nLab quasi-coproduct

Idea

Noticing hat

1. every set is the coproduct over its singleton elements,

2. every skeletal groupoid is the coproduct over its connected component delooping groupoids, hence is a “set of points with automorphism groups”

3. a coproduct is a colimit over a diagram of the shape of a discrete category (a set) Set $\hookrightarrow$ 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 $Grpd_{skl} \hookrightarrow Cat$ which are degreewise free actions, in a suitable sense, such that these colimits tend to behave like homotopy colimits.

Definition

Quasi-coproducts

Definition

A quasi-coproduct in a category $\mathcal{C}$ is a colimit over a diagram

$$\mathscr{V}_{(-)} \,\colon\, \mathcal{G} \longrightarrow \mathcal{C}$$

of the shape of a skeletal groupoid $\mathcal{G} \,\in\, Grpd_{skl} \hookrightarrow$ such that the following condition is satisfied

• for all connected components $\mathbf{B}G_i \hookrightarrow \mathcal{G}$ and all non-initial objects $\mathscr{W} \,\in\, \mathcal{C}$ the group action in Set given by the composite with the hom-functor $\mathcal{C}(\mathscr{W},-)$
  $$\mathbf{B}G_i \hookrightarrow \mathcal{G} \overset{ \mathscr{V}_{(-)} }{\longrightarrow} \mathcal{C} \overset{\mathcal{C}(\mathscr{W},-)}{\longrightarrow} Set$$

  is a free action.

[Hu & Tholen (1996), §1.3]

Homotopy quasi-coproducts

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 $\infty$-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).

Definition

(conventions for delooping groupoids)
For $(G, \mu, \mathrm{e})$ a group, we write

• $\mathbf{B}G \,\equiv\, (G \rightrightarrows pt)$

  for its delooping groupoid with composition given by reverse multiplication in $G$:

  $$\array{ && pt \\ & \mathllap{{}^{g_{12}}}\nearrow && \searrow\mathrlap{{}^{g_{23}}} \\ pt && \underset{\mu(g_{23},\,g_{12})}{\longrightarrow} && pt }$$

  so that (ordinary, left) $G$-actions (group representations)

  $$\rho \,\colon\, G \longrightarrow Hom_{\mathcal{C}}(\mathscr{V}, \mathscr{V})$$

  are identified with functors of the form

  (1)$$\array{ \mathllap{ \rho \,\colon\, } \mathbf{B}G &\longrightarrow& \mathcal{C} \\ pt &\mapsto& \mathscr{V} \\ \Big\downarrow\mathrlap{{}^{g}} && \Big\downarrow\mathrlap{{}^{ \rho(g) }} \\ pt &\mapsto& \mathscr{V} }$$

• $\mathbf{E}G \,\equiv\, (G \times G \rightrightarrows G)$

  for the action groupoid of $G$ acting on itself by left multiplication, so that we have the evident forgetful functor

  $$\array{ \mathbf{E}G &\longrightarrow& \mathbf{B}G \\ g &\mapsto& pt \\ \Big\downarrow\mathrlap{{}^{ g_{12} }} && \Big\downarrow\mathrlap{{}^{ g_{12} }} \\ \mu(g_{12},g) &\mapsto& pt }$$

and the remaining right inverse multiplication action of $G$ on $\mathbf{E}G$ defines a group action

$$\array{ \mathbf{B}G &\longrightarrow& Grpd \\ pt &\mapsto& \mathbf{E}G \\ \Big\downarrow\mathrlap{{}^{ g }} && \Big\downarrow\mathrlap{{}^{ \mu(\text{-},g^{-1}) }} \\ pt &\mapsto& \mathbf{E}G }$$

whose quotient object is

$$\mathbf{B}G \;\simeq\; \big(\mathbf{E}G\big)/G \,.$$

(See also at universal principal bundle and at simplicial classifying space and at Borel construction.)

Definition

A category with homotopy quasi-coproducts is a category $\mathcal{C}$

• equipped with a tensoring over the 1-category Grpd

  (2)$$Grpd \times \mathcal{C} \xrightarrow{(-)\cdot(-)} \mathcal{C}$$

• which has all quasi-coproducts (Def. ) of the form

  (3)$$\array{ \mathcal{G} &\longrightarrow& \mathcal{C} \\ \mathllap{{}^{=}}\Big\uparrow && \Big\uparrow\mathrlap{ (-)\cdot(-) } \\ \underset{i}{\coprod} \mathbf{B}G_i &\underset{ \big( \mathbf{E}G_i ,\, \mathscr{V}_{(-)} \big)_{i\in I} }{\longrightarrow}& Grpd \times \mathcal{C} }$$

Definition

(free homotopy quasi-coproduct completion)
For $\mathcal{C}$ a category, its free homotopy quasi-coproduct completion is

• a category $QC(\mathcal{C})$ with homotopy quasi-products (Def. )

equipped with

• a full subcategory inclusion $\mathcal{C} \hookrightarrow QC(\mathcal{C})$,

such that

• functors$QC(\mathcal{C}) \longrightarrow \mathcal{D}$ to another category $\mathcal{D}$ with homotopy quasi-coproducts which preserve

  1. the $Grpd$-tensoring (2)

  2. the quasi-coproducts (3)

  are fixed by the restriction of the underlying functor along $\mathcal{C} \hookrightarrow QC(\mathcal{C})$.

Properties

Free quasi-coproduct-completion

In generalization of how

• the free coproduct completion of a category $\mathcal{C}$ is the Grothendieck construction

  $$Fam_{\mathcal{C}} \;\; \coloneqq \;\; \underset {X \in Set} {\textstyle{\int}} \mathcal{C}^X$$

  (see there)

it ought to be the case that

• the free homotopy quasi-coproduct completion of $\mathcal{C}$ is the Grothendieck construction

  $$Loc_{\mathcal{C}} \;\; \coloneqq \;\; \underset{\mathcal{X} \in Grpd}{\textstyle{\int}} \mathcal{C}^{\mathcal{X}}$$

This we make precise now.

Definition

For $\mathcal{C}$ a category, we write

(4)$$Loc_{\mathcal{C}} \;\; \coloneqq \;\; \underset{\mathcal{X} \in Grpd}{\textstyle{\int}} \mathcal{C}^{\mathcal{X}}$$

for the Grothendieck construction on the pseudofunctor which sends groupoids $\mathcal{X} \in$ Grpd to the functor category $\mathcal{C}^{\mathcal{X}} \,\coloneqq\,Func(\mathcal{X}, \mathcal{C})$ and functors $f \,\colon\,\mathcal{X} \to \mathcal{X}'$ to the precomposition operation $f^\ast \,\coloneqq\, (-) \circ f$:

(5)$$\array{ Grpd &\longrightarrow& Cat \\ \mathcal{X} &\mapsto& \mathcal{C}^{\mathcal{X}} \\ \Big\downarrow\mathrlap{{}^{f}} && \Big\uparrow\mathrlap{{}^{f^\ast}} \\ \mathcal{X}' &\mapsto& \mathcal{C}^{\mathcal{X}'} }$$

There is a canonical full subcategory-empbedding

(6)$$\array{ \mathcal{C} &\longrightarrow& Loc_{\mathcal{C}} \\ \mathscr{V} &\mapsto& \mathscr{V}_{pt} }$$

by regarding objects of $\mathcal{C}$ as (necessarily constant) functors on the terminal groupoid $pt$.

Example

For $\mathbb{K}$ a field and $\mathcal{C} \,\equiv\,$ $Mod_{\mathbb{K}}$ $\equiv$ $Vect_{\mathbb{K}}$ its category of vector spaces, a functor $\mathcal{X} \longrightarrow Mod_{\mathbb{C}}$ — for $\mathcal{X}$ thought of as the fundamental groupoid of some topological space $X$ —, is (equivalently a flat vector bundle on $X$ but) also known as a local system $X$. Therefore in this case the category (4)

$$Loc_{\mathbb{K}} \;\coloneqq\; Loc_{Mod_{\mathbb{K}}}$$

may be thought of as the category of “local systems over varying base spaces”, whence the notation in Def. .

Remark

If $\mathscr{C}$ has all small colimits, then the Grothendieck construction of Def.

$$\array{ Loc_{\mathcal{C}} \\ \Big\downarrow \\ Grpd }$$

is actually a bifibration, induced from the $Cat_{adj}$-valued pseudofunctor

(7)$$\array{ Grpd_{skl} &\longrightarrow& Cat_{adj} \\ \mathcal{X} &\mapsto& \mathcal{C}^{\mathcal{X}} \\ \Big\downarrow\mathrlap{{}^{f}} && \mathllap{{}^{f_!}}\Big\downarrow {}^{\dashv} \Big\uparrow\mathrlap{{}^{f^\ast}} \\ \mathcal{X}' &\mapsto& \mathcal{C}^{\mathcal{X}'} }$$

whose left adjoint-components $f_!$ are given by left Kan extension.

In this case, all colimits over diagrams $\mathscr{V}_{\mathcal{X}} \,\colon\, \mathcal{I} \longrightarrow Loc_{\mathcal{C}}$ in the Grothendieck construction $Loc_{\mathcal{C}}$ (4) exist and (by the discussion there) are given on underlying groupoids as the corresponding underlying colimit $\underset{\longrightarrow}{lim} \mathcal{X}$ in Grpd and on components in $\mathcal{C}$ the colimit in $\mathcal{C}^{ \underset{\longrightarrow}{lim} \mathcal{X}}$ of the diagram of pushforwards $q_!$ (7) of morphisms along the underlying coprojections $\mathcal{X}_i \xrightarrow{q_i} \underset{\longrightarrow}{lim} \mathcal{X}$:

$$\underset{\longrightarrow}{\lim} \big(\mathscr{V}_{\mathcal{X}}\big) \;\; \simeq \;\; \big( \underset{\longrightarrow}{lim} q_!\mathscr{V} \big)_{ \underset{\longrightarrow}{lim} \mathcal{X} } \;\;\;\; \in \; Loc_{\mathcal{C}} \,.$$

Remark

(tensoring of local systems over groupoids)
If $\mathcal{C}$ carries the structure of a symmetric monoidal category such that the pullback functors $f^\ast$ (5) are strong monoidal functors, then $Loc_{\mathcal{C}}$ (4) inherits the corresponding external tensor product, which we denote

$$Loc_{\mathcal{C}} \times Loc_{\mathcal{C}} \xrightarrow{\;\; \boxtimes \;\;} Loc_{\mathcal{C}} \,.$$

Notice that for any $\mathcal{X} \,\in\, Grpd$ the external tensor product with $\mathbb{1}_{\mathcal{X}} \,\coloneqq\, (p_{\mathcal{X}})^\ast \mathbb{1}_{pt}$ (i.e. with the tensor unit over $\mathcal{X}$, which is the constant functor $\mathcal{X} \to \ast \xrightarrow{\mathbb{1}} \mathcal{C}$) is equivalently the base change operation along the projection out of the product groupoid $pr_{\mathcal{Y}} \,\colon\, \mathcal{X} \times \mathcal{Y} \longrightarrow \mathcal{Y}$

(8)$$\mathbb{1}_{\mathcal{X}} \,\boxtimes\, \mathscr{V}_{\mathcal{Y}} \;\;\; \simeq \;\;\; \big( (pr_{\mathcal{Y}})^\ast \mathscr{V} \big)_{\mathcal{X} \times \mathcal{Y}} \,.$$

Since here the expression on the right hand side does not actually invoke the monoidal structure of $\mathcal{C}$ it is expedient to allow ourselves to use the notation on the left even when $\mathcal{C}$ is not equipped with monoidal structure.

This is the canonical tensoring of $Loc_{\mathcal{C}}$ over Grpd:

(9)$$\array{ Grpd \times Loc_{\mathcal{C}} &\overset{(-)\cdot(-)}{\longrightarrow}& Loc_{\mathcal{C}} \\ \big( \mathcal{X} ,\, \mathscr{V}_{\mathcal{Y}} \big) &\mapsto& \big( (pr_{\mathcal{Y}})^\ast \mathscr{V} \big)_{ \mathcal{X} \times \mathcal{Y} } }$$

Lemma

Given an object $\mathscr{V}_{\mathcal{X}} \,\in\, Loc_{\mathbb{C}}$ (4) it is isomorphic to a coproduct of tensorings with codiscrete groupoids of objects over delooping groupoids

$$\mathscr{V}_{\mathcal{X}} \;\; \simeq \;\; \underset{i \in \pi_0(\mathcal{C})}{\coprod} \Big( CoDisc\big( Obj(\mathcal{X}_i) \big) \cdot \big( \iota_{\mathbf{B}G_i}^\ast \mathscr{V} \big)_{\mathbf{B}G_i} \Big) \,.$$

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:

$$\mathscr{V}_{\mathcal{X}} \,\in\, Loc_{\mathcal{C}} \;\;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\;\; \mathscr{V}_{\mathcal{X}} \simeq \bigg( \underset{i \in \mathcal{X}_i}{\amalg} \Big( CoDisc\big(\mathrm{Obj}(\mathcal{X}_i)\big) \cdot \big( \iota_{\mathbf{B}G_i}^\ast \mathscr{V} \big)_{ \mathbf{B}G_i } \Big) \Bigg) \overset{ \underset{i \in I}{\amalg} \Big( p_i \cdot \big( \iota_{\mathbf{B}G_i}^\ast \mathscr{V} \big)_{ \mathbf{B}G_i } \Big) }{\longrightarrow} \Big( \underset{i \in \mathcal{X}_i}{\amalg} \big( \iota_{\mathbf{B}G_i}^\ast \mathscr{V} \big)_{ \mathbf{B}G_i } \Big)$$

Proof

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 $CoDisc\big(Obj(\mathcal{X}_i)\big) \times \mathbf{B}G_i$ is isomorphic to one pulled back from $\mathbf{B}G_i$, i.e. to one which constant along $CoDisc\big(Obj(\mathcal{X}_i)\big)$. For this choose any $x_i \in Obj(\mathcal{X}_i)$ and consider the evident natural isomorphism

$$\array{ CoDisc\big(Obj(\mathcal{X}_i)\big) &\overset{}{\longrightarrow}& \mathcal{C} \\ x &\mapsto& \mathscr{V}_x &\overset{\mathscr{V}_{(x,x_i)}}{\longrightarrow}& \mathscr{V}_{x_i} \\ \Big\downarrow && \Big\downarrow\mathrlap{{}^{ \mathscr{V}_{(x,x')} }} && \Big\downarrow\mathrlap{{}^{id}} \\ x' &\mapsto& \mathscr{V}_{x'} &\underset{\mathscr{V}_{(x',x_i)}}{\longrightarrow}& \mathscr{V}_{x_i} \mathrlap{\,.} }$$

Lemma

(group representations are quasi-coproducts of constant local systems)
If $\mathcal{C}$ has all colimits them for $G \in Grp$, any $G$-representation $\mathscr{V}_{\mathbf{B}G} \,\colon\, \mathbf{B}G \overset{\rho}{\longrightarrow} \mathcal{C}$ (1), regarded as an object of $Loc_{\mathcal{C}}$ (4), is the colimit over the diagram

$$\array{ \mathbf{B}G &\overset{\;}{\longrightarrow}& Loc_{\mathcal{C}} \\ pt &\mapsto& \mathbf{E}G \cdot \mathscr{V}_{pt} \\ \Big\downarrow\mathrlap{{}^g} && \Big\downarrow\mathrlap{{}^{ \mu(\text{-},g^{-1}) \,\cdot\, {\rho(g)}_{pt} }} \\ pt &\mapsto& \mathbf{E}G \cdot \mathscr{V}_{pt} }$$

Proof

We need to exhibit a colimiting cocone of the following form, where we find it helpful to now use the $\boxtimes$-notation (8) for the $Grpd$-tensoring:

$$\array{ \mathbb{1}_{\mathbf{E}G} \boxtimes \mathscr{V}_{pt} &&\overset{ \mathbb{1}_{\mu(\text{-},g^{-1})} \,\boxtimes\, \rho(g)_{pt} }{\longrightarrow}&& \mathbb{1}_{\mathbf{E}G} \boxtimes \mathscr{V}_{pt} \\ & \mathllap{{}_{\mathscr{q}_q}}\searrow && \swarrow\mathrlap{{\mathfrak{q}}_{q}} \\ && \mathscr{V}_{\mathbf{B}G} }$$

Due to the assumption that $\mathcal{C}$ 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:

$$\array{ \mathbf{E}G && \overset{\mu(\text{-},g^{-1})}{\longrightarrow} && \mathbf{E}G \\ & \mathllap{{}_q}\searrow && \swarrow\mathrlap{{}_q} \\ && \mathbf{B}G }$$

and the $\mathcal{C}$-component over that is the colimit in $\mathcal{C}$ of the diagram of images of morphism under pushforward $q_!$ along the underlying coprojection $q$.

Now observe that

$$\array{ q_! \big( \mathbb{1}_{\mathbf{E}G} \,\boxtimes\, \mathscr{V}_{pt} \big) & \simeq & (G\cdot\mathbb{1})_{\mathbf{B}G} \boxtimes \mathscr{V}_{pt} \\ \mathllap{{}^{ q_!\big( \mathbb{1}_{\mu(\text{-},g^{-1})} \,\boxtimes\, \rho(g)_{pt} \big) }} \Big\downarrow && \Big\downarrow\mathrlap{{}^{ \big( \mu(\text{-},g^{-1}) \cdot \mathbb{1} \big)_{\mathbf{B}G} \,\boxtimes\, {\rho(g)}_{pt} }} \\ q_! \big( \mathbb{1}_{\mathbf{E}G} \,\boxtimes\, \mathscr{V}_{pt} \big) & \simeq & (G\cdot\mathbb{1})_{\mathbf{B}G} \boxtimes \mathscr{V}_{pt} }$$

and the colimit over this system of morphisms is clearly $\mathscr{V}$ with its $G$-action, with coprojection

$$\array{ G \cdot \mathscr{V} &\xrightarrow{\;\; \mathscr{q} \;\;}& \mathscr{V} \\ (g,v) &\mapsto& \rho(g)(v) }$$

It follows that:

Theorem

If $\mathcal{C}$ has all colimits then $Loc_{\mathcal{C}}$ (Def. ) is its free homotopy quasi-coproduct completion (Def. ).

Proof

First, $Loc_{\mathcal{C}}$ is canonically tensored over $Grpd$ (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 $Loc_{\mathcal{C}}$ is a coproduct of $Grpd$-tensorings of quasi-coproducts of constant local systems (6). Therefore, a functor out of $Loc_{\mathcal{C}}$ which preserves the $Grpd$-tensoring and the quasi-coproducts is fixed by its restriction to constant local systems.

References

The notion of quasi-coproducts (Def. ) is due to:

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