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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:
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:
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.
The 1-category of skeletal groupoids is (see there) equivalently the free coproduct completion of the category Grp of groups.
Discussion of colimits over certain diagrams of the shape of skeletal groupoids and regarded as generalized coproducts:
Characterization of skeletal groupoids as the “trivial objects” with respect to a pretorsion theory on Cat:
Last revised on May 25, 2023 at 12:33:29. See the history of this page for a list of all contributions to it.