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Stream: theory: category theory

Topic: monadification

sarahzrf (May 04 2020 at 00:20):

is there a name for the monad of the semantics-structure adjunction? (i.e., the one which takes a functor's codensity monad and then gives the forgetful functor from the eilenberg-moore category) i figure "monadification" sounds appropriate if not ("monadicification" is probably more strictly correct, but also terrible)
does anyone have cool examples of interesting things to apply it to? i found out that the monadification of the inclusion FinSet → Set is the forgetful functor CHaus → Set (up to equivalence, anyway) and now i wanna know what else it might do

fosco (May 04 2020 at 14:52):

Deep down, the reason why the functor Ran_FF is a monad is that you can think Ran_FF as an hom-like construction〈F,F〉
So, the reason why 〈F,F〉is a monad is the same that tells you that the set of self maps End(A) = hom(A,A) for A \in C is a monoid.
This is the ballpark I would try to move into, if I had to find a name for the structure-semantics adjunction, different from "structure-semantics adjunction".

sarahzrf (May 04 2020 at 15:02):

well, i mean a name for its monad, not a name for the adjunction itself

sarahzrf (May 04 2020 at 15:03):

the monad which turns a codensity-monad-admitting functor into a monadic functor

Mike Shulman (May 04 2020 at 19:25):

"codensity algebras"?

Oscar Cunningham (May 04 2020 at 21:20):

I'd say codensity modules.
I've been wondering if things where people say 'An X is a Y which is like a Z' are all examples. For example people say that 'A Lie Algebra is a vector space equipped with a bilinear map which is like a commutator'. So are Lie algebras the codensity modules of the commutator functor from the category of associative algebras to the category of vector spaces with bilinear maps? Likewise 'Locales are like the poset of open sets of a topological space', so are locales the codensity modules of $\mathcal O:\mathbf{Top}^\mathrm{op}\to\mathbf{Poset}$ ?

Oscar Cunningham (May 04 2020 at 21:21):

'A scheme is a ringed space which is like an affine scheme'?

sarahzrf (May 05 2020 at 06:55):

well, regarding the last one... if we have any poset, we can take the set of completely prime filters and make a topological space, right? does that give a left adjoint to $\mathcal O$ ?

sarahzrf (May 05 2020 at 06:56):

or only to $\mathcal O : \mathbf{Top}^\mathrm{op} \to \mathbf{Frm}$ ?

Oscar Cunningham (May 05 2020 at 10:15):

Good point. I think $\mathcal O$ does have a left adjoint in this case, since it preserves limits (although checking that involves thinking about coequalizers in $\mathbf{Top}$ , which make my brain hurt). I think there's an adjoint in my first example too. So these are just trivial examples of codensity monads.

sarahzrf (May 05 2020 at 14:06):

well, i don't know about the others :sweat_smile:

sarahzrf (May 05 2020 at 14:06):

have not thought about them at all

sarahzrf (May 05 2020 at 14:09):

actually, even if that is an adjunction, it just establishes what the codensity monad is—does that tell you that the category of algebras of the codensity monad in question is the category of frames?

Oscar Cunningham (May 05 2020 at 15:37):

I'm trying to work out what the left adjoint actually is. Let $P$ be a poset. Then $(1\to LP)\simeq(P\to\mathcal O1)$ , and $\mathcal O1$ is just the total order on two points. So the points of $LP$ are just all filters of $P$ . I can't see what the open set need to be though.

Jens Hemelaer (May 06 2020 at 12:24):

For each element $p \in P$ , you can define the set of filters:
$U_p = \{ x : p \in x \}$
You can take these sets $U_p$ as the basis for a topology on the set of filters.
This is the natural topology, in the sense that if $X_P$ is the space of filters, with the above topology, then
$\mathbf{PSh}(P)\simeq\mathbf{Sh}(X_P)$ .
I didn't check that this is the right topology on $LP$ , but this would be my best guess.