Is the following of any relevance? Dubuc, Eduardo J.; Kelly, G. M. (1983). "A presentation of topoi as algebraic relative to categories or graphs" <http://www.sciencedirect.com/science/article/pii/0021869383901977>. *J. Algebra*. *81* (2): 420–433. doi <https://en.wikipedia.org/wiki/Digital_object_identifier>: 10.1016/0021-8693(83)90197-7 <https://doi.org/10.1016%2F0021-8693%2883%2990197-7>. On Tue, Nov 7, 2017 at 6:57 PM, <ptj@maths.cam.ac.uk> wrote:
1) I'm not sure what Mike means by `those monads that correspond to toposes' since most toposes don't correspond to monads on anything. I did investigate those toposes which are monadic over Set, or a power of Set, in my papers `When is a variety a topos?', Algebra Universalis 21 (1985), 198--212, and `Collapsed toposes and cartesian closed varieties', J. Algebra 129 (1990), 446--480.
2) A possible answer to this question is that the (2-)category of finitely presented minimal toposes (and logical functors) is equivalent to the dual of the free topos (on no generators), where a topos is said to be minimal if it has no proper full logical subtoposes. This is a result of Peter Freyd, but I don't know whether he ever published it.
Peter Johnstone
On Nov 7 2017, Mike Stay wrote:
1) Finitary monads correspond to Lawvere theories. Is there a name
for those monads that correspond to toposes?
2) In topos theory is there any analogous result to Lawvere's theorem that the opposite of the category of free finitely generated gadgets is equivalent to the Lawvere theory of gadgets? Something like "the opposite of the category of fooable gadgets is equivalent to the topos of gadgets"?
3) nLab says a sketch is a small category T equipped with subsets (L,C) of its limit cones and colimit cocones. A model of a sketch is a Set-valued functor preserving the specified limits and colimits. Is preserving limits and colimits like a ring homomorphism? Preserving both limits and colimits sounds like it ought to involve profunctors, but maybe I'm level slipping.
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