Operads is just another name for signatures (sorts and operators). Operads emphasize arity a bit more, which might invite operads and signatures e.g. "over monoidal categories" to try out different paths, but "over Set" they are the same. Operads tend to "overlook" the distinction between constant and variable, as seen e.g. in tree automata. Anyway, terms (from signatures) have been formally constructed by means of monads over monoidal categories, but sentence functors are rare. In predicate calculus, the (informal!) quantifier symbols are problematic, as they disable a functorial sentence construction. G??del treats those symbols as formal since they are part of his numbering. I've often said that self-referentiality is what makes incompleteness theorems dubious, but treating quantifier symbols as formal ones and disabling categorical constructions is probably the root of that "G??del's success story". The concept of 'theory' in universal algebra (Lawvere theories) and in generalized logic (Goguen, Meseguer, and as we have generalized it to fully embrace signatures) is related but not the same. Let me indeed underline that when terms and sentences are categorically constructed, they are not just "strings of symbols". Monoids are not that universal. Let me also restate that in our developments of these things (including practical applications) we prefer to say that logic must be 'lative' in the sense that terms are constructed in order to enable sentence construction, and so on, and later in that "lative chain" comes 'theory', and many other things making 'logic' a categorical object. In that category, still to be fully unravelled, morphisms are really challenging and fascinating. We feel there is a growing interest in such 'lative' approaches, and we do invite the CT community to be more curious about such foundational things. Best, Patrik On 2017-11-10 03:55, David Yetter wrote:
In what way is defining a theory dependent on defining a sentence?
There are lots of categorical definitions of theories of various sorts:
Operads, PROPs, Lawvere theories,..., even monads.
One of the points of category theory, esp. higher category theory, has
been to liberate mathematical thought from being bound to strings of
symbols like terms and sentences.
Best Thougths,
David Yetter
Professor of Mathematics
Kansas State University
________________________________ From: Patrik Eklund <peklund@cs.umu.se> Sent: Thursday, November 9, 2017 1:13 AM To: categories@mta.ca Subject: categories: How can we have a categorical definition of "theory" ...
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.
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