n-category = (n,n)-category
n-groupoid = (n,0)-category
The general notion of 2-groupoid is also called weak -groupoid to distinguish from the special case of strict -groupoids.
A strict -groupoid is equivalently:
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 -groupoids embed into all -groupoids (modeled by bigroupoids) by regarding a strict -category as a special case of a bicategory. They embed into all -groupoids modeled as Kan complexes via the omega-nerve.
A strict 2-groupoid can also be identified with a crossed complex of the form .
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:
\phi: H\to G
Just as a function between sets, defines as an equivalence relation on by if and only if , so here we get an equivalence relation on the group . 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 , and a group of arrows, given by , the pullback of along itself. Working this out, clearly this group consists of the pairs of elements having the same image in .Such a pair goes from to . 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, . All this is happening within that same group theoretic context and there is a group multiplication on each of the sets given by . With this the maps giving the source and target of an ‘arrow’ are homomorphisms as is the composition.