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 rigidification functor $\mathfrak{C}$ (the left adjoint to the homotopy coherent nerve) to a Dwyer-Kan equivalence (a weak equivalence in the Dwyer-Kan-Bergner model structure on sSet-enriched 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
The image under the Cartesian product-functor of two weak categorical equivalences (Def. ) is again a weak categorical equivalence.
The model structure for quasi-categories is a monoidal model category with respect to cartesian product (hence a cartesian closed model category) 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 both of the following conditions hold:
it is an inner fibration;
it is an isofibration:
in that 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 (2009), cor. 2.4.6.5]
So every fibration in $sSet_{Joyal}$ is an inner fibration, but the converse is in general false. A notable 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.
[Lurie (2009), theorem 2.4.6.1]
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 $\mathbb{L} Id \colon (sSet_{Joyal})^\circ \to (sSet_{Quillen})^\circ$ 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 (Joyal & Tierney (2007), 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 (Joyal & Tierney (2007), 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 never became 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:
On the relation to the model structure for complete Segal spaces:
Discussion with an eye towards Cisinski model structures and the model structure on cellular sets is in
See also
On a model structure for (infinity,2)-sheaves with values in quasicategories:
On transfer of the Joyal model structure to reduced simplicial sets, modelling quasi-categories with a single object:
Last revised on May 31, 2023 at 14:00:12. See the history of this page for a list of all contributions to it.