As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as generalizing topological groups. Functors between groupoids are the generalization of group homomorphisms. Hence my question is if $\infty$-functors of $\infty$-groupoids generalize continous group-homomorphisms? For instance, if $G,H$ are topological groups, and $BG,BH$ denote the associated topological groupoid/$\infty$-groupoids, do we have $$\text{Hom}_{\infty\text{-gpd}}(BG,BH)\cong \text{Hom}_{cont}(G,H)?$$
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1$\begingroup$ Presumably you want functors that intertwine the basepoints in $BG$ and $BH$. Otherwise this fails even for finite groups and 1-groupoids. But even with that correction, one issue is that $\infty$-functors are looser than strict group maps: you would want instead to talk about maps of "homotopy groups" (with the word "coherent" put in for historical reasons). $\endgroup$– Theo Johnson-FreydCommented Sep 24, 2020 at 18:20
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$\begingroup$ Specifically, when people say "$\infty$-categories can be models as $top$-enriched categories", what they have in mind is to start with the 1-category of $top$-enriched categories, and then invert (in some $\infty$- or model-categorical way) the "weak equivalences" of $top$-enriched categories (morally: the continuous functors which are essentially surjective and are "homotopy fully faithful" in that they induce homotopy equivalences on all hom spaces). $\endgroup$– Theo Johnson-FreydCommented Sep 24, 2020 at 18:22
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1$\begingroup$ It can be quite misleading to think of ∞-categories as topologically enriched categories. Rather, ∞-categories are enriched in ∞-groupoids. ∞-groupoids are the objects of the classical homotopy category, which can be defined in many different ways and doesn't have that much to do with topological spaces and continuous maps. $\endgroup$– Marc HoyoisCommented Sep 24, 2020 at 19:02
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Only for special kinds of topological groups (like Lie groups or groups with an underlying topological manifold). This fails horribly for e.g. profinite groups, which are detected as discrete groups by the topological nerve.
answered Sep 24, 2020 at 18:10
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$\begingroup$ That's to bad, I was mostly thinking of profinite groups. Could you maybe elaborate a bit? My confusion mostly stems for my assumption that $\infty$-functors should correspond to continuous functors $\endgroup$ Commented Sep 24, 2020 at 18:13
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$\begingroup$ @curiousmathguy There is a solution to this problem, Pyknotic/Condensed sets, spaces, ∞-categories. (See here: arxiv.org/abs/1904.09966 ). The pyknotic nerve of the associated topological groupoid any compactly generated topological group will contain all of the information you wanted. $\endgroup$ Commented Sep 24, 2020 at 19:27
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$\begingroup$ The point is that any map $|\Delta^n|\to G$ ($G$ profinite) must be constant, so the continuous map $G^{dis}\to G$ induces an equivalence of $\infty$-groupoids $BG^{dis}\to BG$, that's what (I think) Harry meant $\endgroup$ Commented Sep 25, 2020 at 12:18