nLab skeletal groupoid

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:

• for every morphism $\phi$ its source equals its target: $s(\phi) = t(\phi)$,

• all morphisms are automorphisms of some object,

• all connected components are delooping groupoids $\mathbf{B} G$, namely of the automorphism group $G$ on the unique object in that connected components,

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.

Related entries

• skeleton

• delooping groupoid

References

See also the general references at skeletal category, such as

• Saunders MacLane, p. 91 of: Categories for the Working mMathematician?, Graduate texts in mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

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

• Hongde Hu, Walter Tholen, Quasi-coproducts and accessible categories with wide pullbacks, Appl Categor Struct 4 (1996) 387–402 [doi:10.1007/BF00122686]

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

• Francis Borceux, Federico Campanini, Marino Gran, Walter Tholen, Groupoids and skeletal categories form a pretorsion theory in $Cat$ [arXiv:2207.08487]