Given a category$Sp$ of spaces equipped with a forgetful functor$\Gamma : Sp \to Set$ to Set thought of as producing for each space its underlying set of points, a codiscrete space (codiscrete object) $Codisc(S)$ on a set$S$ is, if it exists, the image under the right adjoint$Codisc : Set \to Sp$ of $\Gamma$.

For $\Gamma : Top \to Set$ the obvious forgetful functor from Top, a codiscrete space is a set with codiscrete topology.

Codiscrete cohesive spaces

A general axiomatization of the notion of space is as an object in a cohesive topos. This comes by definition with an underlying-set-functor (or similar) and a left adjoint that produces discrete cohesive structure. See there for details.

Revised on May 3, 2017 08:48:36
by Urs Schreiber
(131.220.184.222)