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every skeletal groupoid is the coproduct over its connected component delooping groupoids, hence is a “set of points with automorphism groups”
a coproduct is a colimit over a diagram of the shape of a discrete category (a set) Set Cat
it makes sense, in homotopy theoretic generalization of coproducts, to consider quasi-coproducts [Hu & Tholen (1996)] to be colimits over diagrams of the shape of skeletal groupoids which are degreewise free actions, in a suitable sense, such that these colimits tend to behave like homotopy colimits.
A quasi-coproduct in a category is a colimit over a diagram
of the shape of a skeletal groupoid such that the following condition is satisfied
is a free action.
The following Def. is a slight variant/specialization of Def. , meant to adapt the notion of quasi-coproducts properly to the context of homotopy theory.
(One may consider other variants, for instance with respect to homotopical categories, and in particular one could consider more general variants, such as with groupoids generalized to sSet-enriched groupoids aka “simplicial groupoids” regarded as models for -groupoids. Maybe later.)
Here, in the vein of speaking obout category theoretic models for homotopy types, throughout by groupoids we mean small strict groupoids (as usual), and we regard the collection Grpd as a 1-category (with functors as the morphisms).
(conventions for delooping groupoids)
For a group, we write
for its delooping groupoid with composition given by reverse multiplication in :
so that (ordinary, left) -actions (group representations)
are identified with functors of the form
for the action groupoid of acting on itself by left multiplication, so that we have the evident forgetful functor
and the remaining right inverse multiplication action of on defines a group action
whose quotient object is
(See also at universal principal bundle and at simplicial classifying space and at Borel construction.)
A category with homotopy quasi-coproducts is a category
equipped with a tensoring over the 1-category Grpd
which has all quasi-coproducts (Def. ) of the form
(free homotopy quasi-coproduct completion)
For a category, its free homotopy quasi-coproduct completion is
equipped with
such that
functors to another category with homotopy quasi-coproducts which preserve
are fixed by the restriction of the underlying functor along .
In generalization of how
the free coproduct completion of a category is the Grothendieck construction
(see there)
it ought to be the case that
the free homotopy quasi-coproduct completion of is the Grothendieck construction
This we make precise now.
For a category, we write
for the Grothendieck construction on the pseudofunctor which sends groupoids Grpd to the functor category and functors to the precomposition operation :
There is a canonical full subcategory-empbedding
by regarding objects of as (necessarily constant) functors on the terminal groupoid .
For a field and its category of vector spaces, a functor — for thought of as the fundamental groupoid of some topological space —, is (equivalently a flat vector bundle on but) also known as a local system . Therefore in this case the category (4)
may be thought of as the category of “local systems over varying base spaces”, whence the notation in Def. .
If has all small colimits, then the Grothendieck construction of Def.
is actually a bifibration, induced from the -valued pseudofunctor
whose left adjoint-components are given by left Kan extension.
In this case, all colimits over diagrams in the Grothendieck construction (4) exist and (by the discussion there) are given on underlying groupoids as the corresponding underlying colimit in Grpd and on components in the colimit in of the diagram of pushforwards (7) of morphisms along the underlying coprojections :
(tensoring of local systems over groupoids)
If carries the structure of a symmetric monoidal category such that the pullback functors (5) are strong monoidal functors, then (4) inherits the corresponding external tensor product, which we denote
Notice that for any the external tensor product with (i.e. with the tensor unit over , which is the constant functor ) is equivalently the base change operation along the projection out of the product groupoid
Since here the expression on the right hand side does not actually invoke the monoidal structure of it is expedient to allow ourselves to use the notation on the left even when is not equipped with monoidal structure.
This is the canonical tensoring of over Grpd:
Given an object (4) it is isomorphic to a coproduct of tensorings with codiscrete groupoids of objects over delooping groupoids
and hence to the domain of a morphism obtained as a coproduct of tensorings (9) of local system over a skeletal groupoid with the terminal maps out of codiscrete groupoids:
On underlying groupoids the isomorphism is induced by a choice of adjoint deformation retraction onto a skeletal groupoid (spelled out here).
It remains to observe that every local system on is isomorphic to one pulled back from , i.e. to one which constant along . For this choose any and consider the evident natural isomorphism
(group representations are quasi-coproducts of constant local systems)
If has all colimits them for , any -representation (1), regarded as an object of (4), is the colimit over the diagram
We need to exhibit a colimiting cocone of the following form, where we find it helpful to now use the -notation (8) for the -tensoring:
Due to the assumption that has colimits, this exists and (Rem. ) by the general formula for colimits in Grothendieck constructions (here) the underlying object in Grpd the is colimiting cocone of the underlying diagram in Grpd:
and the -component over that is the colimit in of the diagram of images of morphism under pushforward along the underlying coprojection .
Now observe that
and the colimit over this system of morphisms is clearly with its -action, with coprojection
It follows that:
If has all colimits then (Def. ) is its free homotopy quasi-coproduct completion (Def. ).
First, is canonically tensored over (by Rem. ) and has all colimits (by Rem. ) and as such is in particular a category with homotopy quasi-coproducts (Def. ).
Moreover, by Lemma and Lemma every object of is a coproduct of -tensorings of quasi-coproducts of constant local systems (6). Therefore, a functor out of which preserves the -tensoring and the quasi-coproducts is fixed by its restriction to constant local systems.
The notion of quasi-coproducts (Def. ) is due to:
Last revised on June 3, 2023 at 16:06:21. See the history of this page for a list of all contributions to it.