If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.

Is there a characterization of limit theories $\mathcal{T}$ such that the category of models of $\mathcal{T}$ is a topos? The category of models of a $\mathcal{T}$ is locally presentable and hence a reflective subcategory of presheaves on $\mathcal{T}$. If there is a characterization of these theories, is it automatic that the reflector is finitely continuous, and hence gives the theories that underly Grothendieck toposes?

I know that Johnstone characterizes the product theories whose models are toposes. Here $\mathcal{T}$ must have no pseudoconstants and be sufficiently unary. I guess I'm curious if anyone has generalized this result.

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    $\begingroup$ I don't know this result of Johnstone you are refering to, could you provide a more precise reference ? that sound interesting. $\endgroup$ Sep 19 '18 at 8:18
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    $\begingroup$ @SimonHenry I imagine Jonathan is referring to Johnstone's paper "When is a variety a topos?", Algebra Universalis 21 (1985), 198-212. An interesting point made in that paper is that alongside the obvious family of examples of finite product theories whose models form a topos (namely, M-sets for a monoid M), there's a not-so-obvious family: the so-called Jónsson-Tarski algebras and generalizations thereof. A JT algebra is a set $A$ equipped with a bijection $A \to A \times A$. $\endgroup$ Sep 19 '18 at 17:13
  • $\begingroup$ @TomLeinster. Yes, that is correct the reference I was referring to. A link I have is link.springer.com/content/pdf/10.1007/BF01188056.pdf but I don't know if it's available without subscription. $\endgroup$ Sep 20 '18 at 1:38
  • $\begingroup$ Thanks ! I knew about the example of Jonsson-Tarski algebras but I wasn't aware of such a characterization. $\endgroup$ Sep 20 '18 at 7:52

My collaborator Julia Ramos González and I are working on this question precisely in these days.

A part of the answer is already cointained in a paper by Carboni, Pedicchio and Rosický: Syntactic characterizations of various classes of locally presentable categories, Journal of Pure and Applied Algebra, 161 (2001) pp 65-90.

Putting together Thm. 5 and 19 one gets that:

A finitely presentable category is a Grothendieck topos if and only if the full subcategory of finitely presentable objects is extensive and pro-exact.

Please, read also the paragraph that comments Thm. 19.

Recall that the full subcategory of finitely presentable objects is essentially the limit theory that presents the locally (finitely) presentable category, i.e. $$\mathcal{K} \cong \text{Lex}(\text{Pres}(\mathcal{K})^{\circ}, \text{Set}). $$

3 March 2019.

As I was mentioning in the previous version of this answer, together with Julia, we worked on a generalization of this statement to the infinitary case and related the site-theoretic presentation with the limit-theory presentation. The result of this investigation, Gabriel-Ulmer duality for topoi and its relation with site presentations, is now on the arXiv:1902.09391.


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