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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.

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    $\begingroup$ Actually, the relational $\beta$-module characterization is due to Barr. It is of course related to the Manes characterization of compact Hausdorff spaces as (functional) $\beta$-algebras. $\endgroup$ – Todd Trimble Jul 26 '17 at 21:28
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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]

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  • $\begingroup$ Ok, great, it's essentially given as a relational algebra, for the functor of filters instead of ultrafilters. Thanks very much for the reference! $\endgroup$ – Kevin Carlson Aug 4 '17 at 16:09
  • $\begingroup$ Possibly a dumb question, but what's the domain and codomain of the filter endofunctor and how does it act on morphisms? $\endgroup$ – goblin Feb 24 at 23:59
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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.

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  • $\begingroup$ Thanks very much, Philippe! I suppose that this constructs spaces as the models of a non-small limit sketch. So I still wonder why Burroni's sketch was cited as an interesting example of a mixed sketch. $\endgroup$ – Kevin Carlson Jul 27 '17 at 4:42
  • $\begingroup$ @KevinCarlson Yes this proves that the topological spaces are the models of a large (i.e. non-small) limit sketch. My answer was about your very last question. You can find an explicit description of the axioms in Example 5.4 of J. Rosický's paper (which is in open access, you just have to google the title). $\endgroup$ – Philippe Gaucher Jul 27 '17 at 7:47
  • $\begingroup$ Ah, great, I understand. $\endgroup$ – Kevin Carlson Jul 27 '17 at 16:59
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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.

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Perhaps Albert's thesis, Esquisses des catégories à limites et des quasi-topologies is still available. If you go to this Cahiers page and follow the link on the left hand page to ESQUISSES MATHEMATIQUES, it says

Most of them are still available and can be obtained by sending an e-mail

to the email address given there. Albert's thesis is no. 5 on the list.

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