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

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.

Related concepts

• adjoint (∞,1)-functor

• Ho(CombModCat)

References

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

• André Joyal, The theory of quasicategories and its applications, lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

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

• Emily Riehl, Dominic Verity, The 2-category theory of quasi-categories, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 (arXiv:1306.5144, doi:10.1016/j.aim.2015.04.021)

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 $\infty$-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, Chapter 1 of: Elements of $\infty$-Category Theory, 2021- (pdf)