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model structure for quasi-categories

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 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.

Details

for the moment, see the corresponding section at model structure on simplicial sets

Relation to the model structure for $\mathrm{\infty }$-groupoids

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

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

where

• $\mathrm{Core}$ is the operation of taking the core, the maximal $\mathrm{\infty }$-groupoid inside an $(\mathrm{\infty },1)$-category;

• $\mathrm{grpdfy}$ is the operation of groupoidification that freely generates an $\mathrm{\infty }$-groupoid on a given $(\mathrm{\infty },1)$-category

(see HTT, around remark 1.2.5.4)

The adjunction $(\mathrm{grpdfy}⊣i)$ is modeled by the left Bousfield localization

$$(\mathrm{Id}⊣\mathrm{Id}):{\mathrm{sSet}}_{\mathrm{Joyal}}\overleftarrow{\to }{\mathrm{sSet}}_{\mathrm{Quillen}}.$$(Id \dashv Id) \; :\; sSet_{Joyal} \stackrel{\leftarrow}{\to} sSet_{Quillen} \,.

Notice that the left derived functor $𝕃\mathrm{Id}:({\mathrm{sSet}}_{\mathrm{Joyal}}{)}^{\circ }\to ({\mathrm{sSet}}_{\mathrm{Quillen}}{)}^{\circ }$ takes a fibrant object on the left – a quasi-category – then does nothing to it but regarding it now as an object in ${\mathrm{sSet}}_{\mathrm{Quillen}}$ 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. There is a Quillen adjunction

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

which arises as nerve and realization for the cosimplicial object

$$k:\Delta \to \mathrm{sSet}:[n]\mapsto \Delta \prime [n],$$k : \Delta \to sSet : [n] \mapsto \Delta'[n] \,,

where ${\Delta }^{\prime }[n]=N(\{0\stackrel{\simeq }{\to }1\stackrel{\simeq }{\to }\cdots \stackrel{\simeq }{\to }n\})$ is the nerve of the groupoid freely generated from the linear quiver $[n]$.

This means that for $X\in \mathrm{SSet}$ we have

• $k^{!}(X{)}_n={\mathrm{Hom}}_{\mathrm{sSet}}(\Delta \prime [n],X)$.

• and $k_{!}(X{)}_n={\int }^{[k]}{X}_k\cdot \Delta \prime [k]$.

Proof This is (JoTi, prop 1.19)

The following proposition shows that $(k_{!}⊣k^{!})$ is indeed a model for $(i⊣\mathrm{Core})$:

Proposition

• For any $X\in \mathrm{sSet}$ the canonical morphism $X\to k_{!}(X)$ is an acyclic cofibration in ${\mathrm{sSet}}_{\mathrm{Quillen}}$;

• for $X\in \mathrm{sSet}$ a quasi-category, the canonical morphism $k^{!}(X)\to \mathrm{Core}(X)$ is an acyclic fibration in ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

Proof This is (JoTi, prop 1.20)

References

The original construction of the Joyal model structure is in

• Andre Joyal, Theory of quasi-categories I

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

• Jacob Lurie, Higher Topos Theory

The relation to the model structure for complete Segal spaces is in

• Andre Joyal, Myles Tierney, Quasi-categories vs. Segal spaces (arXiv)