Contents

category theory

# Contents

## Idea

A skeletal groupoid is a (usually: small) strict category (i.e. equipped with a definite set of objects) which is both a groupoid and skeletal category.

In other words, this means that a skeletal groupoid is a (strict) groupoid for which, equivalently:

and so skeletal groupoids $\mathcal{X}$ are exactly (namely: up to isomorphism) the disjoint unions of delooping groupoids:

$\mathcal{X} \,\in\, Grpd_{strct} \;\;\;\;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\;\;\;\; \mathcal{X} \;\text{skeletal} \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \mathcal{X} \;\;\underset{\color{red}iso}{\simeq}\;\; \underset{ x \in Obj(\mathcal{X}) }{\coprod} \mathbf{B} Aut_{\mathcal{X}}(x) \,.$

As discussed at skeletal category (and in much detail here at Introduction to Topology – 2), if the axiom of choice holds in the underlying category of Sets then every groupoid is equivalent as a category — hence homotopy equivalent as a homotopy 1-type — to a skeletal groupoid, and to an essentially unique one, up to isomorphism of strict groupoids:

$\mathcal{X} \,\in\, Grpd_{strct} \;\;\;\;\;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\;\;\;\;\; \mathcal{X} \;\;\;\; \underset{\color{red}equiv}{\simeq} \;\;\;\; \underset{ x \in \pi_0(\mathcal{X}) }{\coprod} \mathbf{B} \pi_1(\mathcal{X}, x) \,.$

As indicated on the right, this skeletalization of $\mathcal{X}$ extracts its homotopy groups: The set of objects of the skeleton is the set $\pi_0(\mathcal{X})$ of connected components, and the automorphism group at a given object is the fundamental group $\pi_1(\mathcal{X},x)$ at that basepoint.

## Properties

### Relation to groups

The 1-category of skeletal groupoids is (see there) equivalently the free coproduct completion of the category Grp of groups.