Dear Mike, Thanks for the clarification, but the same comment applies: most higher- order theories don't correspond to anything as simply described as a monad (just as most first-order theories don't, but only the very special class of algebraic theories). For first-order theories (including infinitary ones) you have the notion of sketch, but that (unlike a monad) is still presentation-dependent (i.e. it varies according to which parts of the structure you're modelling you regard as primitive); if you want something that doesn't depend on the presentation, you need to go all the way to the syntactic category, which essentially consists of `everything that has to be present in a category containing a model of the theory under consideration' (just as a Lawvere theory contains all the operations which have to be present in a model of an algebraic theory). Syntactic categories can also be constructed for higher-order theories, and they are in fact toposes; but they tend to be insanely complicated (even the free topos, which corresponds to the empty theory, has an incredibly rich structure). Peter Johnstone On Nov 9 2017, Mike Stay wrote:
On Tue, Nov 7, 2017 at 4:57 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'm sorry, I meant "those monads that correspond to higher-order theories", where a higher-order theory gets interpreted in a topos in the same way that a Lawvere theory gets interpreted in a category with finite products.
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