strict 2-groupoid

- 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

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

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 $k \gt 2$ are identities).

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 $(G_2 \to G_1 \stackrel{\to}{\to} G_0)$.

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:

$\phi: H\to G$

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.

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