nLab
indiscrete category

Indiscrete categories

Definitions

An indiscrete category, also called a chaotic category, is a category CC in which there is a unique morphism from each object xx to each object yy:

x,yObj(C):C(x,y)=*, \forall x,y \in Obj(C) : C(x,y) = * \,,

where ** is the point.

Properties

This means that * an indiscrete category is in fact a groupoid; * any inhabited indiscrete category is equivalent to the terminal category.

Therefore, up to equivalence, an indiscrete category is simply a truth value.

The functor Ind:SetStrCatInd\colon Set \to Str Cat sending a set XX to the indiscrete category with XX as its set of objects (viewed as a strict category, that is up to isomorphism) is right adjoint to the forgetful functor Ob:StrCatSetOb\colon Str Cat \to Set sending a category to its set of objects. (The left adjoint DiscDisc to this forgetful functor sends a set XX to the discrete category on XX.)

Of course, we can compose IndInd (or DiscDisc) with the forgetful functor from StrCatStr Cat to the 2-category CatCat, in which we consider categories up to equivalence, as usual. Then the composite

SetIndStrCatCat Set \overset{Ind}\to Str Cat \to Cat

is naturally equivalent to the composite

SetInhTVSubsSetDiscStrCatCat, Set \overset{Inh}\to TV \overset{Subs}\to Set \overset{Disc}\to Str Cat \to Cat ,

where TVTV is the set (viewed a 22-category) of truth values, InhInh takes a set to the truth value of the statement that it is inhabited, PtPt takes a truth value to a subsingleton (left adjoint to InhInh), and DiscDisc is as above.

Revised on April 30, 2013 00:03:56 by Toby Bartels (64.89.53.107)