# nLab homotopy 2-category of (∞,1)-categories

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

## Models

#### 2-Category theory

2-category theory

# Contents

## Idea

The homotopy 2-category $Ho_2\big((\infty,1)Cat\big)$ of the (∞,2)-category (∞,1)Cat of (∞,1)-categories has been argued (Riehl & Verity 13, following Joyal 08 p. 158) to provide a useful context for (∞,1)-category theory (in the spirit of John Gray‘s “formal category theory” in the 2-category Cat of plain categories).

For example, the notion of adjoint (∞,1)-functors turns out to equivalently reduce to plain adjunctions in this homotopy 2-category (Joyal 08 p. 159, Riehl & Verity 13, Sec 4 see there for more).

## Properties

### Relation to homotopy 2-category of model categories?

One might expect (by the discussion there) that the homotopy 2-category of locally presentable (∞,1)-categories (Joyal 08 p. 348) is equivalent to the localization of the 2-category of combinatorial model categories at the Quillen equivalences, Ho(CombModCat).

A full proof of this assertion, if indeed correct, seems not to have been written out.

However, on a closely related note, Renaudin 09 proves an equivalence between Ho(CombModCat) and the homotopy 2-category of locally presentable derivators.

## References

With (∞,1)-categories modeled as quasi-categories, their homotopy 2-category was considered first in

and then developed further (in terms of ∞-cosmoi) in:

Review and further discussion in:

Last revised on June 10, 2021 at 09:28:54. See the history of this page for a list of all contributions to it.