Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The homotopy 2-category 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.
An adjunction in the homotopy 2-category of -categories is equivalently a pair of adjoint -functors.
In view of its classical analog (this Prop.), the remarkable aspect of Prop. is that the homotopy 2-category of -categories is sufficient to detect adjointness of -functors, which would, a priori, be defined as a kind of homotopy-coherent adjointness in the full -category . For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see Riehl & Verity 2016, Thm. 4.3.11, 4.4.11.
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 -categories this is proven in Pavlov 2021 and for presentable derivators it is proven in Renaudin 2006.
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:
Emily Riehl, Chapter 18 of: Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)
Emily Riehl, Dominic Verity, Section 3 of: Fibrations and Yoneda’s lemma in an -cosmos, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)
Emily Riehl, Dominic Verity, Section 1.3 in: Infinity category theory from scratch, Higher Structures Vol 4, No 1 (2020) (arXiv:1608.05314, pdf)
Emily Riehl, Dominic Verity, Elements of ∞-Category Theory, Cambridge studies in advanced mathematics 194, Cambridge University Press (2022) doi:10.1017/9781108936880, ISBN:978-1-108-83798-9, pdf
Last revised on May 8, 2022 at 07:15:20. See the history of this page for a list of all contributions to it.