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
equivalences in/of $(\infty,1)$-categories
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 $sSet_{Joyal}$ on sSet has
cofibrations are the monomorphisms
weak equivalences are those maps that are taken by the left adjoint of the homotopy coherent nerve to a weak equivalence in the model structure on simplicial categories.
The model structure for quasi-categories is the Cisinski model structure on sSet induced by the localizer which consists of the spine inclusions $\{Sp^n \hookrightarrow \Delta^n\}$. See (Ara).
The model structure for quasi-categories is
It is also a monoidal model category and is naturally an enriched model category over itself, hence is $sSet_{Joyal}$-enriched (reflecting the fact that it tends to present an (infinity,2)-category). It is however not $sSet_{Quillen}$-enriched and thus not a “simplicial model category”.
For $p \colon \mathcal{C} \to \mathcal{D}$ a morphism of simplicial sets such that $\mathcal{D}$ is a quasi-category. Then $p$ is a fibration in $sSet_{Joyal}$ precisely if
it is an inner fibration;
it is an “isofibration”: for everey equivalence in $\mathcal{D}$ and a lift of its domain through $p$, there is also a lift of the whole equivalence through $p$ to an equivalence in $\mathcal{C}$.
This is due to Joyal. (Lurie, cor. 2.4.6.5).
So ever fibration in $sSet_{Joyal}$ is an inner fibration, but the converse is in general false. A notably exception are the fibrations to the point:
The fibrant objects in $sSet_{Joyal}$ are precisely those that are inner fibrant over the point, hence those simplicial sets which are quasi-categories.
The inclusion of (∞,1)-catgeories ∞ Grpd $\stackrel{i}{\hookrightarrow}$ (∞,1)Cat has a left and a right adjoint (∞,1)-functor
where
$Core$ is the operation of taking the core, the maximal $\infty$-groupoid inside an $(\infty,1)$-category;
$grpdfy$ is the operation of groupoidification that freely generates an $\infty$-groupoid on a given $(\infty,1)$-category
(see HTT, around remark 1.2.5.4)
The adjunction $(grpdfy \dashv i)$ is modeled by the left Bousfield localization
Notice that the left derived functor $\mathbb{L} Id : (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ takes a fibrant object on the left – a quasi-category – then does nothing to it but regarding it now as an object in $sSet_{Quillen}$ 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 $\Delta^'[n] = N(\{0 \stackrel{\simeq}{\to} 1 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n\})$ is the nerve of the groupoid freely generated from the linear quiver $[n]$.
This means that for $X \in SSet$ we have
$k^!(X)_n = Hom_{sSet}(\Delta'[n],X)$.
and $k_!(X)_n = \int^{[k]} X_k \cdot \Delta'[k]$.
This is (JoTi, prop 1.19)
The following proposition shows that $(k_! \dashv k^!)$ is indeed a model for $(i \dashv Core)$:
For any $X \in sSet$ the canonical morphism $X \to k_!(X)$ is an acyclic cofibration in $sSet_{Quillen}$;
for $X \in sSet$ a quasi-category, the canonical morphism $k^!(X) \to Core(X)$ is an acyclic fibration in $sSet_{Quillen}$.
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 2.2.5.1 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