nLab
(2,1)-category

Idea

By the general rules of $(n,r)$-categories, a $(2,1)$-category is an $\infty$-category such that

• any $j$-morphism is an equivalence, for $j \gt 1$;

• any two parallel $j$-morphisms are equivalent, for $j \gt 2$.

You can start from any notion of $\infty$-category, strict or weak; up to equivalence, the result can always be understood as a locally groupoidal $2$-category.

Models

So, a (2,1)-category is in particular modeled by

• a 2-category in which all 2-morphisms are invertible;

• an (∞,1)-category that is 2-truncated.

Related concepts

• hom-groupoid

• strict (2,1)-category

• equivalence of (2,1)-categories

• monoidal (2,1)-category, symmetric monoidal (2,1)-category

References

The special case of strict (2,1)-categories, motivated from the homotopy 2-category of topological spaces:

• 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)