# nLab model structure for quasi-categories

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

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

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

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.

## Definition

###### Definition

The model structure for quasi-categories or Joyal model structure $sSet_{Joyal}$ on sSet has

## Properties

### As a Cisinski model structure

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).

### General properties

###### Proposition

The model structure for quasi-categories is

###### Remark

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.

###### Proposition

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:

1. it is an inner fibration;

2. 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, 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:

###### Proposition

The fibrant objects in $sSet_{Joyal}$ are precisely those that are inner fibrant over the point, hence those simplicial sets which are quasi-categories.

### Relation to the model structure for $\infty$-groupoids

The inclusion of (∞,1)-categories ∞Grpd $\stackrel{i}{\hookrightarrow}$ (∞,1)Cat has a left and a right adjoint (∞,1)-functor

$(grpdfy \dashv i \dashv Core) \;\; : \;\; (\infty,1)Cat \stackrel{\overset{grpdfy}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\overset{Core}{\to}}} \infty Grpd \,,$

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

The adjunction $(grpdfy \dashv i)$ is modeled by the left Bousfield localization

$(Id \dashv Id) \; :\; sSet_{Joyal} \stackrel{\leftarrow}{\to} sSet_{Quillen} \,.$

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

###### Proposition

$(k_! \dashv k^!) \;\; : sSet_{Quillen} \stackrel{\overset{k^!}{\leftarrow}}{\overset{k_!}{\to}} sSet_{Joyal}$

which arises as nerve and realization for the cosimplicial object

$k : \Delta \to sSet : [n] \mapsto \Delta'[n] \,,$

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 $(i \dashv Core)$:

###### Proposition
• 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)

## History

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):

## References

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