn{Hom}(d. b_i) \ar[r]^-{\alpha_d} \ar[r] & \tn{Hom}(d, b)
}
$$
where $\alpha_c$ (resp.~$\alpha_d$) is the tautological map and both vertical compositions are identity. Since $\alpha_c$ is a bijection, the top commutative square shows that $\alpha_d$ is injective and the bottom commutative square shows that $\alpha_d$ is surjective. Hence $\alpha_d$ is a bijection as well.
\end{proof}

\begin{proof}[(1) $\implies$ (2)]
Let $d$ be a compact object of $\tn{Ind}(\cal C)$. By definition of $\tn{Ind}(\cal C)$, there is a filtered system $d_i$ in $\cal C$ ($i\in\cal I$) with $d \cong \underset{i\in\cal I}{\tn{colim}}\, d_i$, where the colimit is taken in $\tn{Ind}(\cal C)$. Since $d$ is compact, the natural map below is bijective:
$$
\underset{i\in \cal I}{\tn{colim}} \, \tn{Hom}(d, d_i) \rightarrow \tn{Hom}(d, d),
$$
so the morphism $\tn{id}_d$ factors as:
$$
d \xrightarrow{\iota} d_i \xrightarrow{r} d
$$
for some $i\in\cal I$. In particular, $e := \iota \circ r$ is an idempotent acting on $d_i \in \cal C$. This identifies $d$ as the image of a split idempotent in $\cal C$.
\end{proof}

\begin{lem}
\label{lem-karoubian-compacts}
Suppose $\cal C$ is idempotent-complete. Then the canonical embedding $\cal C\hookrightarrow\tn{Ind}(\cal C)^{\tn{cpt}}$ is an equivalence.
\end{lem}
\begin{proof}
This follows from Lemma \ref{lem-compacts-in-ind}.
\end{proof}

\begin{lem}
\label{lem-abelian-compacts}
Suppose $\cal C$ is abelian. Then the canonical embedding $\cal C \hookrightarrow\tn{Ind}(\cal C)^{\tn{cpt}}$ is an equivalence.
\end{lem}
\begin{proof}
This follows form Lemma \ref{lem-abelian-karoubian} and Lemma \ref{lem-karoubian-compacts}.
\end{proof}

%%

\subsubsection{} Suppose $(\cal C, \otimes, \mathbf 1)$ is a symmetric monoidal category.

\begin{lem}
\label{lem-dualizable-compact}
Suppose $\mathbf 1\in\cal C$ is compact and tensoring with any object $-\otimes c : \cal C \rightarrow \cal C$ preserves filtered colimits. Then any dualizable object of $\cal C$ is compact.
\end{lem}
\begin{proof}
Let $c\in \cal C$ be a dualizable object. Consider any filtered system $d_i \in \cal C$ ($i\in \cal I$) whose colimit exists, and is denoted by $d$. We have a commutative diagram:
$$
\xysmall{
	\underset{i\in \cal I}{\tn{colim}} \, \tn{Hom}(c, d_i) \ar[rr]\ar[d]^{\cong} & & \tn{Hom}(c, d) \ar[d]^{\cong} \\
	\underset{i\in \cal I}{\tn{colim}} \, \tn{Hom}(\mathbf 1, c^{\vee}\otimes d_i) \ar[r] & \tn{Hom}(\mathbf 1, \underset{i\in \cal I}{\tn{colim}}(c^{\vee}\otimes d_i)) \ar[r] & \tn{Hom}(\mathbf 1, c^{\vee}\otimes d)
}
$$
The two lower horizontal arrows are bijective, because $\mathbf 1$ is compact and tensoring with $c^{\vee}$ preserves filtered colimits. Thus the upper horizontal arrow is bijective.
\end{proof}

\begin{lem}
Suppose $(\cal C, \otimes, \mathbf 1)$ is a symmetric monoidal abelian category in which every $c\in \cal C$ is dualizable. Then the canonical embedding of symmetric monoidal categories
$$
(\cal C, \otimes, \mathbf 1) \hookrightarrow (\tn{Ind}(\cal C), \tn{Ind}\,\otimes, \mathbf 1)
$$
identifies $\cal C$ with the full subcategory of $\tn{Ind}(\cal C)$ consisting of dualizable objects.
\end{lem}
\begin{proof}
Since $\cal C$ is abelian, the embedding $\cal C\hookrightarrow\tn{Ind}(\cal C)$ identifies $\cal C$ with the compact objects in $\tn{Ind}(\cal C)$ by Lemma \ref{lem-abelian-compacts}. Since every object in $\cal C$ is dualizable (as an object of $\cal C$, hence also as an object of $\tn{Ind}(\cal C)$), we have a containement:
\begin{equation}
\label{eq-compact-dualizable}
\tn{Ind}(\cal C)^{\tn{cpt}} \subset \tn{Ind}(\cal C)^{\tn{dualizable}}
\end{equation}
On the other hand, $\mathbf 1$ is compact in $\tn{Ind}(\cal C)$ and $\tn{Ind}\,\otimes$ preserves filtered colimits [Amelie, \S1.4], so we may apply Lemma \ref{lem-dualizable-compact} to $\tn{Ind}(\cal C)$ and conclude that \eqref{eq-compact-dualizable} is an equivalence. 
\end{proof}

%%

%%

\subsection{Twisted arrows}

\subsubsection{} Let $\cal C$ be a category (resp.~a $k$-linear category for a fixed ring $k$.) Write $\tn{Tw}(\cal C)$ for the category whose objects are morphisms $\varphi : c \rightarrow d$ in $\cal C$, and a morphism $\varphi \rightarrow \varphi'$ is given by a commutative diagram:
$$
\xysmall{
	c \ar[r]^-{\varphi} \ar[d] & d \\
	c' \ar[r]^-{\varphi'} & d' \ar[u]
}
$$
Clearly, a morphism $\varphi \rightarrow \varphi'$ in $\tn{Tw}(\cal C)^{\tn{op}}$ is a commutative diagram:
$$
\xysmall{
	c \ar[r]^-{\varphi} & d \ar[d] \\
	c' \ar[u]\ar[r]^-{\varphi'} & d'
}
$$

%%

\subsubsection{} The twisted arrow category allows one to express morphisms between functors as a suitable limit.

\begin{lem}
\label{lem-twisted-arrow}
Suppose $F, G : \cal C \rightarrow \cal D$ are functors. Then there is a natural bijection:
$$
\tn{Hom}(F, G) \xrightarrow{\sim} \underset{(c\rightarrow d)\in \tn{Tw}(\cal C)^{\tn{op}}}{\tn{lim}} \tn{Hom}_{\cal D}(F(c), G(d)).
$$
\end{lem}

%%

\subsubsection{}
\label{sec-twisted-arrow-tensor} Suppose $(\cal C, \otimes, \mathbf 1)$ is a symmetric monoidal category. Then $\tn{Tw}(\cal C)$ inherits a symmetric monoidal structure defined as follows. For any $\varphi : c\rightarrow d$, $\varphi' : c'\rightarrow d'$, we write $\varphi\otimes\varphi'$ for the monoidal product:
$$
\varphi\otimes\varphi' : c\otimes c' \rightarrow d\otimes d'.
$$
This product is clearly functorial in $\varphi$ and $\varphi'$. The unit in $\tn{Tw}(\cal C)$ is define to be the identity arrow on $\mathbf 1\in\cal C$. The natural isomorphisms:
$$
\mathbf 1\otimes - \xrightarrow{\sim}\tn{id}_{\tn{Tw}(\cal C)},\quad -\otimes \mathbf 1 \xrightarrow{\sim} \tn{id}_{\tn{Tw}(\cal C)}
$$
are the obvious ones.

The associativity constraint is inherited from $\cal C$ as the commutative diagram:
$$
\xymatrix@R=1.5em@C=4em{
	(c \otimes c')\otimes c'' \ar[d]^{\cong}\ar[r]^{(\varphi\otimes\varphi')\otimes\varphi''} & (d\otimes d')\otimes d'' \\
	c \otimes (c'\otimes c'') \ar[r]^{\varphi\otimes(\varphi'\otimes\varphi'')} & d \otimes (d'\otimes d'') \ar[u]_{\cong}
}
$$
and the commutativity constraint is inherited from $\cal C$ as the commutative diagram:
$$
\xymatrix@R=1.5em@C=4em{
	c \otimes c' \ar[d]^{\cong}\ar[r]^{\varphi'\otimes\varphi} & d\otimes d' \\
	c' \otimes c \ar[r]^{\varphi\otimes\varphi'} & d' \otimes d \ar[u]_{\cong}
}
$$
We omit verifying the coherence conditions satisfied by these data.

%%

%%

\subsection{(Co)monadicity}

\subsubsection{} Let $\cal C$ be a category (resp.~$k$-linera category). A \emph{comonad} acting on $\cal C$ is an endofunctor $T : \cal C\rightarrow \cal C$ equipped with the additional data:
\begin{enumerate}
	\item a comultiplication $\Delta : T \rightarrow T\circ T$;
	\item a counit $\varepsilon : T \rightarrow \tn{id}_{\cal C}$,
\end{enumerate}
satisfying the coassociative and counital conditions. In other words, a comonad is a coassciative coalgebra in the monoidal category $(\tn{End}(\cal C), \circ, \tn{id}_{\cal C})$.

%%

\subsubsection{} Suppose $(T, \Delta, \varepsilon)$ is a comonad acting on $\cal C$. We slightly abuse the notation and refer to the endofunctor $T$ as the comonad. A \emph{$T$-comodule} in $\cal C$ is an object $c\in \cal C$ equipped with a coaction:
$$
\alpha : c \rightarrow T(c),
$$
satisfying the following two conditions:
\begin{enumerate}
	\item the two compositions are equal:
	$$
	\xysmall{
		c \ar[r]^-{\alpha} & T(c)  \ar@<0.5ex>[r]^-{\Delta}\ar@<-0.5ex>[r]_-{T(\alpha)} & T\circ T(c).
	}
	$$
	\item the composition $c \xrightarrow{\alpha} T(c) \xrightarrow{\varepsilon} c$ is the identity on $c$.
\end{enumerate}
In other words, a $T$-comodule in $\cal C$ is a comodule object in $\cal C$ with respect to the canonical $\tn{End}(\cal C)$-action on $\cal C$ and the coassociative coalgebra object $T\in\tn{End}(\cal C)$.

%%

\subsubsection{Set-up}
\label{set-up-adjunction} Suppose we are given an adjuction:
$$
\xysmall{
	F : \cal C \ar@<0.5ex>[r] & \cal D : G. \ar@<0.5ex>[l]
}
$$
Then the endofunctor $T:= F\circ G$ has a canonical structure of a comonad acting on $\cal D$. Furthermore, the functor $F$ canonically factors as follows:
$$
\xysmall{
	\cal C \ar[r]^-{\widetilde F}\ar[dr]_F & T\Comod(\cal D) \ar[d]^{\tn{oblv}} \\
	& \cal D
}
$$
where $\tn{oblv}$ is the forgetful functor.

%%

\subsubsection{} We call a functor $F : \cal C \rightarrow \cal D$ \emph{conservative} if for every morphism $\varphi : c_1 \rightarrow c_2$ in $\cal C$, if $F(\varphi)$ is an isomorphism, then so is $\varphi$. The following is a form of the Barr--Beck theorem.

\begin{lem}
\label{lem-barr-beck}
In the set-up \S\ref{set-up-adjunction}, the functor $\widetilde F$ is an equivalence if and only if the following conditions are satisfied.
\begin{enumerate}
	\item $F$ is conservative;
	\item $\cal C$ contains $F$-cosplit equalizers and $F$ preserves them.
\end{enumerate}
\end{lem}


%%

%%

\subsection{Miscellaneous facts}

\subsubsection{} A fact about obtaining commutative algebras:

\begin{lem}
\label{lem-colimit-commutative-algebra}
Suppose $(\cal C, \otimes, \mathbf 1)$ is a symmetric monoidal category, in which $\otimes$ commutes with all colimits. Let $F : (\cal I, \otimes, \mathbf 1) \rightarrow (\cal C, \otimes, \mathbf 1)$ be a symmetric monoidal functor. Then the natural maps:
\begin{align*}
\mathbf 1 &\xrightarrow{\sim} F(\mathbf 1) \\
F(i) \otimes F(j) &\xrightarrow{\sim} F(i\otimes j)
\end{align*}
define a commutative algebra structure on $\underset{\cal I}{\tn{colim}}\,F \in \cal C$. \qed
\end{lem}

%%

\subsubsection{} Let $\cal C$ be a category (resp.~a $k$-linear category).

\begin{lem}
\label{lem-ind-finite-colimit}
The canonical embedding $\cal C \hookrightarrow\tn{Ind}(\cal C)$ preserves finite colimits.
\end{lem}
\begin{proof}
This follows from the fact that in $\tn{Set}$ (resp.~$k\Mod$), finite limits commute with filtered colimits.
\end{proof}

\begin{lem}
\label{lem-ind-all-colimits}
Suppose $\cal C$ contains finite colimits. Then
\begin{enumerate}
	\item $\tn{Ind}(\cal C)$ contains all colimits.
	\item For any functor $F : \cal C \rightarrow\cal D$ preserving finite colimits, where $\cal D$ is a cocomplete category, the ind-extension
	$$
	\tn{Ind}(F) : \tn{Ind}(\cal C) \rightarrow \cal D
	$$
	preserves all colimits.
\end{enumerate}
\end{lem}
\begin{proof}
This follows from Lemma \ref{lem-ind-finite-colimit} together with the fact that all colimits are generated by finite and filtered ones.
\end{proof}



\end{document}