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%opening
\title{A Note on Ind-Completion}
\author{Simon Felten}

\begin{document}

\maketitle

Let $C$ be a category. Then the \emph{ind-completion} $\mathrm{Ind}(C)$ is the full subcategory of $\mathrm{PSh}(C)$ of objects which are filtered colimits of representable presheaves. The Yoneda construction yields a fully faithful embedding $C \subset \mathrm{Ind}(C)$. This embedding has the following universal property:

\begin{lemma}
 Let $D$ be a category which has all filtered colimits, and let $F: C \to D$ be a functor. Then there is, up to isomorphism, a unique functor $\tilde F: \mathrm{Ind}(C) \to D$ such that $\tilde F|_C = F$ and $\tilde F$ preserves filtered colimits.
\end{lemma}
\begin{lemma}
 $\mathrm{Ind}(C)$ has all filtered colimits, and the inclusion $\mathrm{Ind}(C) \subset \mathrm{PSh}(C)$ preserves filtered colimits.
\end{lemma}

\begin{ex}
 Let $A \in \mathrm{Ab}$ be an abelian group and consider the functor 
 $$F(-) = \mathrm{Hom}(A,-): \mathrm{Ab} \to \mathrm{Ab};$$
 since $\mathrm{Ab}$ is cocomplete, we get a factorization $\tilde F: \mathrm{Ind}(\mathrm{Ab}) \to \mathrm{Ab}$ which preserves filtered colimits. In particular, $\mathrm{Ab} \subset \mathrm{Ind}(\mathrm{Ab})$ does \emph{not} preserve filtered colimits, for $F(-)$ does not. Moreover, we see that $\mathrm{Ind}(C)$ of a category which admits all filtered colimits need not be equivalent (via the natural embedding) to $C$.
\end{ex}
\begin{ex}
 Let $k$ be a field, let $\mathrm{Vec}_k$ be the category of $k$-vector spaces, and let $\mathrm{FVec}_k$ be the category of finite-dimensional $k$-vector spaces. We have a canonical functor 
 $$Y: \mathrm{Vec}_k \to \mathrm{PSh}(\mathrm{FVec}_k), \quad V \mapsto (T \mapsto \mathrm{Hom}_k(T,V)),$$
 which is fully (?) faithful. It preserves filtered colimits (but I cannot prove it ...).
 
 We have a factorization $G: \mathrm{Ind}(\mathrm{FVec}_k) \to \mathrm{Vec}_k$ because $\mathrm{Vec}_k$ has all filtered colimits, and the composition $Y \circ G$ must coincide, up to isomorphism, with the canonical embedding $\mathrm{Ind}(\mathrm{FVec}_k) \to \mathrm{PSh}(\mathrm{FVec}_k)$ because $Y$ preserves filtered colimits. Thus, $G$ is an equivalence of categories. We see that, in this example, the embedding $C \to \mathrm{Ind}(C)$ preserves \emph{finite} colimits.
\end{ex}
\begin{ex}
 Let 
 $$F: \mathrm{FVec}_k \to \mathrm{Vec}_k, \quad V \mapsto \mathrm{Hom}(\bigoplus_{i = 0}^\infty k,V);$$
 then there is a factorization $\tilde F: \mathrm{Vec}_k \to \mathrm{Vec}_k$ which preserves filtered colimits. Since $\mathrm{Hom}(\bigoplus_{i = 0}^\infty k,-)$ does \emph{not} preserve filtered colimits, it is not equal to $\tilde F$.
\end{ex}

\begin{lemma}
 The inclusion $C \subset \mathrm{Ind}(C)$ preserves \emph{finite} colimits.
\end{lemma}
\begin{proof}
 In the category of sets, finite limits commute with filtered colimits.
\end{proof}
\begin{ex}
 In general, the inclusion $\mathrm{Ind}(C) \subset \mathrm{PSh}(C)$ does \emph{not} preserve finite colimits. Namely, let $C = \mathrm{Ab}$. Then the inclusion $C \subset \mathrm{PSh}(C)$ is given by $A \mapsto (T \mapsto \mathrm{Hom}(T,A))$, so it does not preserve cokernels.
\end{ex}











\end{document}
