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Stream: learning: questions

Topic: Structured cospans

Vinay Madhusudanan (Mar 29 2020 at 16:33):

Very basic question. According to the definition here: http://math.ucr.edu/home/baez/structured/,
given a functor $L: A \to X$, a structured cospan is a diagram in $X$ of the form $L(a) \rightarrow x \leftarrow L(b)$.

At first this looks like a co-cone with apex $x$, but obviously it's not, because $L$ is a functor from $A$, not from a two-element discrete (shape) category (except when $A$ itself is that). Nevertheless, is there a way of seeing it as a co-cone?

Vinay Madhusudanan (Mar 29 2020 at 16:36):

(I felt this would be out of place in the post-talk discussion, since it's quite basic and probably moves away from the actual focus of the talk, so I'm asking it now)

Thomas Read (Mar 29 2020 at 16:54):

If $\mathbf{2}$ is the discrete category with 2 elements, and $F : \mathbf{2} \to A$ is the diagram that picks out $a$ and $b$ then a cocone under $L \circ F$ is a diagram of the form $L(a) \to x \leftarrow L(b)$.

I'm not sure if that really answers your question though.

Vinay Madhusudanan (Mar 29 2020 at 16:56):

Ah, that's perfect. I thought a composition might work, but I kept thinking (even now after seeing your answer) that $L$ would be somehow lost in the process

Vinay Madhusudanan (Mar 29 2020 at 16:57):

Not $L$ exactly, but the information from $A$. Hm, I see it now. Thank you!

John Baez (Mar 29 2020 at 20:50):

There are different things to say about this but here's one: given any functor $L \colon \mathsf{A} \to \mathsf{X}$ there's a category of structured cospans, i.e. diagrams $L(a) \to x \leftarrow L(b)$ and the sort of obvious morphisms between such diagrams. Moreover this category is a comma category. Daniel Cicala showed this category under some conditions is a topos:

• Rewriting strutured cospans, Section 3: structured cospans form a topos.

Notice this is a category where structured cospans are objects, whereas I'm more interested in treating them as morphisms. But both viewpoints are reconciled in the double category I'll discuss in my talk!

David Egolf (Mar 30 2020 at 00:51):

This is another question about structured cospans, so I thought it might fit here. Let me know if it should be moved to a new topic.
(Disclaimer: I'm very new to category theory...any corrections/insights are much appreciated)

If I understand correctly, a structured cospan $L(a) \to x \leftarrow L(b)$ can be thought of as some object $x$ together with "allowed connections" or "connection interfaces" given by $i: L(a) \to x$ and $o: L(b) \to x$. We can then stick these objects together along their interfaces (provided they have compatible interfaces) by working in a category where these cospans are morphisms and composition corresponds to taking a pushout.

In applications, some objects have more than two allowed connections. For example, consider an electrical circuit that has the following three connections: input, output, and reset. Does this structured cospan approach support modelling of these kinds of networks?

(My initial guess is "yes, but it gets more complicated": I guess the new morphisms would consist of a circuit object $x$ together with three maps $L(\textrm{input interface}) \to x$, $L(\textrm{output interface}) \to x$ and, $L(\textrm{reset interface}) \to x$. Maybe composition would work in the same way (using a pushout with respect to the connecting interface), although we do have the additional wrinkle that the number of interfaces a circuit has can change upon being composed with another circuit. For example, if circuit A has input, output and reset interfaces, and circuit B also has input, output and reset interfaces, then connecting the output of circuit A to the input of circuit B yields a new circuit with four interfaces: input_A, reset_A, output_B, reset_B. So, I guess if we want to support flexible connections between circuits with three interfaces, we need to be able to support connection between circuits with any finite number of interfaces (?).)

John Baez (Mar 30 2020 at 00:53):

This is a good place for talking about structured cospans, I'd say!

John Baez (Mar 30 2020 at 00:58):

If I understand correctly, a structured cospan $L(a) \to x \leftarrow L(b)$ can be thought of as some object $x$ together with "allowed connections" or "connection interfaces" given by $i: L(a) \to x$ and $o: L(b) \to x$. We can then stick these objects together along their interfaces (provided they have compatible interfaces) by working in a category where these cospans are morphisms and composition corresponds to taking a pushout.

That's exactly right.

In applications, some objects have more than two allowed connections. For example, consider an electrical circuit that has the following three connections: input, output, and reset. Does this structured cospan approach support modelling of these kinds of network?

Yes! And that's crucial, because for almost any kind of network that comes up in practice we need to allow objects with more than two connections.

We handle this using the fact that when $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ is a left adjoint (the case worth keeping in mind), the structured cospan category ${}_L \mathsf{Csp}(\mathsf{X})$ is a monoidal category. This lets talk about morphisms from a finite set of objects to a finite set of objects.

Furthermore, this category is compact closed. This lets us change our mind and reinterpret some inputs as outputs or vice versa if we feel like it, and also permute the order of inputs or outputs.

Even better this category is a hypergraph category. This lets us do even more fun things that we often want to do with networks!

All this is proved in the paper Structured cospans.

David Egolf (Mar 30 2020 at 01:12):

Thanks! (I'll have to do some reading to more fully understand.)

John Baez (Mar 30 2020 at 01:13):

All these concepts (which I just added links to, so now they're in blue) are really important for understanding categories where the morphisms are networks. Brendan Fong's thesis is a good introduction to all of them, I think - he defines them all and explains why they're important.

David Michael Roberts (Mar 30 2020 at 01:33):

That's two talks on 1 April I need to watch, then (the other is Kevin Buzzard's, advertised in the Lean and HoTT Zulip chat rooms/spaces/etc). All this ability to watch seminars from around the world is taking up so much time! ;-)

John Baez (Mar 30 2020 at 01:57):

If I succeed in recording my talk and putting it on YouTube, I'll record my talk and put it on YouTube.