nLab codiscrete space

Idea

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 $\Gamma$.

Sometimes the codiscrete topology is also called the chaotic topology.

The dual concept is that of discrete space. For their relation see at discrete and codiscrete topology.

Examples

Codiscrete topological spaces

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.

References

The terminology chaotic topology is motivated (see also at chaos) in

• William Lawvere, Functorial remarks on the general concept of chaos IMA preprint #87, 1984 (pdf)

via footnote 3 in

• William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (tac:tr9, pdf)

and appears for instance in

• The Stacks Project, Example 7.6.6