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We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that the $\hom$ functor sends colimits in the first slot to limits. I am confronted with a situation where I would like the opposite to be true, which sparked my interest about the most general context where this would happen.

Consider a diagram $D$ in the full subcategory category of chain complexes given by chain complexes "of finite type", by which I will mean chain complexes that have degrees bounded below (or: bounded above) and that are finite dimensional in every degree. In particular, these chain complexes have the property that they are isomorphic to their double duals through the canonical inclusion $v\mapsto\langle-, v\rangle$.

Suppose that this diagram $D$ has a limit $\lim D$ in the category of chain complexes and that this limit is in the full subcategory of chain complexes of finite type. Examples would be finite products (trivial in what I want to do) or kernels (which interest me much more).

Then, the dual of the limit of the diagram is the colimit of the dual of the diagram. Indeed, we can consider the colimit $\operatorname{colim}D^\vee$ of the dual diagram $D^\vee$ and take its dual, which gives us the limit of the double dual diagram. But $$\left(\operatorname{colim}D^\vee\right)^\vee\cong\lim D^{\vee\vee}\cong\lim D$$ since $D$ is in the subcategory of finite type. Notice that his in particular implies that $\operatorname{colim}D^\vee$ is of finite type (as taking duals can only increase dimensions). Then $$\left(\lim D\right)^\vee\cong\left(\operatorname{colim}D^\vee\right)^{\vee\vee}\cong \operatorname{colim}D^\vee.$$ Is there a nice and general category theoretical explanation for this phenomenon? What are (reasonably) general situations where something like this occurs, and could someone provide a reference?

Also more generally: when are duals of limits the colimit of the duals?

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    $\begingroup$ Perhaps I am missing something, but is this phenomenon not just happening because the dual functor gives an equivalence of categories $C_{ft} \simeq C_{ft}^{op}$? So it takes (co)limits in $C_{ft}$ to (co)limits in $C_{ft}^{op}$. $\endgroup$
    – Sam Gunningham
    Commented Jun 18, 2020 at 12:29
  • $\begingroup$ @SamGunningham The point is that I am not taking the limits and colimits in the category of finite type chain complexes but in the category of all chain complexes instead and asking for $\lim D$ to have finite type. I'll try to make this clearer in the OP. Also a question would be what are possible good definitions of "duality" and "objects of finite type" in other categories. $\endgroup$
    – Daniel Robert-Nicoud
    Commented Jun 18, 2020 at 16:03
  • $\begingroup$ @SamGunningham Of course, limits in $C_{ft}$ are the same as limits in $C$ (when they exist). But what happens in more general cases? Is there a clean explanation of this phenomenon? (Am I doing something wrong? Am I missing something obvious?) $\endgroup$
    – Daniel Robert-Nicoud
    Commented Jun 18, 2020 at 16:14
  • $\begingroup$ Not an answer in the context you want, but, in the category of locally convex topological (complex) vector spaces, I think it is not always true that the dual of a limit is the colimit of the duals (even just as sets or as vector spaces without topologies). Namely, the (only) proof that I know seems to require that the limitands in the limit be Banach spaces. I do not have a counter-example to prove the necessity of this condition, because I've not needed a more general statement (yet?) Anyway, it does not seem true for "general" reasons. $\endgroup$
    – paul garrett
    Commented Jun 18, 2020 at 17:56
  • $\begingroup$ @paulgarrett Thanks, that's an interesting example :) $\endgroup$
    – Daniel Robert-Nicoud
    Commented Jun 23, 2020 at 21:46

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