nLab quasi-coproduct



Category theory

Limits and colimits

Homotopy theory

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

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models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 sklCatGrpd_{skl} \hookrightarrow Cat 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 𝒞\mathcal{C} is a colimit over a diagram

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

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

  • for all connected components BG i𝒢\mathbf{B}G_i \hookrightarrow \mathcal{G} and all non-initial objects 𝒲𝒞\mathscr{W} \,\in\, \mathcal{C} the group action in Set given by the composite with the hom-functor 𝒞(𝒲,)\mathcal{C}(\mathscr{W},-)
    BG i𝒢𝒱 ()𝒞𝒞(𝒲,)Set \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).


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

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

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

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

    so that (ordinary, left) GG-actions (group representations)

    ρ:GHom 𝒞(𝒱,𝒱) \rho \,\colon\, G \longrightarrow Hom_{\mathcal{C}}(\mathscr{V}, \mathscr{V})

    are identified with functors of the form

    (1)ρ:BG 𝒞 pt 𝒱 g ρ(g) pt 𝒱 \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} }
  • EG(G×GG)\mathbf{E}G \,\equiv\, (G \times G \rightrightarrows G)

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

    EG BG g pt g 12 g 12 μ(g 12,g) pt \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 GG on EG\mathbf{E}G defines a group action

BG Grpd pt EG g μ(-,g 1) pt EG \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

BG(EG)/G. \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.)


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

  • equipped with a tensoring over the 1-category Grpd

    (2)Grpd×𝒞()()𝒞 Grpd \times \mathcal{C} \xrightarrow{(-)\cdot(-)} \mathcal{C}
  • which has all quasi-coproducts (Def. ) of the form

    (3)𝒢 𝒞 = ()() iBG i (EG i,𝒱 ()) iI Grpd×𝒞 \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} }


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

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

equipped with

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

such that

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

    1. the GrpdGrpd-tensoring (2)

    2. the quasi-coproducts (3)

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


Free quasi-coproduct-completion

In generalization of how

it ought to be the case that

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

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

This we make precise now.


For 𝒞\mathcal{C} a category, we write

(4)Loc 𝒞𝒳Grpd𝒞 𝒳 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 𝒳\mathcal{X} \in Grpd to the functor category 𝒞 𝒳Func(𝒳,𝒞)\mathcal{C}^{\mathcal{X}} \,\coloneqq\,Func(\mathcal{X}, \mathcal{C}) and functors f:𝒳𝒳f \,\colon\,\mathcal{X} \to \mathcal{X}' to the precomposition operation f *()ff^\ast \,\coloneqq\, (-) \circ f:

(5)Grpd Cat 𝒳 𝒞 𝒳 f f * 𝒳 𝒞 𝒳 \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)𝒞 Loc 𝒞 𝒱 𝒱 pt \array{ \mathcal{C} &\longrightarrow& Loc_{\mathcal{C}} \\ \mathscr{V} &\mapsto& \mathscr{V}_{pt} }

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


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

Loc 𝕂Loc Mod 𝕂 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. .


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

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

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

(7)Grpd skl Cat adj 𝒳 𝒞 𝒳 f f ! f * 𝒳 𝒞 𝒳 \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 !f_! are given by left Kan extension.

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

lim(𝒱 𝒳)(limq !𝒱) lim𝒳Loc 𝒞. \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}} \,.


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

Loc 𝒞×Loc 𝒞Loc 𝒞. Loc_{\mathcal{C}} \times Loc_{\mathcal{C}} \xrightarrow{\;\; \boxtimes \;\;} Loc_{\mathcal{C}} \,.

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

(8)𝟙 𝒳𝒱 𝒴((pr 𝒴) *𝒱) 𝒳×𝒴. \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 𝒞\mathcal{C} it is expedient to allow ourselves to use the notation on the left even when 𝒞\mathcal{C} is not equipped with monoidal structure.

This is the canonical tensoring of Loc 𝒞Loc_{\mathcal{C}} over Grpd:

(9)Grpd×Loc 𝒞 ()() Loc 𝒞 (𝒳,𝒱 𝒴) ((pr 𝒴) *𝒱) 𝒳×𝒴 \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} } }


Given an object 𝒱 𝒳Loc \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

𝒱 𝒳iπ 0(𝒞)(CoDisc(Obj(𝒳 i))(ι BG i *𝒱) BG i). \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:

𝒱 𝒳Loc 𝒞𝒱 𝒳(⨿i𝒳 i(CoDisc(Obj(𝒳 i))(ι BG i *𝒱) BG i))⨿iI(p i(ι BG i *𝒱) BG i)(⨿i𝒳 i(ι BG i *𝒱) BG i) \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)


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

CoDisc(Obj(𝒳 i)) 𝒞 x 𝒱 x 𝒱 (x,x i) 𝒱 x i 𝒱 (x,x) id x 𝒱 x 𝒱 (x,x i) 𝒱 x i. \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{\,.} }


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

BG Loc 𝒞 pt EG𝒱 pt g μ(-,g 1)ρ(g) pt pt EG𝒱 pt \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} }


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 GrpdGrpd-tensoring:

𝟙 EG𝒱 pt 𝟙 μ(-,g 1)ρ(g) pt 𝟙 EG𝒱 pt 𝓆 q 𝔮 q 𝒱 BG \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 𝒞\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:

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

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

Now observe that

q !(𝟙 EG𝒱 pt) (G𝟙) BG𝒱 pt q !(𝟙 μ(-,g 1)ρ(g) pt) (μ(-,g 1)𝟙) BGρ(g) pt q !(𝟙 EG𝒱 pt) (G𝟙) BG𝒱 pt \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 𝒱\mathscr{V} with its GG-action, with coprojection

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

It follows that:


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


First, Loc 𝒞Loc_{\mathcal{C}} is canonically tensored over GrpdGrpd (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 𝒞Loc_{\mathcal{C}} is a coproduct of GrpdGrpd-tensorings of quasi-coproducts of constant local systems (6). Therefore, a functor out of Loc 𝒞Loc_{\mathcal{C}} which preserves the GrpdGrpd-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.