nLab skeletal groupoid



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Category theory



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

𝒳Grpd strct𝒳skeletal𝒳isoxObj(𝒳)BAut 𝒳(x). \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:

𝒳Grpd strct𝒳equivxπ 0(𝒳)Bπ 1(𝒳,x). \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 𝒳\mathcal{X} extracts its homotopy groups: The set of objects of the skeleton is the set π 0(𝒳)\pi_0(\mathcal{X}) of connected components, and the automorphism group at a given object is the fundamental group π 1(𝒳,x)\pi_1(\mathcal{X},x) at that basepoint.


Relation to groups

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


See also the general references at skeletal category, such as

Discussion of colimits over certain diagrams of the shape of skeletal groupoids and regarded as generalized coproducts (quasi-coproducts):

Characterization of skeletal groupoids as the “trivial objects” with respect to a pretorsion theory on Cat:

Last revised on June 5, 2023 at 11:41:17. See the history of this page for a list of all contributions to it.