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In Sketches: Outline with References 4.3, Wells cites the result that sketches are sketchable by a finite limit sketch. I can't find the Burroni 1970a paper, and I am having a lot of trouble with Lair 1974, since I'm not so good with mathematical French, the result is at the end of the paper, and uses notation that is either referenced earlier in the paper or in other papers. The other two references, as far as I can tell (which isn't saying much), don't contain the result.

Is there an English reference? What is the construction?

I've been trying to figure it out myself, and the main difficulty I'm having is associating the class of diagrams to composites of arrows. It seems like given a sketch with an underlying graph $G$, you want to have maps $s,t: D \to F(G)_1$ where $D$ is a class of two-sided diagrams, $F(G)_1$ is the class of arrows in the free category generated by $G$, and $s$ and $t$ send each diagram to the composite of arrows on each side of the diagram. But $F(G)_1$, as far as I can tell, isn't an object that you can specify in the sketch.

This is an instance of a more general issue that arises when you ask if there is a sketch for presentations relative to a free functor. Given two FL sketches $S, T$ and an interpretation $I : S \to th(T)$ ($th$ sends sketches to classifying categories with finite limits), there is an induced free functor $F_I: Mod(S) \to Mod(T)$, (for simplicity, models take values in Set) and you can construct the category of presentations relative to that free functor, which consists of pairs of objects $(X,R)$ in $Mod(S)$ and a map (perhaps a monomorphism) $R \to F(X) \times F(X)$, and morphisms $f = (f_0, f_1) : (X,R) \to (Y,P)$ of presentations are maps $f_0 : X \to Y$ and $f_1: R \to P$ making the obvious diagram commute.

In the case of groups, this gives exactly group presentations. There is a sketchable theory of essentially algebraic signatures and interpretation from this theory into the theory of categories with chosen finite limits which should give sketches as presentations relative to the induced free functor. The issue again for this construction is the map $R \to F(X) \times F(X)$ which includes this object $F(X)$. Perhaps this is not the way to go about this though.

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  • $\begingroup$ this is answered here at MO; Burroni's paper is online here; a summary, in French, is here, with an English translation in the first MO link. $\endgroup$
    – Carlo Beenakker
    Mar 24, 2019 at 21:35
  • $\begingroup$ Perhaps I am missing something, but this doesn't seem to answer my question, or perhaps, this reference doesn't seem to answer my question (although I can't find the specific title "Esquisses des categories a limites et des quasi-topologies" in any of those links which is what is cited in Wells). The sketch is a sketch for topologies, and I want one for sketches. $\endgroup$
    – Daniel Satanove
    Mar 24, 2019 at 21:46
  • $\begingroup$ "Esquisses des categories a limites et des quasi-topologies" is the title of Burroni's thesis, I understood from the MO posting that the link was to the published version in the Comptes Rendus... $\endgroup$
    – Carlo Beenakker
    Mar 24, 2019 at 21:58
  • $\begingroup$ Ah, sorry, I got the two links mixed up. The summary doesn't answer my question. I will look at Burroni's Paper. $\endgroup$
    – Daniel Satanove
    Mar 24, 2019 at 22:15
  • $\begingroup$ That link, from the third MO answer, is just to a 3 page note. It looks like I would have to do what the fourth answer suggests which involves emailing someone to get the whole paper. $\endgroup$
    – Daniel Satanove
    Mar 24, 2019 at 22:28

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