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

Contents

Context

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

2-Category theory

Contents

Idea

The homotopy 2-category Ho 2((,1)Cat)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 below.

Properties

Adjunctions and adjoint (,1)(\infty,1)-functors

(Riehl & Verity 2015, Rem. 4.4.5; Riehl & Verity 2022, Prop. F.5.6; see there for more).

Remark

In view of its classical analog (this Prop.), the remarkable aspect of Prop. is that the homotopy 2-category of \infty -categories is sufficient to detect adjointness of \infty -functors, which would, a priori, be defined as a kind of homotopy-coherent adjointness in the full ( , 2 ) (\infty,2) -category Cat (,1) Cat_{(\infty,1)} . For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see Riehl & Verity 2016, Thm. 4.3.11, 4.4.11.

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).

At least for the homotopy (2,1)(2,1)-categories this is proven in Pavlov 2021 and for presentable derivators it is proven in Renaudin 2006.

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 October 15, 2021 at 03:59:00. See the history of this page for a list of all contributions to it.