category theory

# Contents

## 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 $X$ and sometimes also the chaotic groupoid (this is explained below), indiscrete groupoid, or coarse groupoid on $X$, in older literature also Brandt groupoid.

## Definition

###### Definition

(codiscrete groupoid)
For $X \in Set$, the codiscrete groupoid of $X$ is the groupoid

$Codisc(X) \;\coloneqq\; X \times X \underoverset {pr_2} {pr_1} {\rightrightarrows}$

whose object of objects is

• $Obj(X) = X$,

whose objects of morphisms is the Cartesian product of $X$ with itself

• $Mor(X) = X \times X$,

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:

$X \times X \times X \xrightarrow{\; (pr_1, pr_3) \; } X \times X$

###### Remark

Def. 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).

## 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 $X$ a finite set of cardinality $n \gt 0$, the category algebra of $Codisc(X)$ is the algebra of $n\times n$ matrices. The contractibility of $Codisc(X)$ 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).

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

Here

$(-)_0 \;\; \colon \;\; \big( X_1 \rightrightarrows X_0 \big) \;\;\; \mapsto \;\;\; X_0$

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

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

is uniquely determined as soon as its component function

$\mathcal{X}_0 \xrightarrow{\;\; F_0 \;\;} CoDisc(S) = S$

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

This association there gives a natural bijection of hom-sets

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

and hence witnesses the claimed adjunction

$CoDisc \;\; \dashv \;\; (-)_0 \,.$

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}

###### Example

(chaotic groupoids as models for universal principal bundles)
For $G \in Grp(Set)$ a group, the pair groupoid on $G$ is isomorphic

$\big( G \times G \underoverset {pr_2} {pr_1} {\rightrightarrows} G \big) \;\; \simeq \;\; \mathbf{E}G$

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

$\mathbf{E}G \;\coloneqq\; G \times G \underoverset {(-) \cdot (-)} {pr_1} {\rightrightarrows} G$

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

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

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

$W G \;=\; N(\mathbf{E}G) \;\;\; \in \; G Act(sSet)$

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

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

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

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

$N(\mathbf{E}G) \;=\; W G \xrightarrow{\;\;} (W G)/G \;=\; \overline{W}G \;=\; N(\mathbf{B}G) \;=\; N\big((\mathbf{E}G)/G\big)$

is the universal simplicial principal bundle with structure group $G \,\in\, Grp(Set) \xhookrightarrow{Disc} Grp(sSet)$ 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 $G$:

$E G \;=\; \big\vert N(\mathbf{E}G) \big\vert \xrightarrow{\;\;} \big\vert N(\mathbf{B}G) \big\vert \;=\; B G$

over its classifying space $B G \simeq K(G,1)$ (which here is an Eilenberg-MacLane space, since $G$ 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 $G$).