What was Burroni's sketch for topological spaces?

If you read french, you can look at page 6 here. It is far from formal but it gives a good idea of the mixed sketch. In my understanding, there is a unique inductive cone in that mixed sketch, which comes from the canonical representation of the filter endofunctor as a colimit of representables. If you read on the rest of the paper, you'll see that this is a quite powerful method as Robert Paré showed that it could serve as a guide to sketch any category with split idempotent.

Here is a rough translation:

When trying to sketch topologies, the difficulties boil down to the problem of getting continuous applications as the natural transformations between models of the sketch. This prevents us to define a topology from its set of open sets, as it would eventually yields open maps instead of continuous one. Inspired by the idea behind Kowalsky's quasi-topologies, for which Ehresmann showed some interest, Albert [Burroni] proves that a topology on a set $E$ is no more than a "convergence" relation [denoted "$\to$"] between the set $\mathbf F(E)$ of filters on $E$ and the set $E$ itself, required to satisfy some axioms. A continuous map is then a map that preserves this notion of "convergence". Among many equivalent others, here is a system of axioms Albert proposed. The identifiers match the ones he used:

• $\mathbf T_2$ if $F \to x$ and $F\subset F'$, then $F' \to x$
• $\mathbf T_1$ $I_E(x) \to x$ ($I_E(x)$ stands for the principal filter on $x$)
• $\mathbf T'_4$ if $\phi\to F$ and $F\to x$, then $K_E(\phi) \to x$ ($K_E(\phi)$ stands for the filter on $E$ obtained by "summing" the elements of the filter $\phi$ on $\mathbf F (E)$)

By describing abstractly this process, he obtains the wanted sketch. It is formed by an "object of points" $e$, an "object of filters" $f(e)$, and an "object of filters on filters" $f(f(e))$, arrows from $e$ to $f(e)$ and from $f(f(e))$ to $f(e)$ to represent the maps $I_E$ and $K_E$, and a monomorphism with target $f(e)\times e$ to represent the convergence relation between filters and points. It is straightforward to translate the axioms above, after which it is just a matter of "typifying" the object $f(e)$ as the set of filters on $e$. To do so, Albert has the idea to decompose the foncteur $\mathbf F : \mathsf{Set} \to \mathsf{Set}$ as a canonical inductive limit of representable functors and then adds the necessary projective (discrete) cones and inductive cone to the skecth. The set-theoritical model of the sketch obtained this way are precisely the topological spaces. Notice that Albert also studies the model of this sketch in $\mathsf{Top}$ to conclude, apparently with a bit of disappointment:

"A topological topology is nothing else than a couple $(\pi,\pi')$ of topologies on a shared set $\theta(\pi)=\theta(\pi')$." [Burroni's thesis, p.65]

The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x))$ where $x$ is a set of variables and $\phi$ and $\psi$ are conjunctions of atomic formulas without equality. The result was proved for topological spaces first by E. Manes in Equational theory of ultrafilter convergence, Alg. Univ. 11 (1980), 163-172 and generalized by J. Rosický to any fibre-small topological category in Concrete categories and infinitary languages, J. Pure Appl. Alg 22 (1981), 309-339. And indeed, the class of axioms is a proper class, otherwise the category of topological spaces would be locally presentable, which is known to be false. Indeed, no non-discrete space is presentable.

The reference for Burroni's sketch of topological spaces is Comptes rendus de l'académie des sciences, 27 juillet 1970 (tome 271), which amazingly can be found online.