nLab relation between quasi-categories and simplicial categories

Idea

As a model for an (∞,1)-categories, sSet-enriched categories (“simplicial categories”) may be thought of as a semi-strictification of a quasi-category, where composition along objects is strictly associative and unital.

The equivalence between quasi-categories and sSet-enriched categories as models for (∞,1)-categories is exhibited by the operation of sending an $sSet$-category $\mathbf{C}$ to the simplicial set $N(\mathcal{C})$ called its homotopy-coherent nerve (in generalization of the simplicial nerve of an ordinary category). Together with its left adjoint operation $\mathfrak{C}$ (“rigidification of quasi-categories”) this constitutes a Quillen equivalence between the two models.

The Quillen equivalence

Our notation partly follows Bergner (2018).

Proposition

There is a Quillen equivalence:

$$sSet\text{-}Cat \underoverset {\underset{\tilde N}{\longrightarrow}} {\overset{\mathfrak{C}}{\longleftarrow}} {\;\;\; \bot \;\;\;} sSet.$$

where:

• $SC$ denotes the category of sSet-enriched categories equipped with the Dwyer–Kan–Bergner model structure,

• $sSet$ denotes the Joyal model structure on simplicial sets,

• $\tilde N$ denotes the homotopy coherent nerve functor,

• $\mathfrak{C}$ denotes rigidification of quasi-categories.

In particular, for $C$ a fibrant SSet-enriched category, the canonical morphism

$$\mathfrak{C}\big(\tilde N(C)\big) \longrightarrow C$$

given by the counit of the above adjunction is derived, hence a Dwyer–Kan weak equivalence of simplicial categories.

For $S$ any simplicial set, the canonical morphism

$$S \longrightarrow \tilde N\Big( R\big( \mathfrak{C}(S) \big) \Big)$$

is a categorical equivalence of simplicial sets, where $R$ denotes a fibrant replacement functor in the Dwyer–Kan–Bergner model structure.

This was claimed and credited to Joyal in Bergner (2007), Thm. 7.8; detailed proof was produced in Lurie (2009), Thm. 2.2.5.1 & p. 91.

For more details on the rigidification of quasi-categories see also Dugger & Spivak (2009a), (2009b).

Relations

Via $\bar W$-construction

We have an evident inclusion

$$sSet Cat \hookrightarrow Cat^{\Delta}$$

of simplicially enriched categories into simplicial objects in Cat.

On the latter the $\bar W$-functor is defined as the composite

$$\bar W \colon Cat^\Delta \stackrel{N^\Delta}{\to} sSet^\Delta \stackrel{}{\to} sSet$$

where first we degreewise form the ordinary nerve of categories and then take the total simplicial set of bisimplicial sets (the right adjoint of pullback along the diagonal $\Delta^n \to \Delta^n \times \Delta^n$).

Proposition

For $C$ a simplicial groupoid there is a weak homotopy equivalence

$$\mathcal{N}(C) \to \bar W(C)$$

from the homotopy coherent nerve

(Hinich)

Related concepts

• straightening and unstraightening

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

There is an operadic analog of the relation between quasi-categories and simplicial categories, involving, correspondingly, dendroidal sets and simplicial operads.

References

The notion of the homotopy coherent nerve (see there for more) goes back to Cordier (1982); Cordier & Porter(1986).

The Quillen equivalence between the model structure for quasi-categories and the model structure on sSet-categories is described in

• Jacob Lurie, §1.1.5 and §2.2 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press (2009) [pup:8957, pdf]

after an unpublished proof by André Joyal was announced in:

• Julie Bergner, Theorem 7.8 in: A survey of $(\infty,1)$-categories, in: John Baez, Peter May (eds.), Towards Higher Categories The IMA Volumes in Mathematics and its Applications 152, Springer (2007) [arXiv:math/0610239, doi:10.1007/978-1-4419-1524-5_2]

Further discussion of rigidification of quasi-categories $\mathfrak{C}$:

• Dan Dugger, David Spivak, Rigidification of quasi-categories, Algebr. Geom. Topol. 11 (2011) 225-261 [arXiv:0910.0814, doi:10.2140/agt.2011.11.225]

• Dan Dugger, David Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263-325 [arXiv:0911.0469, doi:10.2140/agt.2011.11.263]

More along these lines is in

• Emily Riehl, On the structure of simplicial categories associated to quasi-categories, Math. Proc. Camb. Phil. Soc. 150 (2011) 489-504 [arXiv:0912.4809, doi:10.1017/S0305004111000053]

An expository account is in Section 7.8

• Julie Bergner, The homotopy theory of (∞,1)-categories, London Mathematical Society Student Texts 90, 2018.

See also

• Vladimir Hinich, Simplicial nerve in Deformation theory (arXiv:0704.2503)