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Stream: learning: questions

Topic: Enriched categories and "monoidal coproducts"?

Paolo Perrone (Jun 21 2021 at 09:07):

In enriched category theory, composition of arrows is not always given by the cartesian product of the hom-objects, sometimes one enriches in a monoidal category instead.
A nice example is given by Lawvere metric spaces, where the enriching category is $[0,\infty]$ ordered downwards, with the sum as the (monoidal) product.
An equivalent example, that can be obtained by applying the function $e^{-x}$ to $[0,\infty]$, is the unit interval $[0,1]$, ordered upwards, with multiplication as operation. (This has been used, for example, in a recent preprint by Tai-Danae Bradley et al.)
In this last example, and in the preprint I referred to, the objects of $[0,1]$ can be interpreted as probabilities, and the composition of arrows gives the product of probabilities, a bit like in a (discrete-state) Markov chain, which is very neat.

Paolo Perrone (Jun 21 2021 at 09:11):

Now consider a finite, weighted, directed graph as follows:

• Vertices are seen as possible "states";
• Weights are contained in $[0,1]$, and for each vertex, the sum of the weights of its outcoming edges is $1$;
• So, in particular, the adjacency matrix is a stochastic matrix (and every stochastic matrix can be represented as such a graph).
  As above, interpret the weights as probabilties: an edge of weight $1/3$ from $x$ to $y$ means a probability of $1/3$ to "transition" from $x$ to $y$, and so on.

Paolo Perrone (Jun 21 2021 at 09:17):

Suppose that we have two vertices $x$ and $y$, which we view as "initial and final state", and that we want to see not quite what's the probability from $x$ to $y$, but rather, what is the probability of getting from $x$ to $y$ in two steps.
For example, suppose there are two "intermediate" states $a$ and $b$, and arrows as follows:

• $p(x\to a) = 1/2$
• $p(x\to b) = 1/3$
• $p(a\to y) = 1/2$
• $p(b\to y) = 2/3$
  (These are not all the arrows of our graph, but suppose that these are the only ones with arrows from $x$ and to $y$.)
  We would like to say that the probability of going from $x$ to $y$ in two steps is

$$p(x\to y) = p(x\to a) \cdot p(a\to y) + p(x\to b) \cdot p(b\to y)$$

$$= 1/2 \cdot 1/2 + 1/3 \cdot 2/3 = 1/4 + 1/2 = 3/4,$$

since we can choose two possible routes: through $a$ and through $b$.

Paolo Perrone (Jun 21 2021 at 09:22):

However: this is not how we usually work in enriched category theory! In particular, if we were to compose arrows in enriched category theory (for example, in the free enriched category theory generated by the graph, as given in Seven Sketches, section 2.5), we would have to use the maximum instead of the sum.
(The interpretation being, we only care about the "possibility" of going from $x$ to $y$, or about the maximal probability.)
Categorically, one would have to take the colimit of the two numbers, and in a quantale, such as $[0,1]$, the colimit commutes with the product, so that we can indeed do a form of matrix multiplication (see again Seven Sketches above, section 2.5.3).

Paolo Perrone (Jun 21 2021 at 09:25):

So now here comes the question: is there a categorical structure that would give us the real sum here?
Has anybody tried to enrich categories not quite in a distributive monoidal category, where the tensor product commutes with colimits, but rather, in a rig or bimonoidal category, where we have two monoidal structures interacting? (Like sum and product?)

Nathanael Arkor (Jun 21 2021 at 09:42):

This looks like a coend formula. We have $\mathbf C(X, Z) \cong \int^Y \mathbf C(X, Y) \otimes \mathbf C(Y, Z)$, where in your case $\otimes$ is multiplication. The coend is equivalently given by the quotient of a coproduct $(\coprod_Y \mathbf C(X, Y) \otimes \mathbf C(Y, Z))/{\approx}$ by a certain equivalence relation $\approx$, which in your example is trivial, and so gives your formula for $p(x \to y)$ (assuming coproducts in the enriching category are given by sums).

Paolo Perrone (Jun 21 2021 at 09:56):

Yep, a coend is a nice way to put it.
That's exactly the problem: coproducts are not given by sums in that category. They are "just another monoidal structure", that happens to have a nice distributive law with the (monoidal) products.

Nathanael Arkor (Jun 21 2021 at 10:22):

Interesting. It's surprising that the coend formula works so well, yet turns out not actually to be appropriate here.

Paolo Perrone (Jun 21 2021 at 10:34):

Exactly! I feel that there must be something going on here. (A generalization lurking in the background?)

James Wood (Jun 21 2021 at 10:57):

How ⅋ distributes over & (the Cartesian product) in a (Cartesian) *-autonomous category might be a decent starting point.

Paolo Perrone (Jun 21 2021 at 11:32):

I feel like this list could help. (Star-autonomous categories give linearly distributive categories, which are further down the list.)
In particular, rig categories seem to be the thing here. Has anybody tried to enrich over them?

Nathanael Arkor (Jun 21 2021 at 11:33):

You can enrich over a rig category by just viewing it as a monoidal category.

Paolo Perrone (Jun 21 2021 at 11:35):

Nathanael Arkor said:

You can enrich over a rig category by just viewing it as a monoidal category.

Yep, however in that case one still uses the colimits of that category when forming composites as in my example above. (Or when forming the free category over a graph.)
Something in some definition still has to be relaxed, but so far I don't know exactly what.

Nathanael Arkor (Jun 21 2021 at 11:38):

There was some discussion of something similar in another thread a while ago (I think about horizontal categorification of rig categories), but I can't find it now.

Paolo Perrone (Jun 21 2021 at 11:39):

Nathanael Arkor said:

There was some discussion of something similar in another thread a while ago (I think about horizontal categorification of rig categories), but I can't find it now.

Oh really? Interesting. Do you remember any other keyword we could search?

Nathanael Arkor (Jun 21 2021 at 11:42):

I've tried searching the keywords that I think might bring it up, but with no success :oh_no:

Paolo Perrone (Jun 21 2021 at 11:43):

Nathanael Arkor said:

I've tried searching the keywords that I think might bring it up, but with no success :oh_no:

Could it be either this thread or this one?

Nathanael Arkor (Jun 21 2021 at 11:43):

Ah, I think the first thread might have been the one I was thinking of, yes!

Nathanael Arkor (Jun 21 2021 at 11:44):

(I don't know how I didn't find that, considering "enriched" and "bimonoidal" appear in the first post!)

Paolo Perrone (Jun 21 2021 at 12:11):

Wow, it sounds like this post, by @Reid Barton, is exactly what I need! (Edit: almost, but not quite, see below.)
I wonder if this idea has had any followups? Especially, are there more examples of this structure?

Nathanael Arkor (Jun 21 2021 at 12:17):

I don't think there were any further conversations on this topic.

Paolo Perrone (Jun 21 2021 at 12:26):

On second thought, this post is not quite what I need. That seems to be more like a 2-category enriched in the 2-category of bimonoidal categories, I need something a bit less sophisticated. Namely, hom-objects have to be the objects of a bimonoidal category, they don't have to be bimonoidal categories themselves.

Paolo Perrone (Jun 21 2021 at 14:01):

By the way, what are nontrivial examples of bimonoidal categories?
(By which I mean, not just rigs, and not just monoidal categories where the sum is given by coproducts.)

Todd Trimble (Jun 21 2021 at 14:20):

Paolo Perrone said:

By the way, what are nontrivial examples of bimonoidal categories?
(By which I mean, not just rigs, and not just monoidal categories where the sum is given by coproducts.)

I answered at the nForum, but maybe I can answer here too: does it work to take a symmetric monoidal category, and look at strong symmetric monoidal endofunctors on that?

Paolo Perrone (Jun 21 2021 at 15:20):

Todd Trimble said:

Paolo Perrone said:

By the way, what are nontrivial examples of bimonoidal categories?
(By which I mean, not just rigs, and not just monoidal categories where the sum is given by coproducts.)

I answered at the nForum, but maybe I can answer here too: does it work to take a symmetric monoidal category, and look at strong symmetric monoidal endofunctors on that?

It seems so! I'm checking the coherence conditions.

Paolo Perrone (Jun 21 2021 at 16:47):

Yep, that works! Very nice!