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Stream: theory: category theory

Topic: Kan extension as lax-functorial

Patrick Nicodemus (Feb 25 2023 at 21:22):

There's a sense in which Kan extension is lax-functorial, I am looking for a source or reference in which this is described, I would like to cite a reference for the associativity and unit conditions.

Patrick Nicodemus (Feb 25 2023 at 21:23):

If nobody knows what i'm talking about i'll write it down haha.

Patrick Nicodemus (Feb 25 2023 at 21:28):

I guess the simplest statement is this. Let $t : A\to B$ be a 1-cell in a 2-category. Let $Hom(A,A)$ be regarded as the strict monoidal 1-category whose objects are 1-cells and whose morphisms are 2-cells, with monoidal product given by composition. Similarly with $Hom(B,B)$.
Then there is is a lax monoidal functor from $Hom(A,A)\to Hom(B,B)$ sending $x: A\to A$ to the right Kan extension of $tx$ along $t$.

Nathanael Arkor (Feb 25 2023 at 22:13):

Essentially the same observation is made in this Math.StackExchange question, though no-one was able to provide a literature reference.

Patrick Nicodemus (Feb 25 2023 at 22:26):

@fosco Did you ever end up finding a reference lol

Patrick Nicodemus (Feb 25 2023 at 22:30):

Something that is kind of interesting about this:
PJ Huber has a paper very early on in the history of monads where he observes that if T is a monad on D, and we have an adjunction L : C -> D, R : D -> C with L -| R, then LTR is a monad on C, i.e., we can transfer the monad from D to C along the adjunction. The usual observation that an adjunction between L and R gives rise to a monad on C is just the special case of T = id_D. Somehow Huber's way of phrasing it - "monads can be transferred along adjunctions" is always less emphasized in the literature than the special case of the identity monad.

Something similar seems to be happening here. Everybody cares about the codensity monad, which is what Fosco is asking about when the original monad is the identity monad, but we cannot even find a source for the more general case.

Nathanael Arkor (Feb 25 2023 at 23:41):

Somehow Huber's way of phrasing it - "monads can be transferred along adjunctions" is always less emphasized in the literature than the special case of the identity monad.

This is because, when every monad admits a resolution (which is true for V-Cat), Huber's observation is a trivial consequence of the fact that adjunctions compose. So the more general statement is only interesting in 2-categories in which not every monad admits a resolution, which is an uncommon setting.

Nathanael Arkor (Feb 25 2023 at 23:56):

Similarly, when every monad admits a codensity monad (which in particular follows when every monad admits a resolution), then I believe a special case of this more general codensity construction follows from the usual one.

Nathanael Arkor (Feb 26 2023 at 00:02):

In particular, suppose a monad $t$ on $C$ and a right-adjoint 1-cell $r : C \to D$, and suppose that $t$ is the codensity monad of a 1-cell $u : B \to C$. Then, if I haven't got mixed up,

$$\mathrm{Ran}_r (r t) \cong \mathrm{Ran}_r (r (\mathrm{Ran}_u u)) \cong \mathrm{Ran}_r (\mathrm{Ran}_u (r u)) \cong \mathrm{Ran}_{r u} (r u)$$

so that this construction is the codensity monad of $r u$.

Nathanael Arkor (Feb 26 2023 at 00:03):

Therefore, the more general construction is only particularly interesting in 2-categories other than V-Cat.

Christian Williams (Feb 26 2023 at 19:23):

@Nathanael Arkor why is $r(\mathrm{Ran}_{u}u) \simeq \mathrm{Ran}_{u}(ru)$?

Christian Williams (Feb 26 2023 at 19:42):

We have $\mathrm{Ran}_{u}u = \prod b{:}B\;\; C(-,u(b))\pitchfork u(b)$, so it is a limit in $[C,C]$.

Then are we using that postcomposition $r\circ - : [C,C]\to [C,D]$ is a right adjoint, and so preserves limits? Because if $r$ is an arbitrary 1-cell, then that means assuming the existence of left lifts. Maybe that's a part of your assumption about the existence of "resolutions"; I'm not sure.

dusko (Feb 26 2023 at 20:09):

Patrick Nicodemus said:

I guess the simplest statement is this. Let $t : A\to B$ be a 1-cell in a 2-category. Let $Hom(A,A)$ be regarded as the strict monoidal 1-category whose objects are 1-cells and whose morphisms are 2-cells, with monoidal product given by composition. Similarly with $Hom(B,B)$.
Then there is is a lax monoidal functor from $Hom(A,A)\to Hom(B,B)$ sending $x: A\to A$ to the right Kan extension of $tx$ along $t$.

one way to interpret the construction is as a restriction of yoneda structures (cf street-walters) to the analogous "cayley structures", where the yoneda embeddings $A\to [A^o, S]$ are replaced by the cayley embeddings $A\to [A, A]$. while the former is the abstraction of the 2-category of $S$-enriched categories, the latter is the abstraction of the 2-category of monoidal categories with their cayley embeddings. (the iso-strength we stumbled on the other day would be an instance where these monoidal categories are categories of monoids, so that each $[A,A]$ is a 2-category of representable monads...)

obviously, having the right kan extensions is a way of saying that the 0-cells are complete. so switching from the yoneda structures to the cayley structures trades-in the assumption that $S$ is total for the assumption that every $B$ is suitably complete. (but "suitably complete" might be equivalent with "total".) so it is a pricey move. for posets some of the constructions seem to require that the complete lattices be complemented (for duality) and atomic (for size and cogenerators), which by stone duality collapses to $\cal Set^o$. for categories the stone duality is... well, makkai has done it, but most of us never succeeded in using it. a host of technical problems seems to explode --- which of course may be worth the price if what you are seeking to axiomatize is even more complicated.

i am willing to bet that the nontrivial examples will be bicategories, not 2-categories. "although
the fundamental constructions of set theory are categorical, the fundamental constructions of category theory are bicategorical."

John Baez (Feb 26 2023 at 20:11):

The devil's pitchfork is a powerful tool when working with Kan extensions.

$\pitchfork$ :smiling_devil: $\pitchfork$

Nathanael Arkor (Feb 26 2023 at 20:17):

Christian Williams said:

Nathanael Arkor why is $r(\mathrm{Ran}_{u}u) \simeq \mathrm{Ran}_{u}(ru)$?

We have bijections of 1-cells, natural in $f$:

• $f \to r \mathrm{Ran}_u u$
• $\ell f \to \mathrm{Ran}_u u$
• $\ell f u \to u$
• $f u \to r u$
• $f \to \mathrm{Ran}_u (r u)$

Patrick Nicodemus (Feb 26 2023 at 20:41):

nvm, I understand.

Patrick Nicodemus (Feb 26 2023 at 20:41):

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Christian Williams (Feb 26 2023 at 20:45):

what is $\ell$?

Christian Williams (Feb 26 2023 at 20:50):

if you're assuming that $r$ is a right adjoint, then I understand. but it looked like you said it was an arbitrary morphism.

Nathanael Arkor (Feb 26 2023 at 21:20):

That's true. I confused myself by calling the 1-cell $r$. I need to think about whether the proof can be adapted to the case when $r$ is an arbitrary 1-cell.