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Treating the ultrafilter monad for sets as a role model, under modest hypotheses on the two parameters involved, we consider some topological notions for T-pre-orders, internal to a category C that comes equipped with a monad T. We call them T-spaces in C, present a rudimentary theory for them, and then focus on complete regularity. The full subcategory of T-spaces with this property is, like the category of all T-spaces, fibred over C, but distinguishes itself by a universal property in CAT/C vis-à-vis the monadic subcategory of compact Hausdorff T-spaces. The generality of the setting offers a broad array of example situations. The talk is based in part on recent wok with Amir Homayoun Nejah. Some references follow, but in the talk we will not assume any prior knowledge of the subject. [01] M. Barr: Relational algebras, SLNM 137, 1970.
[02] R. Börger: Coproducts and ultrafilters, JPAA 46, 1987.
[03] A. Burroni: T-catégories (catégories dans un triple), Cahiers 12, 1971.
[04] M.M. Clementino and W. Tholen: Metric, topology and multicategory-a common approach, JPAA 179, 2003.
[05] D. Hofmann, G.J. Seal and W. Tholen (eds.): Monoidal Topology, Cambridge University Press, 2014.
[06] G. Janelidze and W. Tholen: Extended Galois theory and dissonant morphisms, JPAA 143, 1999.
[07] T. Leinster: Codensity and the ultrafilter monad, TAC 28, 2013.
[08] E.G. Manes: Compact Hausdorff objects, Top. Appl. 4, 1974.
[09] A.H. Nejah and W. Tholen: A categorical review of complete regularity, in preparation.
[10] R.J. Perry: Completely regular relational algebras, Cahiers 17, 1976.
[11] W. Tholen and L. Yeganeh: The comprehensive factorization of Burroni's T-functors, TAC 36, 2021.
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