Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
A quasi-category is a simplicial set satisfying weak Kan filler conditions that make it behave like the nerve of an (∞,1)-category.
There is a model category structure on the category SSet – the Joyal model structure or model structure on quasi-categories – such that the fibrant objects are precisely the quasi-categories and the weak equivalences precisely the correct categorical equivalences that generalize the notion of equivalence of categories.
The model structure for quasi-categories or Joyal model structure on sSet has
As a Cisinski model strucre
The model structure for quasi-categories is the Cisinski model structure on sSet induced by the localizer which consists of the spine inclusions . See (Ara).
The model structure for quasi-categories is
For a morphism of simplicial sets such that is a quasi-category. Then is a fibration in precisely if
it is an inner fibration;
it is an “isofibration”: for everey equivalence in and a lift of its domain through , there is also a lift of the whole equivalence through to an equivalence in .
This is due to Joyal. (Lurie, cor. 18.104.22.168).
So ever fibration in is an inner fibration, but the converse is in general false. A notably exception are the fibrations to the point:
(Lurie, theorem 22.214.171.124)
Relation to the model structure for -groupoids
The inclusion of (∞,1)-catgeories ∞ Grpd (∞,1)Cat has a left and a right adjoint (∞,1)-functor
is the operation of taking the core, the maximal -groupoid inside an -category;
is the operation of groupoidification that freely generates an -groupoid on a given -category
(see HTT, around remark 126.96.36.199)
The adjunction is modeled by the left Bousfield localization
Notice that the left derived functor takes a fibrant object on the left – a quasi-category – then does nothing to it but regarding it now as an object in and then producing its fibrant replacement there, which is Kan fibrant replacement. This is indeed the operation of groupoidification .
The other adjunction is given by the following
There is a Quillen adjunction
which arises as nerve and realization for the cosimplicial object
where is the nerve of the groupoid freely generated from the linear quiver .
This means that for we have
This is (JoTi, prop 1.19)
The following proposition shows that is indeed a model for :
For any the canonical morphism is an acyclic cofibration in ;
for a quasi-category, the canonical morphism is an acyclic fibration in .
This is (JoTi, prop 1.20)
A similar model for (∞,n)-categories is disucssed at
The original construction of the Joyal model structure is in
Unfortunately, this is still not publically available.
A proof that proceeds via homotopy coherent nerve and simplicially enriched categories is given in detail following theorem 188.8.131.52 in
The relation to the model structure for complete Segal spaces is in
Discussion with an eye towards Cisinski model structures and the model structure on cellular sets is in