How can we have a categorical definition of "theory" ...

[Patrik Eklund <peklund@cs.umu.se>]

How can we have a categorical definition of 'theory', when we do not
have a categorical definition of 'sentence'?

Institutions (Goguen) and Entailment Systems (Meseguer) do have a
Sentence functor but their underlying category of signatures is
abstract, so without sort (over a category?) and operators (potentially
over another category?).

Many-valuedness and its (algebraic) foundations is looking into these
things.

Just wondering if anybody is thinking along these lines. Uwe at least, I
guess.

Cheers,

Patrik

PS Why many-valuedness? Well, for one thing, everything in and around
Google and Facebook is many-valued.



On 2017-11-08 01:57, 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.
>>
>>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]