Higher category theory
higher category theory
Extra properties and structure
A bigroupoid is an algebraic model for (general, weak) 2-groupoids along the lines of a bicategory.
A bigroupoid is a bicategory in which every morphism is an equivalence and every 2-morphism is an isomorphism.
More explicitly, a bigroupoid consists of:
- A collection of objects , also called -cells;
- For each pair of -cells , a groupoid , whose objects are called morphisms or -cells and whose morphisms are called 2-morphisms or -cells;
- For each -cell , a distinguished -cell called the identity morphism or identity -cell at ;
- For each triple of -cells , a functor called horizontal composition;
- For each pair of -cells , a functor called the inverse operation;
- For each pair of -cells , two natural isomorphisms called unitors: ;
- For each quadruple of -cells , a natural isomorphism called the associator between the two functors from to built out of ; and
- For each triple of -cells , two natural isomorphisms called the unit and counit between the two composites of and and the constant functors on the relevant identity morphisms;
The Duskin nerve operation identifies bigroupoids with 3-coskeletal Kan complexes.
Revised on September 15, 2010 05:25:51
by Urs Schreiber