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

(codiscrete groupoid)
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

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

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. . 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}

Last revised on March 5, 2024 at 06:24:47. See the history of this page for a list of all contributions to it.