model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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 for 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 whose class of weak equivalences is the localizer generated by the spine inclusions $\{Sp^n \hookrightarrow \Delta^n\}$. See (Ara).
The model structure for quasi-categories is
It is also a monoidal model category with respect to cartesian product and thus 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” with respect to this enrichment.
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 every 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 every 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)-categories ∞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)
André Joyal on the history of the Joyal model structure (also on MathOverflow):
I became interested in quasi-categories (without the name) around 1980 after attending a talk by Jon Beck on the work of Boardman and Vogt. I wondered if category theory could be extended to quasi-categories. In my mind, a crucial test was to show that a quasi-category is a Kan complex if its homotopy category is a groupoid. All my attempts at showing this have failed for about 15 years, until I stopped trying hard! I found a proof after extending to quasi-categories a few basic notions of category theory. This was around 1995. The model structure for quasi-categories was discovered soon after. I did not publish it immediately because I wanted to show that it could be used for proving something new in homotopy theory. I am a bit of a perfectionist (and overly ambitious?). I was hoping to develop a synthesis between category theory and homotopy theory (hence the name quasi-categories). I met Lurie at a conference organised by Carlos Simpson in Nice (in 2001?). I gave a talk on the model structure and Lurie asked for a copy of my notes afterward.
A similar model for (∞,n)-categories is discussed at
There are analogues of the Joyal model structure for cubical sets (with or without connection):
The original construction of the Joyal model structure is in
Unfortunately, this is still not publicly available, but see the lecture notes:
or the construction of the model structure in Cisinski’s book
which closely follows Joyal’s original construction.
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
See also
A model structure for (infinity,2)-sheaves with values in quasicategories is discussed in
Last revised on July 30, 2022 at 12:12:22. See the history of this page for a list of all contributions to it.