In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni's Université de Paris thesis, and I don't see any evidence of it having ever been published more, well, publically.

It seems that this must be a large sketch, since categories sketchable by a small sketch are accessible, and in any case topological spaces would be locally presentable if they were accessible, and thus sketchable by a limit sketch.

Regarding large sketches, one has the large limit sketch of Spanier giving quasi-topological spaces. Burroni's thesis title includes the word "quasi-topologies," although it's not clear to me whether that is meant in Spanier's sense. Could Burroni have done something similar, adding some colimit cones to Spanier's topology on compact Hausdorff spaces to restrict to precisely ordinary spaces?

In any case, does anyone know his construction, or another sketch for spaces, or a reference in which one is written down? The only comparable thing I can think of is (**EDIT:**) Barr's theorem on spaces as relational $\beta$-algebras, ($\beta$ being the ultrafilter monad) and I doubt there's going to be some theorem relating relational algebras and large-sketchable categories.

compact Hausdorff spacesas (functional) $\beta$-algebras. $\endgroup$ – Todd Trimble♦ Jul 26 '17 at 21:28