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: questions

Topic: Categories with multiple forms of composition?

Shea Levy (Nov 21 2020 at 13:49):

Are there studied cases where there there are multiple ways to compose arrows instead of just one? e.g. enriching over some bimonoidal category instead of just a monoidal one?

Fabrizio Genovese (Nov 21 2020 at 21:37):

What do you mean precisely? You could see a monoidal category as a category where you have two different ways of composing things. More generally, in a n-category you have n ways of composing n-cell (along 0-cells, 1-cells and so on) I guess you have a different intuition in mind?

Shea Levy (Nov 21 2020 at 21:40):

I mean two "vertical" compositions.

If you see monoidal categories as allowing vertical composition of zero-cells (as well as a horizontal composition of one-cells), then in a bimonoidal category you have two vertical compositions of zero-cells (with some relationship between them). Is there anything like this beyond the zero-cell level?

Fabrizio Genovese (Nov 21 2020 at 21:42):

The natural question for me would be "what is a 2-bimonoidal category?"

Fabrizio Genovese (Nov 21 2020 at 21:42):

It may end up doing this on 1-cells too

Fabrizio Genovese (Nov 21 2020 at 21:43):

So I guess this is similar to what you were asking in the beginning, indeed. :smile:

Matteo Capucci (he/him) (Nov 21 2020 at 21:47):

Double categories are sort of categories with two composition laws :thinking: but a category doubly enriched in a bimonoidal category seems more interesting, though unheard of afaik

Matteo Capucci (he/him) (Nov 21 2020 at 21:52):

An observation: if $(V, I, \otimes, O, \oplus)$ is a bimonoidal cat whose identity is lax monoidal, and $C$ is a cat with enrichment in both monoidal structures, then base change along the identity of $V$ transports one enrichment into the other.
For example you have this situation when $\otimes = \times$ and $\oplus = +$, because there's always a canonical morphism $A \times B \to A+B$ (EDIT: no, unless $V$ has biproducts)

Shea Levy (Nov 21 2020 at 21:52):

Hmm thinking through the enrichment in $Set$, we need an interesting composition for $Hom(A, B) + Hom(B, C)$?

Matteo Capucci (he/him) (Nov 21 2020 at 21:53):

Yeah, weird

Matteo Capucci (he/him) (Nov 21 2020 at 21:54):

Maybe $Ab$ with direct sum and tensor product is more interesting

Shea Levy (Nov 21 2020 at 21:55):

Shea Levy said:

Hmm thinking through the enrichment in $Set$, we need an interesting composition for $Hom(A, B) + Hom(B, C)$?

There's an element of $Hom(A + B, C)$ you could get...

Shea Levy (Nov 21 2020 at 21:55):

Wait no sorry

Shea Levy (Nov 21 2020 at 21:56):

I guess $A \times B \to B + C$ :shrug:

Matteo Capucci (he/him) (Nov 21 2020 at 21:57):

Matteo Capucci said:

Maybe $Ab$ with direct sum and tensor product is more interesting

Such a double enrichment on a category $C$ would amount to the structure of abelian group on every hom set $\mathbf C(A,B)$, a 'normal' composition $\mathbf C(A,B) \oplus \mathbf C(B, C) = \mathbf C(A,B) \times \mathbf C(B,C) \to \mathbf C(A,C)$ plus a bilinear composition $\mathbf C(A,B) \otimes \mathbf C(B,C) \to \mathbf C(A,C)$

Matteo Capucci (he/him) (Nov 21 2020 at 22:03):

Mmmh maybe rings are such categories? They are one object categories whose set of morphisms is their set of elements. Then $\oplus$-composition is given by $+$ and $\otimes$-composition is given by $\cdot$

Shea Levy (Nov 21 2020 at 22:22):

Is there some property that captures the sense in which $+$ in $Set$ is "too weak" while $\oplus$ isn't?

Shea Levy (Nov 21 2020 at 22:24):

Oh wait I think I confused myself between enriching over $Set$ vs treating $Set$ itself as one of these categories. There's nothing that prohibits $C(A, B) + C(B, C) \to C(A, C)$ in general if C is a set, only if they're function sets.

Matteo Capucci (he/him) (Nov 21 2020 at 22:37):

Exactly, though I can't come up with non-trivial examples of that.

Matteo Capucci (he/him) (Nov 21 2020 at 22:38):

Shea Levy said:

Is there some property that captures the sense in which $+$ in $Set$ is "too weak" while $\oplus$ isn't?

This is a good question, I don't know. Surely $+$ is a coproduct while $\oplus$ is (also) a product.

Nathanael Arkor (Nov 21 2020 at 23:46):

Matteo Capucci said:

Mmmh maybe rings are such categories? They are one object categories whose set of morphisms is their set of elements. Then $\oplus$-composition is given by $+$ and $\otimes$-composition is given by $\cdot$

Yes, Ab-enriched categories are called "ringoids" and are horizontal categorifications of rings (in the same sense categories may be considered "monoidoids", i.e. multi-object monoids).

Shea Levy (Nov 21 2020 at 23:50):

Nathanael Arkor said:

Matteo Capucci said:

Mmmh maybe rings are such categories? They are one object categories whose set of morphisms is their set of elements. Then $\oplus$-composition is given by $+$ and $\otimes$-composition is given by $\cdot$

Yes, Ab-enriched categories are called "ringoids" and are horizontal categorifications of rings (in the same sense categories may be considered "monoidoids", i.e. multi-object monoids).

But are ringoids usually considered to have two kinds of arrow composition? From n-lab it looks like we just use the bilinear composition

Nathanael Arkor (Nov 22 2020 at 00:01):

I'm not sure a ringoid is what you were asking for in your question. For that, you could take two categories $\mathbf C$ and $\mathbf D$ sharing object-classes and hom-classes, and then impose conditions on the two composition and unit operations of the categories.

Nathanael Arkor (Nov 22 2020 at 00:02):

But it may not be obvious how to generalise the original laws to the composition structure, because the usual equations for single-sorted structures may not make sense when the elements are now arrows (i.e. the equations may no longer type-check).

Nathanael Arkor (Nov 22 2020 at 00:06):

In particular, it only seems obvious how to generalise ordered equations to this setting (i.e. those in which each variable appears the same number of times, in the same order, on the LHS and RHS of an equation — so only the operations are permitted to change).

Reid Barton (Nov 22 2020 at 13:43):

Just speaking heuristically, "multiplication" operations (written $\times$ or $\otimes$) are likely to correspond to some kind of composition but "addition" operations (written $+$ or $\oplus$) are not. In a setting where it makes sense to "add" two 1-morphisms, they should probably be parallel 1-morphisms and the sum will again be a 1-morphism between the same pair of objects.

Reid Barton (Nov 22 2020 at 13:47):

For an example with an "enriched in bimonoidal" flavor, we could consider the 2-category of "rings and bimodules":

• an object is a ring $R$ (not necessarily commutative)
• for fixed $R$ and $S$, the Hom-category from $R$ to $S$ is the category of $(R, S)$-bimodules. So in particular it has a binary direct sum operation which it makes sense to denote by $\oplus$.
• the composition of an $(R,S)$-bimodule $M$ and an $(S,T)$-bimodule $N$ is the $(R,T)$-bimodule $M \otimes_S N$. Then, we have laws (technically more structure) like $M \otimes_S (N_1 \oplus N_2) \cong M \otimes_S N_1 \oplus M \otimes_S N_2$ and so on.

Reid Barton (Nov 22 2020 at 13:48):

If you look at the Hom-category from $\mathbb{Z}$ to itself, you see a copy of the bimonoidal category of abelian groups, with direct sum and tensor product. But in general $\oplus$ and $\otimes$ have different "types", like Nathanael was suggesting.

Reid Barton (Nov 22 2020 at 13:51):

If you have a particular situation which seems to be an example where two kinds of composition do make sense, then it's probably better to study this example and generalize from it, since there are a lot of potential categorical structures out there among which several could be the answer to "category with multiple kinds of composition".

Amar Hadzihasanovic (Nov 22 2020 at 14:35):

Cockett, Koslowski and @Robert Seely defined something called a linear bicategory, so that

linear bicategory : linearly distributive category = bicategory : monoidal category.

This has two different kinds of composition of 1-morphisms which in the single-object case correspond to the tensor and par of a linearly distributive category. Not sure if this has gone far since then, but they have a few examples in the first section.

Paolo Perrone (Jun 21 2021 at 12:14):

Interesting thread! I think I have found another example of category enriched in a bimonoidal category, in this thread.