nLab
indiscrete category

Definitions

An indiscrete category, also called a chaotic category, is a category $C$ in which there is a unique morphism from each object $x$ to each object $y$:

$$\forall x,y\in \mathrm{Obj}(C):C(x,y)=*,$$\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 $\mathrm{Ind}:\mathrm{Set}\to \mathrm{Str}\mathrm{Cat}$ sending a set $X$ to the indiscrete category with $X$ as its set of objects (viewed as a strict category, that is up to isomorphism) is right adjoint to the forgetful functor $\mathrm{Ob}:\mathrm{Str}\mathrm{Cat}\to \mathrm{Set}$ sending a category to its set of objects. (The left adjoint $\mathrm{Disc}$ to this forgetful functor sends a set $X$ to the discrete category on $X$.)

Of course, we can compose $\mathrm{Ind}$ (or $\mathrm{Disc}$) with the forgetful functor from $\mathrm{Str}\mathrm{Cat}$ to the 2-category $\mathrm{Cat}$, in which we consider categories up to equivalence, as usual. Then the composite

$$\mathrm{Set}\stackrel{\mathrm{Ind}}{\to }\mathrm{Str}\mathrm{Cat}\to \mathrm{Cat}$$Set \overset{Ind}\to Str Cat \to Cat

is naturally equivalent to the composite

$$\mathrm{Set}\stackrel{\mathrm{Inh}}{\to }\mathrm{TV}\stackrel{\mathrm{Subs}}{\to }\mathrm{Set}\stackrel{\mathrm{Disc}}{\to }\mathrm{Str}\mathrm{Cat}\to \mathrm{Cat},$$Set \overset{Inh}\to TV \overset{Subs}\to Set \overset{Disc}\to Str Cat \to Cat ,

where $\mathrm{TV}$ is the set (viewed a $2$-category) of truth values, $\mathrm{Inh}$ takes a set to the truth value of the statement that it is inhabited, $\mathrm{Pt}$ takes a truth value to a subsingleton (left adjoint to $\mathrm{Inh}$), and $\mathrm{Disc}$ is as above.