nLab rigidification of quasi-categories

Idea

Both quasi-categories as well as sSet-enriched categories serve as models for $(\infty,1)$-categories (exhibited by the Joyal model structure for quasi-categories and the Dwyer–Kan–Bergner model structure on sSet-enriched categories, respectively) and in fact as Quillen equivalent models (see the relation between quasi-categories and simplicial categories) whose right Quillen functor is the homotopy coherent nerve-operation. Its left adjoint

$$\mathfrak{C} \;\colon\; sSet \longrightarrow sSet\text{-}Cat$$

[Lurie (2009), pp. 20] thus encodes any quasi-category equivalently in an sSet-enriched category. Since in the latter model, but not in the former, horizontal composition is a strictly associative and unital operation, one may think of $\mathfrak{C}(-)$ as (partially) “rigidifying” the data in a quasi-category, in the sense of semi-strictification of $\infty$-categories, whence it is also known as the operation of rigidification of quasi-categories [Dugger & Spivak (2011a)].

Definition

The Joyal rigidification functor

The Joyal rigidification functor $\mathscr{C}$ is defined as the left adjoint of the Cordier–Vogt homotopy coherent nerve functor $N$:

(1)$$sSet \underoverset {\underset{N}{\longleftarrow}} {\overset{\mathfrak{C}}{\longrightarrow}} {\;\;\; \bot \;\;\;} sSet\text{-}Cat$$

This notion is briefly mentioned and attributed to André Joyal by Bergner (2007), above Theorem 7.8; the first detailed account is due to Lurie (2009), §1.1.5 and §2.2.

The Dugger–Spivak rigidification functor

The Dugger–Spivak rigidification functor [Dugger & Spivak (2011a)] provides a more explicit model for the same (∞,1)-functor, by virtue of writing down an explicit formula that does not use colimits. The resulting functor is connected to the Joyal functor by a zigzag of natural weak equivalences.

Specifically, given a simplicial set $S$ (not necessarily fibrant in the Joyal model structure), we construct the Dugger–Spivak rigidification $\mathfrak{C}^{\mathrm{nec}}S$ as the following simplicial category:

Objects are vertices of $S$.

The simplicial set of morphisms $x\to y$ is the nerve of the category of necklaces in $S$ (introduced by Hans-Joachim Baues). A necklace is a simplicial map

$$\Delta^{n_1}\vee \cdots \vee \Delta^{n_k}\to S,$$

where $\vee$ glues the final vertex of the preceding simplex to the initial vertex of the following simplex. Morphisms are commutative triangles of simplicial maps that preserve the initial and final vertex of the entire necklace.

Composition is defined by concatenating necklaces. The resulting functor from simplicial sets admits a zigzag of weak equivalences to the Joyal rigidification functor.

Properties

Respect for products

The rigidification operation does not strictly preserve Cartesian products (cf. products of simplicial sets) but it does so up to equivalence:

Proposition

For $S,\, S' \,\in\, sSet$, the natural transformation

$$\mathfrak{C}(S \times S') \overset{ \big( \mathfrak{C}(pr_S) ,\, \mathfrak{C}(pr_{S'}) \big) }{\longrightarrow} \mathfrak{C}(S) \times \mathfrak{C}(S')$$

(induced from the projections $S \overset{pr_S}{\leftarrow} S \times S' \overset{pr_{S'}}{\rightarrow} S'$) is a Dwyer-Kan equivalence.

[Lurie (2009), Cor. 2.2.5.6; Dugger & Spivak (2011a), Prop. 6.2]

Relation to Dwyer-Kan groupoids

Let

(2)$$L \,\colon\, sSet\text{-}Cat \longrightarrow sSet\text{-}Grp$$

denote the functor from sSet-enriched categories to sSet-enriched groupoids (Dwyer-Kan simplicial groupoids) which is degreewise given by the localization operation left adjoint to the full subcategory inclusion Grpd $\hookrightarrow$ Cat.

Let

(3)$$\mathcal{G} \,\colon\, sSet \longrightarrow sSet\text{-}Grpd \hookrightarrow sSet\text{-}Cat$$

denote the Dwyer-Kan fundamental simplicial groupoid-construction with values in sSet-enriched groupoids (Dwyer-Kan simplicial groupoids), here to be regarded among sSet-enriched categories.

Proposition

For $S \,\in\, sSet$ there is a natural transformation

$$L \circ \mathfrak{C}(S) \longrightarrow \mathcal{G}(S)$$

which is a Dwyer-Kan equivalence (from the localization (2) of the rigidification (1) to the Dwyer-Kan fundamental groupoid (3)).

[Minichiello, Rivera & Zeinalian (2023), Thm. 1.1 (Cor. 4.2)]

Related concepts

• homotopy coherent nerve

• relation between quasi-categories and simplicial categories

• Dwyer-Kan fundamental simplicial groupoid

References

The construction is briefly mentioned and attributed to André Joyal 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]

First detailed discussion (not using a name beyond the symbol “$\mathfrak{C}$”) is due to:

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

Further discussion and the terminology “rigidification” is due to

• 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]

aimed at discussing quasi-categorical hom-spaces:

• 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]

See also:

• 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]

• Emilio Minichiello, Manuel Rivera, Mahmoud Zeinalian, §3.2 in: Categorical models for path spaces, Advances in Mathematics 415 (2023) 108898 [arXiv:2201.03046, doi:10.1016/j.aim.2023.108898]