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Stream: learning: questions

Topic: Enrichment relations categorically

Asad Saeeduddin (Sep 15 2020 at 04:07):

Is there some way in which the relationship of enrichment forms some kind of category like structure? Like, where self-enrichment is an identity and if C is enriched in D, and D is enriched in E, the structure of E still somehow transitively "shows up" in C

Asad Saeeduddin (Sep 15 2020 at 04:12):

the transitivity doesn't quite seem to work out, but this prompts the question of whether there is some sense in which it is possible to "dissolve" intermediate enrichments and consider a category C that is enriched in D which is in turn enriched in E to merely be enriched in E

Asad Saeeduddin (Sep 15 2020 at 04:20):

Identities don't seem to work either, really. I guess this is a nonsense

John Baez (Sep 15 2020 at 04:26):

It sounds like you might enjoy understanding "change of base". A category can only be enriched in a monoidal category V. If you have a monoidal functor f: V $\to$ W between monoidal categories, you can take any category enriched in V and turn it into a category enriched in W. This is called change of base. It gives a 2-functor from VCat (the 2-category of categories enriched in V) to WCat.

John Baez (Sep 15 2020 at 04:28):

For example there's a monoidal functor f: Set $\to$ Vect sending any set to the vector space having that set as basis. Change of base using this turns categories into linear categories (that is, categories enriched in Vect).

Asad Saeeduddin (Sep 15 2020 at 04:28):

that sounds really interesting, thanks!

John Baez (Sep 15 2020 at 04:29):

I think there's a lot about change of base in Kelly's book.

Asad Saeeduddin (Sep 15 2020 at 04:30):

this is probably a question for a different thread, but I was wondering if there is such a thing as a "free pseudofunctor to the bicategory of pseudomonoids" for any monoidal bicategory. if so, since Cat is a monoidal bicategory, does the free pseudofunctor to the bicategory of monoids of Cat give us on 1-cells (functors) the corresponding "free monoidal functor" in some sense?

Asad Saeeduddin (Sep 15 2020 at 04:30):

this is only tangentially related to what you just explained, i guess i'll ask it in a different thread

Asad Saeeduddin (Sep 15 2020 at 04:31):

more strictly related to what you just explained, by monoidal functor do you mean a strong/strict monoidal functor?

John Baez (Sep 15 2020 at 04:31):

John Baez said:

I think there's a lot about change of base in Kelly's book.

No, there's not: in the introduction he says he doesn't discuss this at all!

John Baez (Sep 15 2020 at 04:32):

Asad Saeeduddin said:

more strictly related to what you just explained, by monoidal functor do you mean a strong/strict monoidal functor?

Strong or even lax (which are more general). Only an evil clown would say "monoidal functor" to mean "strict monoidal functor", because most monoidal functors aren't strict.

John Baez (Sep 15 2020 at 04:33):

Most people use "monoidal functor" to mean strong, and that's what I do, but I just remembered that base change works fine for lax monoidal functors.

Asad Saeeduddin (Sep 15 2020 at 04:34):

i see, so there is actually no difference in the end product if you only have a lax monoidal functor?

Asad Saeeduddin (Sep 15 2020 at 04:34):

that's somewhat surprising

Asad Saeeduddin (Sep 15 2020 at 04:34):

or do you somehow manage to transport less stuff given the weaker relationship?

John Baez (Sep 15 2020 at 04:39):

Asad Saeeduddin said:

this is probably a question for a different thread, but I was wondering if there is such a thing as a "free pseudofunctor to the bicategory of pseudomonoids" for any monoidal bicategory. if so, since Cat is a monoidal bicategory, does the free pseudofunctor to the bicategory of monoids of Cat give us on 1-cells (functors) the corresponding "free monoidal functor" in some sense?

I'm not sure I understand this. Let's see. In any monoidal bicategory C we can define pseudomonoids, and we get a bicategory (not monoidal!) of pseudomonoids in C - call it Ps(C).

There's always a forgetful functor U: Ps(C) $\to$ C since any pseudomonoid has an underlying object: it's an object together with some other stuff.

I imagine you're asking if U has a left adjoint F: C $\to$ Ps(C) sending any object to the free pseudomonoid on that object.

This clearly won't work unless C has enough 2-colimits!

It's probably best to start one level down: in any monoidal category C we can define monoids, and we get a category Mon(C) and a forgetful functor U: Mon(C) $\to$ C, and we can ask if this has a left adjoint F: C $\to$ Mon(C).

If C = Set with its cartesian monoidal structure, the answer is "yes", and we have

$F(S) = 1 + S + S^2 + S^3 \cdots$

So here we are taking a coproduct. In the category of finite sets this wouldn't work!

John Baez (Sep 15 2020 at 04:41):

Asad Saeeduddin said:

i see, so there is actually no difference in the end product if you only have a lax monoidal functor?

I don't know what that means. The base change you get depends on the lax monoidal functor you use, so clearly the "end product" will be different if I use some strong monoidal functor and you use some lax one.

But if you just look at what base change does, you'll see a lax monoidal functor is all you need.

Asad Saeeduddin (Sep 15 2020 at 04:42):

@John Baez Is this the appropriate definition of base change: https://ncatlab.org/nlab/show/base+change?

Asad Saeeduddin (Sep 15 2020 at 04:42):

I can't recognize anything about monoidal categories or functors in there, I guess I'd need to appropriately substitute some definitions

Asad Saeeduddin (Sep 15 2020 at 04:43):

ah it's this: https://ncatlab.org/nlab/show/change+of+enriching+category

John Baez (Sep 15 2020 at 04:44):

Yes, that sounds better. The first page is about a completely unrelated thing called "base change". There are a number of things in math called "base change".

Asad Saeeduddin (Sep 15 2020 at 04:44):

now the thing I'm wondering is, are V1-Cat and V2-Cat as referenced in that page referring to strict 2-categories?

John Baez (Sep 15 2020 at 04:45):

Yes, the 2-category of V-enriched categories is strict since V-functors compose associatively - they work pretty much just like ordinary functors. The only difference is that V-categories have "hom-objects" which are objects in V instead of "hom-sets".

Asad Saeeduddin (Sep 15 2020 at 04:50):

I see, ok. Does anything change if we have further structure in the category that we're enriching in? i.e. we don't have monoidal _categories_ that we're enriching in, but instead monoidal _bicategories_, and presumably we want a change of base _pseudofunctor_ instead of a 2-functor. Does something like that still work? If so, do we perhaps end up needing a strong monoidal functor instead of a lax one?

Asad Saeeduddin (Sep 15 2020 at 04:51):

I'll try and witness some of this in Agda, maybe the answer will fall out after I play with it a bit. In any case, thanks for the pointer!

Asad Saeeduddin (Sep 15 2020 at 04:53):

If so, do we perhaps end up needing a strong monoidal functor instead of a lax one?

actually this question is malformed. what i meant to ask is whether we end up needing equivalences where the laxators for a lax monoidal functor go

John Baez (Sep 15 2020 at 04:53):

i.e. we don't have monoidal _categories_ that we're enriching in, but instead monoidal _bicategories_, and presumably we want a change of base _pseudofunctor_ instead of a 2-functor.

Yes, that must work too, though I don't know if anyone has written it up yet. Someone should have.... but there's so much of this stuff to do, some things remain undone.

John Baez (Sep 15 2020 at 04:54):

If so, do we perhaps end up needing a strong monoidal functor instead of a lax one?

You later changed your mind about this, but anyway: a lax monoidal pseudofunctor will certainly suffice for this 2-categorical version of base change.

John Baez (Sep 15 2020 at 04:55):

It's best to really dig into how base change works - it's not very complicated - and then you can guess how all this stuff is gonna go.

Asad Saeeduddin (Sep 15 2020 at 04:55):

will do. thanks again

John Baez (Sep 15 2020 at 04:55):

Sure!

fosco (Sep 15 2020 at 10:31):

John Baez said:

John Baez said:

I think there's a lot about change of base in Kelly's book.

No, there's not: in the introduction he says he doesn't discuss this at all!

RE change of base: look here instead http://www.tac.mta.ca/tac/volumes/10/10/10-10.pdf