Contents

model category

for ∞-groupoids

Contents

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

References

Last revised on June 10, 2021 at 07:35:53. See the history of this page for a list of all contributions to it.