Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R : UX ⇀ X between ultrafilters and elements of a set X is the convergence relation for a quasi-locally-compact (that is, exponentiable) topology on X if and only if the following conditions are satisfied: 1. id ⊆ R ◦ η 2. R ◦ UR = R ◦ μ where η : X → UX and μ : U(UX) → UX are the unit and the multiplication of the ultrafilter monad, and U : Rel → Rel extends the ultrafilter functor U : Set → Set to the category of sets and relations. (U , η, μ) fails to be a monad on Rel only because η is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on Rel . Strict algebras are exponentiable and T1 spaces.
[1]
Donald Yau,et al.
Categories
,
2021,
2-Dimensional Categories.
[2]
G. M. Kelly,et al.
On topological quotient maps preserved by pullbacks or products
,
1970,
Mathematical Proceedings of the Cambridge Philosophical Society.
[3]
K. Hofmann,et al.
A Compendium of Continuous Lattices
,
1980
.
[4]
Alan Day.
Filter monads, continuous lattices and closure systems
,
1975
.
[5]
J. Isbell,et al.
General function spaces, products and continuous lattices
,
1986,
Mathematical Proceedings of the Cambridge Philosophical Society.
[6]
Oswald Wyler.
Algebraic theories of continuous lattices
,
1981
.
[7]
M. Potter,et al.
Sets: An Introduction
,
1990
.