An indiscrete category, also called a chaotic category, is a category in which there is a unique morphism from each object to each object :
where is the point.
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 sending a set to the indiscrete category with as its set of objects (viewed as a strict category, that is up to isomorphism) is right adjoint to the forgetful functor sending a category to its set of objects. (The left adjoint to this forgetful functor sends a set to the discrete category on .)
Of course, we can compose (or ) with the forgetful functor from to the 2-category , in which we consider categories up to equivalence, as usual. Then the composite
is naturally equivalent to the composite
where is the set (viewed a -category) of truth values, takes a set to the truth value of the statement that it is inhabited, takes a truth value to a subsingleton (left adjoint to ), and is as above.
Revised on April 30, 2013 00:03:56
by Toby Bartels