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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 4th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI put the opening material into an Idea section, and replaced several '?'s with ∞. <a href="https://ncatlab.org/nlab/revision/diff/Infinity-Grpd/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/Infinity-Grpd/20">v20</a>, <a href="https://ncatlab.org/nlab/show/Infinity-Grpd">current</a>

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

   diff, v20, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 20th 2020
   • (edited Feb 20th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn 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]]. [[equivalence of (∞,1)-categories|Equivalently]] this means all of the following: 1. $\infty Grpd$ is the [[simplicial localization]] of the [[category]] [[Top]]${}_k$ of ([[weakly Hausdorff topological space|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 [[presentable (∞,1)-category|presented]] by the [[classical model structure on topological spaces]]: $\infty Grpd \simeq L_{whe} Top_k$. 1. $\infty Grpd$ 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 [[presentable (∞,1)-category|presented]] by the [[classical model structure on simplicial sets]]: $\infty Grpd \simeq L_{whe} sSet$. Hence, as a [[simplicially enriched category|Kan-complex enriched category]] (a [[fibrant object]] in the [[model structure on sSet-categories]]) $\infty Grpd$ is the [[full subcategory|full]] [[simplicially enriched category|sSet enriched]]-[[subcategory]] in [[sSet]] on those that are [[Kan complexes]]. 1. $\infty Grpd$ is the [[full sub-(∞,1)-category]] of [[(∞,1)Cat]] on those [[(∞,1)-categories]] that are [[∞-groupoids]]. *** <a href="https://ncatlab.org/nlab/revision/diff/Infinity-Grpd/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/Infinity-Grpd/22">v22</a>, <a href="https://ncatlab.org/nlab/show/Infinity-Grpd">current</a>

   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.

   ---

   diff, v22, current