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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeDec 14th 2010

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?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeDec 14th 2010

Yes, right.

• CommentRowNumber3.
• CommentAuthorHarry Gindi
• CommentTimeDec 14th 2010

The notions are equivalent though, no? (I’m thinking about simplicial sets vs simplicial spaces as models for classical homotopy theory)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 14th 2010
• (edited Dec 14th 2010)

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 $C$.

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeDec 14th 2010

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.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeDec 14th 2010

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.