codiscrete groupoid (changes) in nLab

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Idea

The codiscrete groupoid on a set is the groupoid whose objects are the elements of the set and which has a unique morphism for every ordered pair of objects.

This is also called the pair groupoid of XX and sometimes also the chaotic groupoid (this is explained below), indiscrete groupoid, or coarse groupoid on XX, in older literature also Brandt groupoid.

Definition

\begin{definition}\label{CodiscreteGroupoid} (codiscrete groupoid) \linebreak For X∈SetX \in Set, the codiscrete groupoid of XX is the groupoid

Codisc(X)≔X×X⇉pr 2pr 1X Codisc(X) \;\coloneqq\; X \times X \underoverset {pr_2} {pr_1} {\rightrightarrows} X

whose object of objects is

• Obj(X)=XObj(X) = X,

whose objects of morphisms is the Cartesian product of XX with itself

• Mor(X)=X×XMor(X) = X \times X,

whose source and target morphism are the two canonical projections out of the product, and whose composition operation is the unique one compatible with this:

X×X×X→(pr 1,pr 3)X×X X \times X \times X \xrightarrow{\; (pr_1, pr_3) \; } X \times X

\end{definition}

\begin{remark} Def. \ref{CodiscreteGroupoid} manifestly makes sense in the generality of internal groupoids internal to any category with finite limits (in fact only finite products are involved in the definition of codiscrete groupoids). \end{remark}

Properties

General

• Every codiscrete groupoid on an inhabited set is contractible: equivalent to the terminal groupoid (the point). More generally, any codiscrete groupoid is equivalent to a truth value.

• For XX a finite set of cardinality n>0n \gt 0, the category algebra of Codisc(X)Codisc(X) is the algebra of n×nn\times n matrices. The contractibility of Codisc(X)Codisc(X) is reflected in the fact that this algebra is Morita equivalent to the ground ring, which is the category algebra of the point.

  This maybe serves to illustrate: even though codiscrete groupoids are pretty trivial, they are not too trivial to be entirely without interest. Often it is useful to have big puffed-up versions of the point available (see cofibrant resolution).

• The underlying directed graph of a codiscrete groupoid is a complete graph (in that there is one and only one edge between any ordered pair of vertices).

Adjointness

The 1-category Grpd of groupoids is related to Set by an adjoint quadruple of functors

\begin{tikzcd} \mathrm{Grpd} \ar[rr, shift left=26pt, { \pi_0 }{description}] \ar[rr, { (-)_0 }{description}] && \mathrm{Set} \ar[ll, shift right=13pt, { \mathrm{Disc} }{description}] \ar[ll, shift left=13pt, { \mathrm{Codisc} }{description}] \end{tikzcd}

Here

(−) 0:(X 1⇉X 0)↦X 0 (-)_0 \;\; \colon \;\; \big( X_1 \rightrightarrows X_0 \big) \;\;\; \mapsto \;\;\; X_0

sends a groupoid to its set of objects.

The right adjoint to this functor sends a set to its codiscrete groupoid according to Def. \ref{CodiscreteGroupoid}. To see this, observe the hom-isomorphism that reflects this adjunction:

For 𝒳=(𝒳 1⇉𝒳 0)∈\mathcal{X} = \big( \mathcal{X}_1 \rightrightarrows \mathcal{X}_0\big)\,\in\, Grpd and for S∈S \,\in\, Set, a morphism of groupoids (i.e. a functor) of the form

𝒳→FCoDisc(S) \mathcal{X} \xrightarrow{\;\; F \;\;} CoDisc(S)

is uniquely determined as soon as its component function

𝒳 0→F 0CoDisc(S)=S \mathcal{X}_0 \xrightarrow{\;\; F_0 \;\;} CoDisc(S) = S

is chose, because for every morphism (x→fy)∈𝒳 1(x \xrightarrow{f} y) \,\in\,\mathcal{X}_1 there is one and only one morphism F 0(x)→F 0(y)F_0(x) \to F_0(y) that it may be sent to, and making this unique choice for each ff does constitute a functor FF for every choice of F 0F_0.

This association there gives a natural bijection of hom-sets

Grpd(𝒳,CoDisc(S))≃Set(𝒳 0,S) Grpd \big( \mathcal{X} ,\, CoDisc(S) \big) \;\; \simeq \;\; Set \big( \mathcal{X}_0 ,\, S \big)

and hence witnesses the claimed adjunction

CoDisc⊣(−) 0. CoDisc \;\; \dashv \;\; (-)_0 \,.

It has been argued in Lawvere 1984 that such codiscrete object-constructions, right adjoint to forgetful functors, deserve to be called “chaotic”.

Correspondingly, nerves of codiscrete groupoids are precisely the codiscrete objects in sSet, regarded as a cohesive topos over Set.

Examples

{Example}

\begin{example}\label{UniversalGPrincipalBundle} (chaotic groupoids as models for universal principal bundles) \linebreak For G∈Grp(Set)G \in Grp(Set) a group, the pair groupoid on GG is isomorphic

(G×G⇉pr 2pr 1G)≃EG \big( G \times G \underoverset {pr_2} {pr_1} {\rightrightarrows} G \big) \;\; \simeq \;\; \mathbf{E}G

to the action groupoid of the right (say) group action of GG on itself by right group-multiplication:

EG≔G×G⇉(−)⋅(−)pr 1G \mathbf{E}G \;\coloneqq\; G \times G \underoverset {(-) \cdot (-)} {pr_1} {\rightrightarrows} G

The nerve of the latter is equal to the standard incarnation (since we chose right action) of the universal principal simplicial complex WG∈sSetW G \, \in \, sSet:

N(EG)=WG. N(\mathbf{E}G) \;\; = \;\; W G \,.

The residual left multiplication action of GG on itself makes EG\mathbf{E}G a GG-action object internal to Grpd

EG∈GAct(Grpd). \mathbf{E}G \;\;\; \in \; G Act(Grpd) \,.

Since the nerve operation is a right adjoint (this Prop.) it preserves action objects, and the result

WG=N(EG)∈GAct(sSet) W G \;=\; N(\mathbf{E}G) \;\;\; \in \; G Act(sSet)

is the standard GG-action on the universal principal simplicial complex of GG.

The quotient of the group action on EG\mathbf{E}G yields the delooping groupoid BG\mathbf{B}G of GG

(1)EG→(EG)/G=BG. \mathbf{E}G \xrightarrow{\;\;} (\mathbf{E}G)/G \;=\; \mathbf{B}G \,.

While the nerve operation does not in general preserve colimits (this Exp.) it does preserve (by this Exp.) this particular colimit coprojection (1). The resulting Kan fibration

N(EG)=WG→(WG)/G=W¯G=N(BG)=N((EG)/G) N(\mathbf{E}G) \;=\; W G \xrightarrow{\;\;} (W G)/G \;=\; \overline{W}G \;=\; N(\mathbf{B}G) \;=\; N\big((\mathbf{E}G)/G\big)

is the universal simplicial principal bundle with structure group G∈Grp(Set)↪DiscGrp(sSet)G \,\in\, Grp(Set) \xhookrightarrow{Disc} Grp(sSet) regarded as a simplicial group.

Finally, the geometric realization of this into compactly generated topological spaces is the standard model for the universal principal bundle of GG:

EG=|N(EG)|→|N(BG)|=BG E G \;=\; \big\vert N(\mathbf{E}G) \big\vert \xrightarrow{\;\;} \big\vert N(\mathbf{B}G) \big\vert \;=\; B G

over its classifying space BG≃K(G,1)B G \simeq K(G,1) (which here is an Eilenberg-MacLane space, since GG was assumed to be discrete group – but this was just for simplicitiy of exposition, the analogous discussion applies to the chaotic topological groupoid of a topological group GG).

\end{example}

• delooping groupoid

• discrete groupoid

• indiscrete category

• category enriched in a bicategory

• complete graph