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strict 2-groupoid

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Homotopy theory

Strict 22-groupoids

Idea

The general notion of 2-groupoid is also called weak 22-groupoid to distinguish from the special case of strict 22-groupoids.

Definitions

A strict 22-groupoid is equivalently:

Properties

A strict 2-groupoid is in particular a strict 2-category and a 2-groupoid, but a 2-groupoid that is a strict 2-category need not be a strict 2-groupoid, since its 1-morphisms might only be weakly invertible.

Strict 22-groupoids embed into all 22-groupoids (modeled by bigroupoids) by regarding a strict 22-category as a special case of a bicategory. They embed into all 22-groupoids modeled as Kan complexes via the omega-nerve.

A strict 2-groupoid can also be identified with a crossed complex of the form (G 2G 1G 0)(G_2 \to G_1 \stackrel{\to}{\to} G_0).

Examples

Amongst the simplest examples will be the strict 2-groups, as these are strict 2-groupoids with a single object. About the simplest example of such an object then comes from a group homomorphism:

ϕ:HG\phi: H\to G

as follows.

Just as a function between sets, f:XYf : X\to Y defines as an equivalence relation on XX by x 0x 1x_0\sim x_1 if and only if ϕx 0=ϕx 1\phi x_0 = \phi x_1, so here we get an equivalence relation on the group HH. That equivalence relation is a congruence so is an internal equivalence relation, that is, it is internal to the category of groups. An equivalence relation is also a groupoid in a well known way, and here we get an internal groupoid within the category of groups. There will be a group of objects, namely HH, and a group of arrows, given by H× GHH \times_G H, the pullback of ϕ\phi along itself. Working this out, clearly this group consists of the pairs (h 0,h 1)(h_0,h_1) of elements having the same image in GG.Such a pair goes from h 0h_0 to h 1h_1. The composition is then very simple:

(h 0,h 1)(h 1,h 2)=(h 0,h 2)(h_0,h_1)\star (h_1,h_2) = (h_0,h_2)

and inverses are given by swapping the two entries, (h 0,h 1) 1=(h 1,h 0)(h_0,h_1)^{-1} = (h_1,h_0). All this is happening within that same group theoretic context and there is a group multiplication on each of the sets given by (h 0,h 1).(h 0 ,h 1 )=(h 0h 0 ,h 1h 1 )(h_0,h_1).(h^\prime_0,h^\prime_1) = (h_0h^\prime_0,h_1h^\prime_1). With this the maps giving the source and target of an ‘arrow’ are homomorphisms as is the composition.

Revised on November 28, 2012 16:07:43 by Urs Schreiber (82.169.65.155)