Re: Some questions about different notions of "theory"

[ptj@maths.cam.ac.uk]

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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