# nLab model structure on relative categories

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

# 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 11:35:53. See the history of this page for a list of all contributions to it.