Indiscrete categories

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 $y$:

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

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 $Ind\colon Set \to Str Cat$ 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 $Ob\colon Str Cat \to Set$ sending a category to its set of objects. (The left adjoint $Disc$ to this forgetful functor sends a set $X$ to the discrete category on $X$.)

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

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

is naturally equivalent to the composite

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

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

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