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Stream: theory: category theory

Topic: functoriality of codensity monad

Christian Williams (Nov 19 2020 at 02:19):

When people write about the construction of codensity monads, they describe it as a functor $(\mathrm{Cat}/A)^{\mathrm{op}} \to \mathrm{Mnd}(A)$ . This is great, but it leaves out the 2-categorical structure of the slice over $A$ .

I've been trying to see how it would act on natural transformations, and it seems like there may be a variance obstruction that makes it impossible. Does anyone have thoughts about whether forming codensity monads can be made into a 2-functor? Thanks.

Christian Williams (Nov 19 2020 at 02:34):

Or a simpler version of the question - given an object $c\in C$ in a cartesian category, we can form the endomorphism operad $\langle c,c\rangle:\mathbb{F}\to \mathrm{Set}$ defined by $\langle c,c\rangle (n) = C(c^n,c)$ . Is this functorial?

Christian Williams (Nov 19 2020 at 02:36):

(These are closely related to a previous thread about the possible functoriality of $X\mapsto [X,X]$ .)

sarahzrf (Nov 19 2020 at 03:33):

all else aside, i wonder if you can make it a functor out of the (2, 1)-category or something

Dan Doel (Nov 19 2020 at 03:53):

The construction I'm familiar with isn't functorial in the sense you're asking about, I think.

Dan Doel (Nov 19 2020 at 03:55):

For $Z X = 0$ , the codensity monad is double negation. For the identity functor, it's the identity. There's a natural transformation from the zero functor to the identity functor, but a transformation between the codensity monads would be double negation elimination.