bigroupoid

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/ω-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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**$x,y,z,\dots$, 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).

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

Revised on September 15, 2010 05:25:51
by Urs Schreiber
(188.20.66.18)