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 all morphisms are strictly invertible.
• a Grpd-enriched category in which all 1-morphisms are strictly invertible.
• a strict omega-groupoid that is $2$-truncated (i.e. in which all k-morphisms for $k>2$ 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 $(G_2\to G_1\overrightarrow{\to }{G}_0)$.

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:

$$\varphi :H\to G$$\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 $\varphi x_0=\varphi x_1$, 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 $\varphi$ 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:

$$(h_0,h_1)\star (h_1,h_2)=(h_0,h_2)$$(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 $(h_0,h_1).(h_0^{\prime },h_1^{\prime })=(h_0{h}_0^{\prime },h_1{h}_1^{\prime })$. 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

• n-groupoid, infinity-groupoid