Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: reading & references

Topic: Codensity monad induced by an object

Vincent Moreau (May 12 2024 at 17:22):

Dear category theorists,

I have found on the codensity monad page something that bugs me. Indeed, Example 3.2 is the following:

Let $d$ be an object in a closed category $C$. Then the codensity monad of the constant functor $d : 1 \to C$ is the double dualization monad associated to $d$, given by $d^{d^-}$.

It seems to me that this is not correct. Indeed, for any category $C$ together with an object $d$ which has powers $S \pitchfork d$ for any set $S$, it seems to me that the codensity monad induced by $d$ is the functor $C(-, d) \pitchfork d : C \to C$. This is not the double dualization as computed with the internal hom in a CCC.

Am I missing something, for example in what it means that $C$ is closed, or on the way to interpret the formula $d^{d^-}$, or is this a typo?

(sorry in advance if this is not the correct stream)

David Kern (May 12 2024 at 17:49):

The formula you give is indeed the correct formula for the codensity monad of the functor selecting the object $d$, which is also known as the endomorphism monad of $d$ since morphisms from some other monad $T$ into it correspond to $T$-algebra structures on $d$. But in a closed category, powers are given by the internal homs, so this does indeed recover the double dualisation.
And since this is a reading and references stream, there is some early writing on the subject by Anders Kock.

Nathanael Arkor (May 12 2024 at 17:52):

I suppose the subtlety is that one must take the $C$-enriched codensity monad, rather than the unenriched codensity monad, so that the power is taken in $C$ rather than in Set. This should be clarified on the nLab page.

Graham Manuell (May 12 2024 at 17:52):

@Vincent Moreau I think you are right. I think the internal double dual would instead be an enriched codensity monad.

In the paper David linked Kock is talking about strong monads, which I think is equivalent to being enriched in this setting.

David Kern (May 12 2024 at 17:55):

Ah yes, that's a good subtlety.

Vincent Moreau (May 12 2024 at 17:56):

I see, thanks all for your replies. Indeed, when trying to prove the fact that $((-) \Rightarrow d) \Rightarrow d$ is the codensity monad, I was blocked as I needed a map $c \Rightarrow c' \to F(c) \Rightarrow F(c')$. Is this given by the enriching condition on functors?

David Kern (May 12 2024 at 18:01):

If $c\Rightarrow c^\prime$ means the internal hom, then yes, that's exactly (part of) the data of an enriched functor.

Vincent Moreau (May 12 2024 at 18:04):

I see, great! Thanks.

John Baez (May 12 2024 at 18:11):

A very minor question: why is this conversation titled "codensity one object"? I don't know what that means.

Vincent Moreau (May 12 2024 at 18:16):

I was wondering what was the codensity monad associated to a functor $1 \to C$, hence coming from one object of $C$.