indiscrete category in nLab (original) (raw)
Indiscrete categories
Indiscrete categories
Definitions
An indiscrete category is a category CC in which there is a unique morphism from each object xx to each object yy:
∀x,y∈Obj(C):C(x,y)=*, \forall x,y \in Obj(C) : C(x,y) = * \,,
where ** is the point.
The terms chaotic category, and codiscrete category are also used.
Properties
This means that
- an indiscrete category is in fact a groupoid, in fact a codiscrete 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:Set→StrCatInd\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:StrCat→SetOb\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
Set→IndStrCat→Cat Set \overset{Ind}\to Str Cat \to Cat
is naturally equivalent to the composite
Set→InhTV→SubsSet→DiscStrCat→Cat, 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.
Last revised on March 2, 2023 at 17:36:35. See the history of this page for a list of all contributions to it.