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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 below.

Properties

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

Proposition

An adjunction in the homotopy 2-category of $(\infty,1)$-categories is equivalently a pair of adjoint $(\infty,1)$-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 $(\infty,2)$-category $Cat_{(\infty,1)}$. 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 $(2,1)$-categories this is proven in Pavlov 2021 and for presentable derivators it is proven in Renaudin 2006.

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, p. 158 (10 of 348) in: The theory of quasicategories and its applications, lectures at: Advanced Course on Simplicial Methods in Higher Categories, Quadern 45 2, Centre de Recerca Matemàtica, Barcelona 2008 (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)