Continuity is an adjoint functor

Abstract

For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X\to Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.

To appear in the American Mathematical Monthly

References in corpus (1)

• Blowing up non-commutative smooth surfaces