15
$\begingroup$

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.

$\endgroup$
  • 4
    $\begingroup$ I don't know this result of Johnstone you are refering to, could you provide a more precise reference ? that sound interesting. $\endgroup$ – Simon Henry Sep 19 '18 at 8:18
  • 2
    $\begingroup$ @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 $A \to A \times A$. $\endgroup$ – Tom Leinster 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. $\endgroup$ – Jonathan Gallagher 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. $\endgroup$ – Simon Henry Sep 20 '18 at 7:52
14
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.