on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
By regarding a simplicial set as an object in the standard model structure on simplicial sets, one effectively identifies it (up to weak equivalence) with that ∞-groupoid that it presents under Kan fibrant replacement.
If the original simplicial set is the nerve of a category, the corresponding Kan fibrant replacement is something like the $\infty$-groupoidification of that category: see geometric realization of categories.
This way each ordinary category models an ∞-groupoid. The Thomason model category structure on Cat exhibits this: in this model category a morphism between two categories is a weak equivalence, precisely if it induces a weak equivalence of the corresponding $\infty$-groupoids.
It turns out the Thomason model structure on Cat is Quillen equivalent to the standard model structure on simplicial sets.
This is remarkable, in that it says that every homotopy type, i. e. every weak equivalence class of $\infty$-groupoids, is obtained by $\infty$-groupoidifying just categories.
In fact, every cofibrant object in this structure is a poset. Since every object in a model category is weakly equivalent to a cofibrant one, this means that even the nerves of just posets are sufficient to model all homotopy types.
This is a rather curious aspect of the Thomason model on Cat: it does not really have anything intrinsically to do with categories, but rather uses these as a way to present ∞-groupoids. It particular, it does not see the equivalences of categories. There is a different model structure on Cat in which weak equivalences are the “true” weak equivalences of categories (not of anything constructed from them). This is called the canonical model structure on Cat.
The original reference is
A correction to this article was made in
There it was clarified that every cofibrant object in the Thomason model structure is a poset.
A useful review of the Thomason model structure and a generalization of the model structure to n-fold categories is given in
The following article gives a large class of categories which are fibrant in the Thomason model structure.
Some posets that are cofibrant in the Thomason model structure are studied in
The analog of the Thomason model structure for the stable case – an equivalence between connective stable homotopy theory and nerves of symmetric monoidal categories – is discussed in