Convergence in exponentiable spaces

Claudio Pisani

Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation $R:{\cal U}X \rightharpoonup X$ 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 \subseteq R\circ\eta $
2.$R\circ {\cal U}R = R\circ\mu $

where $\eta : X \to {\cal U}X$ and $\mu : {\cal U}({\cal U}X) \to {\cal U}X$ are the unit and the multiplication of the ultrafilter monad, and ${\cal U} : \bi{Rel} \to \bi{Rel}$ extends the ultrafilter functor ${\cal U} : \bi{Set} \to \bi{Set}$ to the category of sets and relations. $({\cal U},\eta,\mu)$ fails to be a monad on $\bi{Rel}$ only because $\eta$ is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on $\bi{Rel}$. Strict algebras are exponentiable and $T_1$ spaces.

Keywords: exponentiable spaces, quasi-local-compactness, convergence, ultrafilter monad, lax monads and algebras, continuous lattices.

1991 MSC: Primary 54A20, 54D45; Secondary 18C15.

Theory and Applications of Categories, Vol. 5, 1999, No. 6, pp 148-162.

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