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The page internal ∞-groupoid claimed that the case of “internal ∞-groupoids in an (∞,1)-category” was discussed in detail at groupoid object in an (∞,1)-category. That doesn’t seem right to me—I think the groupoid objects on the latter page are really only internal 1-groupoids, not internal ∞-groupoids. They’re “∞” in that their composition is associative and unital only up to higher homotopies, but those are homotopies in the ambient (∞,1)-category; they themselves contain no “higher cells” as additional data. In particular, if the ambient (∞,1)-category is a 1-category, then an internal groupoid in the sense of groupoid object in an (∞,1)-category is just an ordinary internal groupoid, no ∞-ness about it. Does that seem right?
Yes, right.
The notions are equivalent though, no? (I’m thinking about simplicial sets vs simplicial spaces as models for classical homotopy theory)
Harry says:
The notions are equivalent though, no?
Which notions precisely?
(I’m thinking about simplicial sets vs simplicial spaces as models for classical homotopy theory)
Right, so a “group object in $Top$” in the sense of the definition at groupoid object in an (infinity,1)-category is equivalently just a (connected) topological space $X$, in that it is equivalent to the Cech nerve of $* \to X$.
So groupoid objects in $Top \simeq \infty Grpd$ are themselves $\infty$-groupoids all right. But given a 1-category $C$ with pullbacks regarded as an $(\infty,1)$-category, a “groupoid object in $C$” in this sense is just an ordinary groupoid object in $C$, not an $\infty$-groupoid object in $C$.
In particular, if $C = Set$, then an $\infty$-groupoid object in $C$ is any $\infty$-groupoid, while a groupoid object in $C$ must be a groupoid.
More generally, I think that in an (∞,1)-topos (or more generally an “exact (∞,1)-category”), you can say that “internal group objects” (groupoid objects on the terminal object) are equivalent to “pointed connected objects” of the topos (those equipped with an effective-epi from the terminal object). Yet more generally, the same correspondence works between groupoid objects on a fixed object X and objects equipped with an effective-epi from X.
However, in general internal groupoids will be different from objects of the category: internal groupoids in the (∞,1)-category of Kan complexes, for instance, are more or less Segal spaces that are “groupoidal” (all morphisms are invertible). In particular they have a “double-category-like” character, so I don’t think one can really say that they “are” ∞-groupoids. Although I think you might expect to make them equivalent to ∞-groupoids if you impose a further condition analogous to “completeness” of your Segal spaces, or if you use some kind of “anafunctor-like” morphisms between them.
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