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Stream: theory: category theory

Topic: Internally compact objects

Brendan Murphy (Jun 25 2024 at 18:40):

Suppose $\mathcal{V}$ is locally finitely presentable, is equipped with a symmetric monoidal closed structure, and that compact objects in $\mathcal{V}$ are dualizable are and closed under binary tensor product. Note that I am not assuming the unit is compact! Under these assumptions dualizability is equivalent to being "internally compact", in that $X$ is dualizable iff the internal hom $[X, -] : \mathcal{V} \to \mathcal{V}$ preserves filtered colimits. But this isn't really a proper version of internal compactness, because we're looking at ordinary colimits instead of enriched ones (and $\mathcal{V}$ isn't lfp in the monoidal sense).

I was wondering if under these assumptions $\mathcal{V}$ is some sort of enriched Ind-completion of its subcategory of dualizable objects. Of course it won't be the ordinary Ind-completion if the unit fails to be compact. Is there any work I should look at related to this? Or is any variant of this unlikely to work for some obvious reason? The cases I'm interested in are actually symmetric monoidal stable $\infty$-categories, but I figure the $1$-categorical case is easier to look at first.

Giacomo (Jul 12 2024 at 15:33):

Even though $\mathcal V$ is not lfp in the monoidal sense, this is still covered by Kelly's paper "Structures defined by finite limits in the enriched context". In certain parts Kelly only needs that $\mathcal V_0$ is lfp and tensor products of finitely presentable objects are finitely presentable.
In this situation, if $\mathcal G$ is the full subcategory of $\mathcal V$ spanned by the enriched finitely presentable objects (i.e. those for which the internal hom preserves filtered colimits), then $\mathcal V\simeq Lex [\mathcal G^{op},\mathcal V]$, where "Lex" denotes preservation of finite weighted limits (finite conical limits + powerrs by $\mathcal G$). By 6.11 of that paper (which holds in this context, see 6.13), one obtains that $\mathcal V$ is the enriched free cocompletion of $\mathcal G$ under filtered colimits.

Brendan Murphy (Jul 12 2024 at 17:18):

Okay, I'll take another look. I found it hard to tell which results did or did not need it but it sounds like things are okay!