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at total category I have added after the definition and after the first remark these two further remarks:
+– {: .num_remark}
Since the Yoneda embedding is a full and faithful functor, a total category induces an idempotent monad on its category of presheaves, hence a modality. One says that is a totally distributive category if this modality is itself the right adjoint of an adjoint modality.
=–
+– {: .num_remark}
The -adjunction of a total category is closely related to the -adjunction discussed at Isbell duality and at function algebras on ∞-stacks. In that context the -modality deserves to be called the affine modality.
=–
The article currently says something to the effect of “cototal categories are more rare than total categories”. But it occurs to me that is cototal by Day’s criterion (it’s complete, mono-complete, and has a cogenerator given by the indiscrete space on two elements). In fact, since being a topological functor is self-dual, and since is cototal, any category which is topological over is cototal – I’ll add this to the article as a class of examples. I don’t know of a reason to expect categories of a more “algebraic” nature to be cototal, but at least this suggests that many categories of “spaces” might be cototal.
Yes, good observation.
It’s known for example that is not cototal, and neither is say the category of commutative rings . An easy way to see this is to produce continuous functors that are not representable, e.g., for , the classical example is the class-indexed product of representables where ranges over all simple groups. (For any group , will be trivial once the simple group has cardinality greater than , so the product of over all simple will still be a set.) A similar example can be cooked up for commutative rings; see e.g. this MO answer. I guess algebraic categories with a plentiful supply of simple objects would be amenable to similar constructions.
I added some more examples to total category. (One is that Ab is cototal as well as total.)
fixed ref
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