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\begin{document}

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\section*{CocommCoalg}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{colimits}{Colimits}\dotfill \pageref*{colimits} \linebreak
\noindent\hyperlink{local_presentability}{Local presentability}\dotfill \pageref*{local_presentability} \linebreak
\noindent\hyperlink{totality_and_completeness}{Totality and completeness}\dotfill \pageref*{totality_and_completeness} \linebreak
\noindent\hyperlink{cartesian_closure}{Cartesian closure}\dotfill \pageref*{cartesian_closure} \linebreak
\noindent\hyperlink{comonadicity_over_}{Comonadicity over $R Mod$}\dotfill \pageref*{comonadicity_over_} \linebreak
\noindent\hyperlink{extensivity}{Extensivity}\dotfill \pageref*{extensivity} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


This page describes some nice properties of the [[category]] $R Cocomm Coalg$ of [[cocommutative coalgebra|cocommutative]] [[coalgebras]] over a [[ground ring]] $R$, in particular details of the proof that it is a [[complete category|complete]], [[cocomplete category|cocomplete]], [[lextensive category|lextensive]], [[cartesian closed category]] with a [[generating set]].

In the sequel, we will simply say ``coalgebra'', although we really mean \emph{cocommutative} (coassociative, counital) coalgebra.

\hypertarget{colimits}{}\subsection*{{Colimits}}\label{colimits}

Colimits in $R Cocomm Coalg$ are created (reflected) by the forgetful functor $U: R Cocomm Coalg \to R Mod$. As the codomain is cocomplete, so is the domain. Thus for example, the coproduct of coalgebras $C, D$ is the direct sum module $C \oplus D$ equipped with the evident comultiplication $C \oplus D \to (C \otimes C) \oplus (D \otimes D) \hookrightarrow (C \oplus D) \otimes (C \oplus D)$.

It is perhaps well to point out explicitly that if $f: C \to D$ is a coalgebra map, then the [[image]] $f(C)$ in $R Mod$ inherits a coalgebra structure from $D$ and provides the image in $R Cocomm Coalg$. This is an easy consequence of the fact that $f(C \otimes C) = f(C) \otimes f(C)$ as submodules of $D \otimes D$.

\hypertarget{local_presentability}{}\subsection*{{Local presentability}}\label{local_presentability}

Most of the nice properties can be derived as consequences of the [[locally presentable category|local presentability]] of $R Cocomm Coalg$. For this we refer to the paper by \hyperlink{Porst}{Porst}.

In the case where $R$ is a [[field]] $k$, the category $k Cocomm Coalg$ is locally finitely presentable. The finitely presentable objects of $k Cocomm Coalg$ are those coalgebras that are [[finite-dimensional vector space|finite-dimensional]] as [[vector spaces]]. The first step towards establishing locally finite presentability is the fundamental theorem of coalgebras, which guarantees that every coalgebra is a filtered colimit of finite-dimensional coalgebras:

\begin{theorem}
\label{}\hypertarget{}{}
Every coalgebra $C$ is the union of its finite-dimensional subcoalgebras, i.e., is the [[directed colimit]] of the system of finite-dimensional subcoalgebras of $C$ and inclusion maps between them.

\end{theorem}
(For the nonce, we define ``subcoalgebra'' of $C$ to mean vector subspace $i: V \hookrightarrow C$ such that the restricted comultiplication $\delta_C \circ i$ is contained in the subspace $V \otimes V$ of $C \otimes C$. Later we will see that subcoalgebras are actually the same thing as [[subobjects]] in $k Cocomm Coalg$ in the sense of equivalence classes of [[monomorphisms]].)

\begin{prop}
\label{}\hypertarget{}{}
Every finite-dimensional coalgebra $C$ is finitely presentable, i.e., $\hom(C, -): k Cocomm Coalg \to Set$ preserves filtered colimits.

\end{prop}
\begin{proof}
If $D = colim_j D_j$ is a directed colimit, then the image of a coalgebra map $f: C \to D$ is a finite-dimensional subcoalgebra inclusion $f(C) \to D$ which, as a finitely presentable vector space, is included in one of the components $D_j \to D$ of the colimit cone; this inclusion is a subcoalgebra inclusion.

\end{proof}
It follows easily from these results and cocompleteness of $k Cocomm Coalg$ that $k Cocomm Coalg$ is locally finitely presentable. As a result we have a [[Gabriel-Ulmer duality|Gabriel-Ulmer]] equivalence

\begin{displaymath}
k Cocomm Coalg \simeq Lex(k Cocomm Coalg_{fd}^{op}, Set)
\end{displaymath}
where the category of finite-dimensional coalgebras is dual to the category of finite-dimensional algebras, so that also

\begin{displaymath}
k Cocomm Coalg \simeq Lex(Comm Alg_{fd}, Set).
\end{displaymath}
Naturally, choosing a representative of each isomorphism class of finite-dimensional coalgebras, we obtain a generating set of $k Cocomm Coalg$.

\hypertarget{totality_and_completeness}{}\subsection*{{Totality and completeness}}\label{totality_and_completeness}

As is the case for any locally presentable category, $R Cocomm Coalg$ is not only cocomplete but is a [[total category|totally cocomplete category]] and is (therefore) [[complete category|complete]] as well.

The construction of [[limits]] can be described explicitly. The [[equalizer]] of two coalgebra maps $f, g: C \stackrel{\to}{\to} D$ is the largest subcoalgebra contained in the $R Mod$ equalizer (the latter coincides with the equalizer as computed in [[Set]]). This can be described even more explicitly in [[Sweedler notation]]: the equalizer of $f, g$ is the set

\begin{displaymath}
E = \{c \in C: c_{(1)} \otimes f(c_{(2)}) \otimes c_{(3)} = c_{(1)} \otimes g(c_{(2)}) \otimes c_{(3)}\}
\end{displaymath}
with the unique structure of coalgebra that makes it a subcoalgebra of $C$.

Assuming the construction of [[cofree coalgebra|cofree cocommutative coalgebra]]s, viz. the right adjoint $K: R Mod \to R Cocomm Coalg$ to the forgetful functor $U: R Cocomm Coalg \to R Mod$ (which we also touch on below), the [[product]] of a family of coalgebras $C_i$ can be described as follows. Consider the product $\prod_i C_i$ taken in $R Mod$, and let $p: U K(\prod_i C_i) \to \prod_i C_i$ be the component of the counit of the adjunction $U \dashv K$ at that product. Then the product $C$ of the $C_i$ taken in $R Cocomm Coalg$ is the largest subcoalgebra $j: C \hookrightarrow K(\prod_i C_i)$ such that each composite $\pi_i p j: C \to C_i$ is a coalgebra map. For a proof, see the \hyperlink{Agore}{article} by Agore.

\hypertarget{cartesian_closure}{}\subsection*{{Cartesian closure}}\label{cartesian_closure}

The [[cartesian product]] of two coalgebras $C, D$ is given by $C \otimes D$ with the evident coalgebra structure. The functor $C \otimes -: R Cocomm Coalg \to R Cocomm Coalg$ is a [[cocontinuous functor]], since colimits are reflected from $R Mod$ and $C \otimes -: R Mod \to R Mod$ is cocontinuous there.

For locally presentable or more generally total categories $\mathbf{C}$, cocontinuity of a functor $F: \mathbf{C} \to \mathbf{C}$ is enough to guarantee that $F$ has a right adjoint. It follows that $R Cocomm Coalg$ is cartesian closed.

\hypertarget{comonadicity_over_}{}\subsection*{{Comonadicity over $R Mod$}}\label{comonadicity_over_}

Again, since $U: R Cocomm Coalg \to R Mod$ is cocontinuous and $R Cocomm Coalg$ is locally presentable, $U$ has a right adjoint $K$. This is described more explicitly at [[cofree coalgebra|cofree cocommutative coalgebra]].

The comonadicity of $U$ is proven in the \hyperlink{Barr}{article} by Barr, section 4.

\hypertarget{extensivity}{}\subsection*{{Extensivity}}\label{extensivity}

We will show that $k Cocomm Coalg$ is a [[lextensive category]].

\begin{prop}
\label{}\hypertarget{}{}
Coproducts of coalgebras are disjoint.

\end{prop}
\begin{proof}
The coproduct of coalgebras $C, D$ is $C \oplus D$, and the pullback of the two coalgebra inclusions $i_C: C \to C \oplus D, i_D: D \to C \oplus D$ is the coalgebra equalizer of the two maps

\begin{displaymath}
\itexarray{
 & & C & &  \\ 
 & _\mathllap{\pi_C} \nearrow & & \searrow _\mathrlap{i_C} \\ 
C \otimes D & & & & C \oplus D \\
 & _\mathllap{\pi_D} \searrow & & \nearrow _\mathrlap{i_D} \\ 
 & & D & &
}
\end{displaymath}
This coalgebra equalizer is constructed as the largest subcoalgebra of $C \otimes D$ contained in the $k Vect$ equalizer. But the $k Vect$ equalizer is easily seen to be $0$ (cf. the fact that $k Vect$ coproducts are themselves disjoint), and so the coalgebra equalizer must be $0$ as well, concluding the proof.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Coproducts of coalgebras are stable under pullback.

\end{prop}
\begin{proof}
Consider the following diagram:

\begin{displaymath}
\itexarray{
& & F & & \\ 
& & \downarrow j & & \\
& & X & & \\ 
& & \downarrow f & & \\ 
C & \underset{i_C}{\to} & C \oplus D & \underset{i_D}{\leftarrow} & D
}
\end{displaymath}
where $f$ is an arbitrary coalgebra map and $j: F \hookrightarrow X$ is a finite-dimensional subcoalgebra inclusion. We will show that the pullback of the coproduct diagram along $f \circ j$ is a coproduct decomposition of $F$.

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Michael Barr, \href{http://www.math.mcgill.ca/barr/ftp/pdffiles/coalgebra.pdf}{Coalgebras over a commutative ring}, J. Alg. 32 (1974), 600--610.

\end{itemize}
\begin{itemize}%
\item Hans-Eberhard Porst, \href{http://dml.cz/bitstream/handle/10338.dmlcz/108017/ArchMathRetro_042-2006-4_7.pdf}{On corings and comodules}, Archivum Mathematicum (2006), No. 4, 419-425.

\end{itemize}
\begin{itemize}%
\item Ana L. Agore, \emph{Limits of Coalgebras, Bialgebras and Hopf Algebras}, arxiv.org/pdf/1003.0318, 2010. (\href{http://arxiv.org/pdf/1003.0318.pdf}{pdf})

\end{itemize}
[[!redirects Cocomm Coalg]] [[!redirects CommCoalg]] [[!redirects Comm Coalg]] [[!redirects category of cocommutative coalgebras]] [[!redirects category of commutative coalgebras]]

category: category



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