nLab (2,1)-category

Contents

Context

2-Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

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

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

  • any two parallel jj-morphisms are equivalent, for j>2j \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 22-category.

Models

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

Properties

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

References

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

algebraic structureoidification
truth valuepreorder
magmamagmoid
pointed magma with an endofunctionsetoid/Bishop set
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on May 15, 2022 at 23:34:26. See the history of this page for a list of all contributions to it.