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Stream: theory: category theory

Topic: Are the Ind and Pro completions of a locally small catego...

Brendan Murphy (Sep 23 2024 at 02:09):

There is a formula $\mathrm{Hom}_{\operatorname{Ind}(\mathcal{C})}("{\varinjlim_{i : I}}" X_i, "{\varinjlim_{j : J}}" X_j) = \varprojlim_{i : I} \varinjlim_{j : J} \operatorname{Hom}_{\mathcal{C}}(X_i, Y_i)$ for hom sets in $\operatorname{Ind}(\mathcal{C})$ which is a cofiltered limit of filtered colimits. Can we turn that into a formal cofiltered limits of formal filtered limits of hom sets and get an enrichment over $\operatorname{Pro}(\operatorname{Ind}(\mathsf{Set}))$? I guess I'm also curious about the enriched case, eg starting with an Ab-enriched category, but I don't really understand how ind/pro completions work there