nLab strict 2-groupoid

Idea

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

Definitions

A strict $2$-groupoid is equivalently:

• a strict 2-category in which both 1-morphisms and 2-morphisms are invertible,

• a Grpd-enriched category in which all 1-morphisms are strictly invertible,

• a Grpd-enriched groupoid,

• a strict omega-groupoid that is $2$-truncated (i.e. in which all k-morphisms for $k \gt 2$ are identities).

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 $2$-groupoids embed into all $2$-groupoids (modeled by bigroupoids) by regarding a strict $2$-category as a special case of a bicategory. They embed into all $2$-groupoids modeled as Kan complexes via the omega-nerve.

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

Strict 2-groupoids still model all homotopy 2-types. See also at homotopy hypothesis – for homotopy 2-types.

Examples

Fundamental 2-groupoid

The fundamental 2-groupoid of a topological space with thin homotopy classes of paths (Hardie, Kapms & Kieboom (2000)).

Similarly the strict path 2-groupoid of a manifold.

Delooping of strict 2-groups

The deloopings of strict 2-groups are the strict 2-groupoids with a single object.

About the simplest example of such an object then comes from a group homomorphism

$$\phi: H\to G$$

as follows.

Just as a function between sets, $f : X\to Y$ defines as an equivalence relation on $X$ by $x_0\sim x_1$ if and only if $\phi x_0 = \phi x_1$, so here we get an equivalence relation on the group $H$. 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 $H$, and a group of arrows, given by $H \times_G H$, the pullback of $\phi$ along itself. Working this out, clearly this group consists of the pairs $(h_0,h_1)$ of elements having the same image in $G$.Such a pair goes from $h_0$ to $h_1$. The composition is then very simple:

$$(h_0,h_1)\star (h_1,h_2) = (h_0,h_2)$$

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

Related concepts

• strict 2-category, strict 2-functor

• strict 2-group

• n-groupoid, infinity-groupoid

References

See also the references at strict 2-category.

The fundamental 2-groupoid of a topological space with thin homotopy classes of paths:

• Keith A. Hardie, Klaus H. Kamps, Rudger Kieboom, A homotopy 2-groupoid of a Hausdorff space. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures 8 (2000) 209-234 [doi:10.1023/A:1008758412196]