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Let $C$ be a (small) category. Recall that the category $pro(C)$ of pro-objects in $C$ is the category that has as objects the diagrams $f:D\to C$ with $D$ a small cofiltered category, and for $f:D\to C$ and $g:E\to C$ the set of morphisms is $$\hom_{pro(C)}(f,g):=\varprojlim_{e\in E}\varinjlim_{d\in D}\hom_C(f(d),g(e))\ .$$ It is the category of "formal limits" in $C$. It is obvious that $pro(C)$ is complete. My question is:

Q1: Under what conditions on $C$, if any, is $pro(C)$ cocomplete?

My question stems from the following two examples:

  1. Let $Vect$ be the category of vector spaces over a fixed field, and let $fVect$ be the category of finite dimensional vector spaces. Every $V\in Vect$ is the colimit of all its finite dimensional subspaces. Therefore, linear duality gives an anti-equivalence of categories $$-^*:Vect\longrightarrow pro(fVect)\ .$$ As $Vect$ is complete, it follows that $pro(fVect)$ is cocomplete (even though $fVect$ is neither complete nor cocomplete).
  2. The same situation is true if we replace $Vect$ by the category of coassociative coalgebras and $fVect$ by the category of finite dimensional associative algebras, see Getzler-Goerss, Prop. 1.7.

Since I am here, I may as well ask a second question that has been nagging at me for some time.

Q2: How to show that the functor $-^*$ is essentially surjective?

If $f:D\to fVect$ is an object of $pro(fVect)$, one can dualize $f$ to get $$f^*:D^{op}\longrightarrow fVect\ ,$$ i.e. $f^*(d):=f(d)^*$ and similarly for arrows, and then take its colimit in $Vect$ (not $fVect$) to obtain a vector space $V_f$. This would be my candidate for a preimage of $f$ (up to isomorphism). However, if we take the diagram of all finite dimensional subspaces of $V_f$ we might find a diagram which is much bigger than the original $f$ (consider e.g. $f:*\to fVect$ a single finite dimensional vector space of dimension greater than $1$...) Can one exhibit an isomorphism between $f$ and the new diagram in an easy way?

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    I think the category of pro-objects in a small category is always locally presentable, which would imply that it's cocomplete. – 2017-02-22
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    @QiaochuYuan Hi. I am not really familiar with the concept. Would you happen to have a good reference by any chance? – 2017-02-22
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    @QiaochuYuan By the way, another question I have in the same context is: is the functor $pro(fVect)\to Vect$ given by taking the limit in $Vect$ an equivalence of categories between $pro(fVect)$ and its image? (But I expect the answer to be no.) – 2017-02-22
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    You can start here: https://ncatlab.org/nlab/show/locally+presentable+category . Also, I lied, I think the correct statement is that ind-categories are always locally presentable, but since locally presentable categories are both complete and cocomplete it still implies the desired result. And no: morphisms between profinite vector spaces can be described as the morphisms which are continuous with respect to a certain topology, so the functor you describe fails to be full. – 2017-02-22
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    Also, maybe I need ind-categories of finitely cocomplete small categories. Not sure. – 2017-02-22
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    @QiaochuYuan Ok, thanks. – 2017-02-22

1 Answers 1

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I thought about it, and I think I have an answer. Suppose that $C$ is category and that $\tilde{C}$ is a full subcategory. Further assume:

  1. The category $C$ is complete and cocomplete.
  2. Every object of $C$ is canonically a colimit of a diagram in $\tilde{C}$. I.e. we suppose that there exists a functor $$\delta:C\longrightarrow ind(\tilde{C})$$ to the category of ind-objects in $\tilde{C}$ such that the functor $\text{colim}\ \delta$ is naturally isomorphic to the identity. Here, $\text{colim}$ is the colimit taken in the category $C$.
  3. Every object in $\tilde{C}$ is compact in $C$. That means that every morphism (in $C$) from an object of $\tilde{C}$ to a colimit of objects in $C$ splits through the inclusions.
  4. We have an anti-equivalence of categories $$-^*:\tilde{C}\longrightarrow\tilde{C}\ .$$

Then $\delta$ and $\text{colim}$ give an equivalence of categories between $C$ and $ind(\tilde{C})$. To prove this, notice the following. If $F:D\to\tilde{C}$ and $G:E\to\tilde{C}$ are two objects in $ind(\tilde{C})$, then \begin{align} ind(\tilde{C})(F,G) :=&\ \lim_{d\in D}\underset{e\in E}{\text{colim}}\ \tilde{C}(F(d),G(e))\\ =&\ \lim_{d\in D}\underset{e\in E}{\text{colim}}\ C(F(d),G(e))\\ \cong&\ \lim_{d\in D}\ C(F(d),\text{colim}\ G)\\ \cong&\ C(\text{colim}\ F,\text{colim}\ G)\ . \end{align} In the second line we used the fact that $\tilde{C}$ is a full subcategory of $C$, in the third one we used assumption (3), and in the fourth one the fact that the hom functor takes colimits in the first slot to limits of sets. We use this to construct a natural transformation $$\delta\ \text{colim}\Rightarrow1_{ind(\tilde{C})}\ .$$ Let $F\in ind(\tilde{C})$, then \begin{align} ind(\tilde{C})(F,\delta\ \text{colim}\ F)\cong&\ C(\text{colim}\ F,\text{colim}\ \delta\ \text{colim}\ F)\\ \cong&\ C(\text{colim}\ F,\text{colim}\ F)\ , \end{align} where we used assumption (2). Therefore, the morphism $1_{\text{colim}\ F}$ provides a canonical element of $ind(\tilde{C})(F,\delta\ \text{colim}\ F)$, and thus a natural transformation $$\delta\ \text{colim}\Rightarrow1_{ind(\tilde{C})}\ .$$ To conclude, the anti-equivalence of categories gives us an equivalence between the categories $pro(\tilde{C})$ and $ind(\tilde{C})$. It follows that the categories $C$ and $pro(\tilde{C})$ are equivalent, and thus the latter category is cocomplete as the first one is complete by assumption (1).

The conditions (1)-(4) may seem restrictive, but they are satisfied in both examples presented in the question (e.g. with $C=Vect$ and $\tilde{C}=fVect$ in the first example).

Useful links: the nlab pages on ind-objects, pro-objects, and compact objects.