nLab bigroupoid

Idea

A bigroupoid is an algebraic model for (general, weak) 2-groupoids along the lines of a bicategory.

Definition

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 $x,y,z,\dots$, also called $0$-cells;
• For each pair of $0$-cells $x,y$, a groupoid $B(x,y)$, whose objects are called morphisms or $1$-cells and whose morphisms are called 2-morphisms or $2$-cells;
• For each $0$-cell $x$, a distinguished $1$-cell $1_x\colon B(x,x)$ called the identity morphism or identity $1$-cell at $x$;
• For each triple of $0$-cells $x,y,z$, a functor ${\circ}\colon B(y,z) \times B(x,y) \to B(x,z)$ called horizontal composition;
• For each pair of $0$-cells $x,y$, a functor ${-}^{-1}\colon B(y,x) \to B(x,y)$ called the inverse operation;
• For each pair of $0$-cells $x,y$, two natural isomorphisms called unitors: $id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y)$;
• For each quadruple of $0$-cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$; and
• For each triple of $0$-cells $x,y,z$, two natural isomorphisms called the unit and counit between the two composites of ${-}^{-1}$ and $id_{B(x,y)}$ and the constant functors on the relevant identity morphisms;

such that

• The unitors and associator satisfy a unitality and associativity coherence law, which we don not write this out here, but the structure is as in a monoidal category; and
• The unit and counit satisfy the same axioms as the constraint isomorphisms in an adjunction (which we also do not write out in full here).

Properties

The Duskin nerve operation identifies bigroupoids with certain 3-coskeletal Kan complexes.