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: theory: category theory

Topic: What is a functor equipped with this thing?

Asad Saeeduddin (Apr 07 2020 at 11:11):

Does this sort of thing look like an anything to anyone:

EDIT: I transcribed this wrong originally, leaving the original incorrect formula below so the conversation still makes sense:

$? : F A \otimes (F B \bullet F C) \otimes F D \to F ((A \otimes B) \bullet (C \otimes D))$

Correct definition:

$? : F A \otimes F (B \bullet C) \otimes F D \to F ((A \otimes B) \bullet (C \otimes D))$

Where $F : \mathcal{C} \to \mathcal{C}$ is an endofunctor, $\otimes : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a symmetric monoidal structure on $\mathcal{C}$ (with a unit I don't care about right now) and so is $\bullet$.

Note that the symmetry of the monoidal structures means all I really need is the slightly simpler (but more asymmetric):

Incorrect original expression:

$?_l : F A \otimes (F B \bullet F C) \to F ((A \otimes B) \bullet C)$

Correct expression:

$?_l : F A \otimes F (B \bullet C) \to F ((A \otimes B) \bullet C)$

which looks kind of strength-y. we can use the symmetries of the tensors to obtain a flipped around $?_r$ version of this, and use the two in concert to recover $?$.

Asad Saeeduddin (Apr 07 2020 at 11:12):

Sorry if this is a question more suited to the "basic questions" channel, I can move it there if more appropriate.

Asad Saeeduddin (Apr 07 2020 at 11:13):

I was trying to imagine this as maybe a laxity or a strength or something for the composition of $F$ inside $\bullet$, but I didn't get anywhere

Matteo Capucci (he/him) (Apr 07 2020 at 11:26):

That looks a lot like a distributivity relation. Idk in that context if there's a better way to see it.

Asad Saeeduddin (Apr 07 2020 at 11:27):

It's strange. It's almost like $F$ is laxly preserving the "tritensor" $- \otimes (- \bullet -) : \mathcal{C} \times \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ instead of the normal tensor of a monoidal structure (which in my limited knowledge is always a bifunctor).

Matteo Capucci (he/him) (Apr 07 2020 at 11:27):

Are $\otimes$ and $\bullet$ related in any way?

Asad Saeeduddin (Apr 07 2020 at 11:28):

like, normally a laxity for an endofunctor goes $F A \otimes F B \to F (A \otimes B)$. It looks like $?_l$ is just doing that for a three argument "tensor" instead

Matteo Capucci (he/him) (Apr 07 2020 at 11:29):

Yeah, that's true

Asad Saeeduddin (Apr 07 2020 at 11:29):

@Matteo Capucci They're loosely related in the very specific situation where I found this (they're product and coproduct respectively) but I'd ideally not want that to bias or prematurely disqualify any candidate concepts

Asad Saeeduddin (Apr 07 2020 at 11:33):

Actually if we want to look at it as a laxity-like thing we need to look at different "tritensors", I didn't notice they're associated differently in $?_l$. So it'd be from $(- \otimes -) \bullet -$ on the domain to $- \otimes (- \bullet -)$ on the codomain.

Matteo Capucci (he/him) (Apr 07 2020 at 11:35):

Maybe the word 'operad' shuld be pronounced, because of the arities

Matteo Capucci (he/him) (Apr 07 2020 at 11:36):

Though I barely know what an operad is so don't take my word for it

Asad Saeeduddin (Apr 07 2020 at 11:37):

Unfortunately I only have the vaguest sense of what an operad is based on some folks' attempts to explain it to me, so if there's some insight in that term I can't access it

Asad Saeeduddin (Apr 07 2020 at 11:43):

Does this maybe have something to do with preserving a "duoidal structure"?

Asad Saeeduddin (Apr 07 2020 at 11:56):

Agh, sorry. I just realized I transcribed it wrong. It's:

$? : F A \otimes F (B \bullet C) \otimes F D \to F ((A \otimes B) \bullet (C \otimes D))$

$?_l : F A \otimes F (B \bullet C) \to F ((A \otimes B) \bullet C)$

$?_r : F (A \bullet B) \otimes F C \to F (A \bullet (B \otimes C))$

Please ignore everything I said above.

Todd Schmid (Apr 07 2020 at 18:16):

It looks to me like a form of medial from deep inference proof theory.
https://www.lix.polytechnique.fr/~lutz/papers/CharMedial.pdf
Is this useful?

Asad Saeeduddin (Apr 07 2020 at 20:48):

@Todd Schmid I'll go through the paper and let you know. Thanks for the pointer!

Asad Saeeduddin (Apr 08 2020 at 10:03):

for the specific situation where $\bullet$ is the coproduct, we can obtain a morphism:

$?_l : F A \otimes F (B \bullet C) \to F ((A \otimes B) \bullet C)$

given a morphism:

$?'_l : F (A \bullet C) \otimes F (B \bullet C) \to F ((A \otimes B) \bullet C)$

By precomposing a mapped injection $F \ i_l \otimes \mathbf{1}$.

The result is a little bit easier to reason about: its existence shows that the functor $F (- \bullet C)$ laxly preserves the $\otimes$ tensor.