nLab
homotopy 2-category

Idea

The homotopy 2-category of an (∞,n)-category $\mathcal{C}$ is the 2-category $Ho_2(\mathcal{C})$ with the same objects and 1-morphisms as $\mathcal{C}$ and with the 2-morphisms being the equivalence classes of 2-morphisms of $\mathcal{C}$.

In other words, for every pair $X,Y$ of objects in $\mathcal{C}$, the hom-category $Ho_2(\mathcal{C})(X,Y)$ is the ordinary homotopy category of the $(\infty,n-1)$-category $\mathcal{C}(X,Y)$.

Examples

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

• homotopy category, homotopy category of a model category

Related concepts

• homotopy category

• localization of a 2-category

• model 2-category

References

The homotopy 2-category of topological spaces, continuous functions and homotopies, regarded as a strict (2,1)-category, hence a Grpd-enriched category:

• Peter H. H. Fantham, Eric J. Moore, Groupoid Enriched Categories and Homotopy Theory, Canadian Journal of Mathematics 35 3 (1983) 385-416 (doi:10.4153/CJM-1983-022-8)

On the Toda bracket understood as homotopy-coherent pasting diagrams in a pointed homotopy 2-category:

• Keith Hardie, Klaus Heiner Kamps, Rudger Kieboom, Higher homotopy groupoids and Toda brackets, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 (euclid:hha/1139840198)

• Keith Hardie, Howard Marcum, Nobuyuki Oda, Bracket operations in the homotopy theory of a 2-category, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) (rendiconti:33/02)

• Keith Hardie, Klaus Heiner Kamps, Howard Marcum, The Toda bracket in the homotopy category of a track bicategory, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

• Howard Marcum, Nobuyuki Oda, Long Box Bracket Operations in Homotopy Theory, Appl Categor Struct 19, 137–173 (doi:10.1007/s10485-009-9186-3)

In the context of the homotopy 2-category of (∞,1)-categories and formal $(\infty,1)$-category theory:

• 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)

• 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)