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Stream: learning: questions

Topic: Quiver in C and Spans

Eric Forgy (Dec 10 2020 at 16:58):

Hi :wave:

The nLab defines a directed graph as a functor

$G: X\to \text{Set},$

where $X$ is a category with two objects $X^0,X^1$ and two morphisms $s,t: X^1\to X^0.$

Then

$\text{DiGraph} = [X,\text{Set}]$

is the functor category of directed graphs.

It says this is naturally extended to a functor category $[X,C]$, which would be referred to as a "Quiver in $C$".

I'm trying to understand this for some common categories, e.g. "Quiver in $\text{Vect}$", "Quiver in $\text{Alg}$", "Quiver in $\text{BiMod}"$, "Quiver in $\text{DGAlg}$", etc.

Starting with $[X,\text{Vect}]$, let

$V: X\to\text{Vect}$

with $V^0 = V(X^0)$ and $V^1=V(X^1)$ and I'll reuse the labels $s$ and $t$ for source and target maps $s,t: V^1\to V^0.$

First question: Is $V^1 = V^0\otimes V^0$?

A "Quiver in $C$" looks like a special "Span in $C$". Is this a useful way to think of it?

Then this article from John looks interesting (but a bit over my head): Bimodules Versus Spans.

I'd love to understand how a "Quiver in $C$" might relate to bimodules.

John Baez (Dec 10 2020 at 17:24):

A quiver in C is the same as an "endospan" in C. A general span looks like

John Baez (Dec 10 2020 at 17:24):

$x \leftarrow a \rightarrow y$

John Baez (Dec 10 2020 at 17:25):

and we call this a span from $x$ to $y$ . So, by "endospan" I mean a span from some object to itself, like this:

John Baez (Dec 10 2020 at 17:25):

$x \leftarrow a \rightarrow x$

John Baez (Dec 10 2020 at 17:26):

But this is just a quiver in C.

John Baez (Dec 10 2020 at 17:27):

Btw, don't say "endospan" unless you're talking to someone who deeply understands category theory: it's not a standard term.

John Baez (Dec 10 2020 at 17:28):

Someone who really knows about this stuff would probably guess what you meant, but don't count on it.

John Baez (Dec 10 2020 at 17:28):

I'd love to understand how a "Quiver in C" might relate to bimodules.

John Baez (Dec 10 2020 at 17:31):

Well, in my article I explain how every object in $\mathsf{Set}^{op}$ is a commutative monoid in $\mathsf{Set}^{op}$. "Everyone" knows this, but I go a bit further:

John Baez (Dec 10 2020 at 17:32):

We can define bimodules not just for algebras (which are modules in Vect) but for monoids in any category... any category that can have monoids in it. And then a span of sets is the exact same thing as a bimodule in $\mathsf{Set}^{op}$.

John Baez (Dec 10 2020 at 17:34):

So, in brief: the concept of bimodule makes sense in a large class of categories, called "monoidal categories". And this generalization of bimodule has spans of sets as a special case.

John Baez (Dec 10 2020 at 17:35):

So in particular a quiver of sets (better known to category theorists as a "graph"), which is the same as an endospan of sets, is a special case of an endo-bimodule.

John Baez (Dec 10 2020 at 17:36):

Again, "endo-bimodule" is not a standard term, but I mean an $(A,B)$-bimodule where $A = B$.

John Baez (Dec 10 2020 at 17:36):

There's a lot more to say about all this, but I will spare you and not say anything more.

Eric Forgy (Dec 10 2020 at 17:38):

I'd love to hear more :heart: This is all fascinating stuff :blush:

Thank you :raised_hands:

John Baez (Dec 10 2020 at 17:39):

You think you'd love to hear more, but you said my article was "too advanced" so I think at some point you'll "hit the wall" - I don't want to overdose you.

John Baez (Dec 10 2020 at 17:40):

I'll just say a couple of things. A category is a quiver in Set with extra structure, namely "identity" arrows and the ability to compose arrows.

John Baez (Dec 10 2020 at 17:40):

So a category is a special sort of span in Set.

John Baez (Dec 10 2020 at 17:41):

And this viewpoint turns out to be very powerful: people in the know say "a category is the same as a monoid in Span(Set)".

John Baez (Dec 10 2020 at 17:41):

That's one thing: it takes work to learn this and adopt this viewpoint, but it's very powerful.

John Baez (Dec 10 2020 at 17:43):

Another thing: I've said how every span in Set can be seen as a bimodule in $\mathsf{Set}^{\mathrm{op}}$; one could ask to generalize this to other categories. It works for all "cartesian" categories - categories with finite products.

John Baez (Dec 10 2020 at 17:45):

One could also wonder if a plain old bimodule of algebras, i.e. a bimodule in $(\mathsf{Vect}, \otimes)$, can be seen as a span somewhere. I don't think so - but if someone knows how to do it, I'd love to hear it.

Eric Forgy (Dec 10 2020 at 17:49):

You think you'd love to hear more, but you said my article was "too advanced"

I see. By "spare" you meant literally "spare me the pain". Sometimes "spare" indicates a perceived lack of disinterest. That definitely isn't the case here, but yeah. I might not yet be able to appreciate. That has never stopped me before though :blush:

John Baez (Dec 10 2020 at 22:23):

I figure it's better to say a bit of stuff you understand than produce a massive wall of text. I actually like being understood - unlike some mathematicians. :upside_down:

Eric Forgy (Dec 10 2020 at 22:38):

I might be going off in the wrong direction, but I have a series of $n$ categories $\text{AlgObj}_n$ of algebraic objects having successively complex structures with a series of forgetful functors $U_n: \text{AlgObj}_n\to\text{AlgObj}_{n-1}$ terminating with

$U_1: \text{AlgObj}_1 \to \text{Set}.$

For each $U_n$, I'm assuming there is a left adjoint $F_n$

Then, I want to try to understand

$[X,\text{AlgObj}_n]$

by successively freely constructing

$X\overset{G}{\to}\text{Set}\overset{F_1}{\to}\text{AlgObj}_1\overset{F_2}{\to}\cdots\overset{F_n}{\to}\text{AlgObj}_n$

by incrementally adding structure.

Eric Forgy (Dec 10 2020 at 23:15):

For example, with

$X\overset{G}{\to}\text{Set}\overset{F_1}{\to}\text{Vect}$

we end up with two vector spaces $V^0, V^1$ and two linear maps $s,t:V^1\to V^0.$

I "think" $V^1 = V^0\otimes V^0$ so that $V^0$ is used to contruct $V^1$, but then we can continue with

$X\overset{G}{\to}\text{Set}\overset{F_1}{\to}\text{Vect}\overset{F_2}{\to}\text{GVect}$

of graded vector spaces where $V^n = \otimes^n V^0$.

Ok, but then I get confused because that starts feeling like a span, so maybe I should have started with

$X\overset{G}{\to}\text{Set}\overset{F_1}{\to}\text{Span(Vect)}$

instead :thinking:

"Like a record baby :dancer:"

Eric Forgy (Dec 10 2020 at 23:26):

At the end, I want to end up with something like graded Hilbert modules.