nLab model structure on relative categories

Idea

The model category structure on the category of categories with weak equivalences is a model for the (∞,1)-category of (∞,1)-categories.

Every category with weak equivalences $C$ presents under Dwyer-Kan simplicial localization a simplicially enriched category or alternatively under Charles Rezk‘s simplicial nerve a Segal space, both of which are incarnations of a corresponding (∞,1)-category $\mathbf{C}$ with the same objects of $C$, at least the 1-morphisms of $C$ and such that every weak equivalence in $C$ becomes a true equivalence (homotopy equivalence) in $\mathbf{C}$.

By the work of Barwick and Kan, there is a model structure on relative categories presenting the (∞,1)-category of simplicial spaces.

Details

For the purposes of the present entry, a category with weak equivalences means the bare minimum of what may reasonably go by that name:

Definition A relative category $(C,W)$ is a category $C$ equipped with a choice of wide subcategory $W$.

A morphism in $W$ is called a weak equivalence in $C$. Notice that we do not require here that these weak equivalence satisfy 2-out-of-3, nor even that they contain all isomorphisms.

A morphism $(C_1,W_1) \to (C_2,W_2)$ of relative catgeories is a functor $C_1 \to C_2$ that preserves weak equivalences.

Write $RelCat$ for the category of relative categories and such morphisms between them.

Model category structure

Theorem

The subdivided nerve of Barwick-Kan induces a right transferred model structure on $RelCat$ such that there is a Quillen equivalence

$$K_\xi : sSet^{\Delta^{\op}}_{Reedy} \leftrightarrows (RelCat, BK) : N_\xi$$

Both functors preserve and reflect weak equivalences, and the adjunction unit and counit are natural weak equivalences. The simplicial nerve $N$ is naturally weakly equivalent to $N_\xi$.

Furthermore, this all remains true if the Reedy model structure on $sSet^{\Delta^{\op}}$ is replaced with any of its Bousfield localizations.

The model structure on RelCat is left proper. It is also right proper when the model structure on $sSet^{\Delta^{\op}}$ is right proper.

Proof

The existence and properness of the model structure and the Quillen equivalence is Theorem 6.1 of Barwick-Kan, as is the fact $N_\xi$ preserves and reflects weak equivalences.

Proposition 10.3 shows the adjunction unit is a natural weak equivalence. That the counit is a natural weak equivalence follows by considering the triangle identity, 3-for-2, and the fact $N_\xi$ reflects weak equivalences

$$N_\xi C \xrightarrow{\eta N_\xi C} N_\xi K_\xi N_\xi C \xrightarrow{N_\xi \varepsilon C} N_\xi C$$

Finally, by considering the diagram

$$\array{ X & \stackrel{\eta}{\to} & N_\xi K_\xi X \\ \downarrow f && \downarrow \\ Y & \stackrel{\eta}{\to} & N_\xi K_\xi Y }$$

we see $f$ is a weak equivalence iff $N_\xi K_\xi f$ is, and iff $K_\xi f$ is.

In particular, Rezk’s complete Segal space structure on bisimplicial sets is a Bousfield localization of the Reedy model structure, so we can define:

Definition

Let $(RelCat, Rezk)$ denote the model structure corresponding to the complete Segal spaces.

Theorem

The localization $L(RelCat, Rezk) \simeq (\infty,1)Cat$ is the ∞-localization functor $(C,W) \mapsto L(C, W)$ inverting the weak equivalences

Proof

We will see below that we can compute this through the map $(C,W) \to (NC, NW)$ to marked simplicial sets. Suppose we’re given a fibrant replcaement $(NC, NW) \to Y^\natural$. Since $NC$ is a quasi-category, composition induces an equivalence for every quasi-category $Z$

$$Fun(Y, Z) \simeq Map^\flat(Y^\natural, Z^\natural) \to Map^\flat((NC, NW), Z^\natural) \simeq Fun_{NW}(NC, Z)$$

thus $Y$ satisfies the universal property of $L(NC, NW)$.

It is shown in Meier that categories of fibrant objects are fibrant in this model structure.

Compatibility with other models for (∞,1)-categories

One often uses the hammock localization $L^H : RelCat \to sSetCat$, where $sSetCat$ is given the Bergner model structure whose weak equivalences are the Dwyer-Kan equivalences: i.e. the local weak homotopy equivalences.

Proposition

A functor $f$ in $RelCat$ is a Rezk equivalence iff $L^H(f)$ is a Dwyer-Kan equivalence.

This is main theorem 1.4 of Barwick-Kan.

The idea underlying marked simplicial is directly analogous to the idea underlying relative categories. In fact, the functor $RelCat \to sSet^+ : (C,W) \to (NC, NW)$ preserves and reflects equivalences, since

Proposition

Let $R$ be a fibrant replacement in $sSetCat$. Then the natural transformation $(NC, NW) \mapsto NRL^H(C,W)^\natural$ of marked simplicial sets is a natural weak equivalence.

This is theorem 1.2.1 of Hinich 13

Nerve functors

The compatibility of the various nerve and simplicial localization functors is in section 1.11 of

• Clark Barwick and Dan Kan, A characterization of simplicial localization functors (arXiv:math/1012.1540)

References

• Clark Barwick and Dan Kan,

  • Relative categories: another model for the homotopy theory of homotopy theories (arXiv:math/1011.1691)

  • A characterization of simplicial localization functors (arXiv:math/1012.1540)

  • In the category of relative categories the Rezk equivalences are exactly the DK-equivalences (arXiv:math/1012.1541)

  • A Thomason-like Quillen equivalence between quasi-categories and relative categories (arXiv:math/1101.0772)

  • Partial model categories and their simplicial nerves (arXiv:math/1102.2512)

• Vladimir HinichDwyer-Kan Localization Revisited,

  Homology, Homotopy and Applications Volume 18 (2016) Number 1 (arXiv:1311.4128, doi:10.4310/HHA.2016.v18.n1.a3)

• Lennart Meier, Fibration Categories are Fibrant Relative Categories, arxiv