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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJul 4th 2018

I put the opening material into an Idea section, and replaced several ’?’s with ∞.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 19th 2020
• (edited Feb 20th 2020)

In diuscussion with laymen I noticed that this old entry didn’t explain very much at all. I have now expanded the Idea-section as follows:

$\infty Grpd$ is the (∞,1)-category of ∞-groupoids, i.e. of (∞,0)-categories. This is the archetypical (∞,1)-topos, the home of classical homotopy theory.

Equivalently this means all of the following:

1. $\infty Grpd$ is the simplicial localization of the category Top${}_k$ of (weakly Hausdorff) locally compact topological spaces at the weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: $\tau_0(\infty Grpd) \simeq$ Ho(Top), itself presented by the classical model structure on topological spaces: $\infty Grpd \simeq L_{whe} Top_k$.

2. $\infty Grpd$ is the simplicial localization of the category sSet of simplicial sets at the simplicial weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: $\tau_0(\infty Grpd) \simeq$ Ho(sSet), itself presented by the classical model structure on simplicial sets: $\infty Grpd \simeq L_{whe} sSet$.

Hence, as a Kan-complex enriched category (a fibrant object in the model structure on sSet-categories) $\infty Grpd$ is the full sSet enriched-subcategory in sSet on those that are Kan complexes.

3. $\infty Grpd$ is the full sub-(∞,1)-category of (∞,1)Cat on those (∞,1)-categories that are ∞-groupoids.