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.

Properties

The oidification of a monoidal groupoid is a (2,1)-category.

Related concepts

• hom-groupoid

• strict (2,1)-category

• equivalence of (2,1)-categories

• concrete (2,1)-category

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

• monoidal groupoid

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)

algebraic structure	oidification
truth value	preorder
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory