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let $\Phi$ be a class of small categories (for example, finite categories). A category $D$ is called $\Phi$-cocomplete if it admits all $\Phi$-colimits, i.e. for diagrams of the form $I \to D$ with $I \in \Phi$. A functor is called $\Phi$-cocontinuous if it preserves $\Phi$-colimits. Now it is a well-known result that every category $C$ admits a $\Phi$-cocompletion $\Phi(C)$, which is a $\Phi$-cocomplete category equipped with a functor $C \to \Phi(C)$ which induces an equivalence of categories between the category of $\Phi$-cocontinuous functors $\Phi(C) \to D$ and all functors $C \to D$, where $D$ is a $\Phi$-cocomplete category.

I want to understand this construction as detailed as possible. I've seen sketches of it in many papers. For details the authors often refer to Kelly's book Basic concepts of enriched categories, in this case section 5.7. But for me this is too complicated and I would prefer a more direct and down-to-earth exposition (hopefully avoiding all the theory of Kan extensions). Besides I'm not interested yet in the specialities of enriched categories. Perhaps someone knows a nice and understandable reference?

Here is what I've already understood: Consider the Yoneda embedding $Y : C \to [C^{op},\mathrm{Set}]$. Close the image under isomorphisms and under $\Phi$-colimits; this is done by a transfinite construction. If $\Phi$ is small, then this construction stops after some ordinal (and otherwise it stops after some ordinal outside of the universe?). So we get a $\Phi$-cocomplete full subcategory $\Phi(C) \subseteq [C^{op},\mathrm{Set}]$. We claim that $Y : C \to \Phi(C)$ is the desired cocompletion. So let $F : C \to D$ be a functor to a $\Phi$-cocomplete category $D$. What is the definition of the induced $\Phi$-cocontinous functor $G : \Phi(C) \to D$? It is clear what to do on $Y(C)$. Probably the rest is transfinite recursion? If $y \in \Phi(C)$, then $y$ is a $\Phi$-colimit of objects $y_i$ for which $G(y_i)$ has already been constructed. Let us also assume that the action on morphisms has been constructed, then we get a diagram and we may take the colimit $G(y) := \mathrm{colim}_i G(y_i)$. This must be the right definition on objects. Now if $y \to y'$ is a morphism, how can we define $G(y) \to G(y')$?

Finally I would like to ask for more specific descriptions of $\Phi(C)$. For example, if $\Phi$ is the class of all finite categories, i.e. we want to adjoin finite colimits, then why is every object of $\Phi(C)$ a coequalizer of two morphisms between finite coproducts of representables? Again this seems to be well-known, but I couldn't find a reference which explains this directly. It seems to be a result concerning distributivity of finite coproducts and coequalizers.

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Actually, I think the approach with Kan extensions and weighted colimits is actually quite insightful. Let me try to explain it briefly.

Let $\mathcal{D}$ be a locally small category. Recall that the colimit of a diagram $X : \mathcal{J} \to \mathcal{D}$ weighted by a presheaf $W : \mathcal{J}^\mathrm{op} \to \mathbf{Set}$ is an object $W \star_\mathcal{J} X$ equipped with bijections $$[\mathcal{J}^\mathrm{op}, \mathbf{Set}](W, \mathcal{D}(X, D)) \cong \mathcal{D}(W \star_\mathcal{J} X, D)$$ that are natural in $D$. The diagram $\mathcal{J}$ need not be small for this definition to make sense. This is sometimes also called the functor tensor product. It is clear that ${-} \star_{\mathcal{J}} {-}$ is functorial and preserves colimits in both variables, and the Yoneda lemma implies $$Y (j) \star_{\mathcal{J}} X \cong X (j)$$ for all $j$ in $\mathcal{J}$. Thus, if $W$ can be expressed as an iterated $\Phi$-colimit of representable presheaves and $\mathcal{D}$ has $\Phi$-colimits, then $W \star_{\mathcal{J}} X$ exists in $\mathcal{D}$.

Now take $\mathcal{J} = \mathcal{C}$, $X = F : \mathcal{C} \to \mathcal{D}$ and assume $\mathcal{D}$ has $\Phi$-colimits. By what we said before, $P \star_{\mathcal{C}} F$ exists in $\mathcal{D}$ as long as $P$ is in $\Phi (\mathcal{C})$, so we obtain a functor ${-} \star_{\mathcal{C}} F : \Phi (\mathcal{C}) \to \mathcal{D}$ which preserves $\Phi$-colimits and which extends $F : \mathcal{C} \to \mathcal{D}$ along the Yoneda embedding $Y : \mathcal{C} \to \Phi (\mathcal{C})$, as required.

Obtaining an explicit description of $\Phi (\mathcal{C})$ seems to be a difficult problem in general and indeed relies on facts about distributivity of colimits in $\mathbf{Set}$. I thought about the finite limit case a few months ago but I didn't find a reference either.

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  • $\begingroup$ I already knew this description using coends in the case that $\Phi$ consists of all small categories and didn't realize that the same works here. Thanks a lot. Somehow I was confused since the formula $\hom(P \otimes_{\mathcal{C}} F,T) \cong \hom(P,\hom(F(-),T))$ suggests that $- \otimes_{\mathcal{C}} F$ is a left adjoint and therefore preserves all colimits. But this seems to be wrong, since $\hom(F(-),T)$ doesn't have to belong to $\Phi(C)$. $\endgroup$ – Martin Brandenburg Aug 19 '13 at 12:43
  • $\begingroup$ As for the remaining question: I've read this as Lemma 5 in arxiv.org/pdf/1212.1545v1.pdf. The author refers to a statement in Kelly's Structures defined by finite limits in the enriched context. Again I don't get it. $\endgroup$ – Martin Brandenburg Aug 19 '13 at 13:12
  • $\begingroup$ Hmmm. I don't see the connection either, but I haven't studied the paper carefully. The way I would do it is to show that the class of presheaves obtained as colimits of finite diagrams of representables is already closed under finite colimits: for finite coproducts this is obvious, and for coequalisers one builds a diagram that looks a bit like a bipartite graph. $\endgroup$ – Zhen Lin Aug 19 '13 at 14:31
  • $\begingroup$ Alright. The reason is basically that every representable is projective. Then every map from it into a cokernel already maps into the domain of the cokernel, so we can add this etc. $\endgroup$ – Martin Brandenburg Aug 20 '13 at 10:19

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