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Stream: learning: questions

Topic: Categories where composition behaves like Those in $$Rel$$

Jack Jia (Jan 16 2026 at 23:34):

I have some thoughts on this that are very hard to state precisely so my apologies if this doesn't make sense.

The composition in the category $Rel$ (sets with relations), $Mat$ (Natural numbers representing dimensions with linear maps) and $Prof$ (Categories with profunctors- I don't know much about co/ends, but I wanted to mention this) all behave alike, as some kind of "Summing (by "sum", I mean $\vee$, $\sum$, and $\int$, respectively) over the middle index".

Now another category where composition behave like this would be the Fukaya category of some symplectic manifold, where the convolution product is give by this type of formula.

I was told that with the homological mirror symmetry in mind, this could be some kind of geometric incarnation of the fact that we have a bi(2?)-category $Ring$ (Rings, bimodules and bimodule homs) on the algebraic side, how can I make sense of this?

Also, it doesn't really parse for me to think of matrices as "generalized maps between numbers", but it makes sense to think of bimodules as generalized ring maps, or relations as generalized set maps. What is the correct way to think about this? And in general, how can I produce the notion of "generalized maps"?

Kevin Carlson (Jan 17 2026 at 00:24):

I'd have to look into your mirror symmetry question, but regarding the last paragraph, matrices are generalized maps between finite sets, just as relations are. Specifically, a relation $A\to B$ generalizes a function in that each element of $A$ may be mapped to no element, or several elements, of $B$; similarly, an $A\times B$-indexed matrix sends each element of $A$ to a linear combination of elements of $B.$

Peva Blanchard (Jan 17 2026 at 08:22):

Continuing on @Kevin Carlson's comment, there is the following analogy

• a relation $A \not\to B$ is like a function $A \to 2^B$
• a matrix $A \not\to B$ is a like a function $A \to \mathbb{R}^B$
• a profunctor $A \not\to B$ is like a functor $A \to \text{Set}^{B^{op}}$

If I am not mistaken, the first two are examples of Kleisli maps, respectively for the powerset monad, and the monad associated with the free-forgetful adjunction between finite sets and (finite dim) vector spaces. The last one seems similar, but goes a level higher, and would probably need 2-categorical treatment.

John Baez (Jan 17 2026 at 08:37):

I would change "is like" to "is", where I'm using "is" in the category theorist's sense.

David Corfield (Jan 17 2026 at 12:36):

There's a move in the symplectic world (and elsewhere) to look beyond functions (here, symplectomorphisms) to something more span-like (correspondences), as in the [[Weinstein symplectic category]]:

image.png

Then some [[span]]-like composition follows, like your summing over the middle index:

image.png

Rémy Tuyéras (Jan 17 2026 at 12:58):

Peva Blanchard said:

Continuing on Kevin Carlson's comment, there is the following analogy

• a relation $A \not\to B$ is like a function $A \to 2^B$
• a matrix $A \not\to B$ is a like a function $A \to \mathbb{R}^B$
• a profunctor $A \not\to B$ is like a functor $A \to \text{Set}^{B^{op}}$

There is something beautiful about this presentation: it suggests thinking of elimination procedures (Gaussian elimination, in the matrix case) as manipulating maps in a way that resembles "homotopies of paths".

In homotopy theory one studies maps like $[0,1] \to X^{[0,1]}$. If I write this as $J \to X^{I}$ with $J = [0,1] = I$, the point is usually not to interpret elements of $J$ as being sent to elements of $I$ via some relational correspondence. Rather, $J$ indexes higher-order data: its elements parametrize "transitions" between two points of $I$.

With that lens, Gaussian elimination feels like passing through homotopies. Concretely, given an abstract morphism $f: A \to X^B$, one can compare $f(a)$ and $f(a')$. In linear algebra, this is row subtraction: replacing (say) the row for $a'$ by the difference $\delta = f(a') - f(a) \in X^B$. In homotopy theory, the analogous move is to compare two 1-paths via a 2-path between them.

The key difference is that in homotopy theory the "2-dimensional" structure is kept explicit, whereas elimination flattens that higher-order structure back down to level 1 in order to reduce the matrix.

Rémy Tuyéras (Jan 17 2026 at 13:12):

David Corfield said:

Then some [[span]]-like composition follows, like your summing over the middle index:

And of course, the homotopy viewpoint leads us back to the span-like viewpoint.

Homotopies, once abstracted, give graph-like structures, which are nothing more than spans over the same objects. A span such as $X^B \rightrightarrows A$ is precisely the kind of object that records sources and targets

Jack Jia (Jan 17 2026 at 20:51):

Thanks a lot! All of these responses are really insightful.

Reading through no post again and I realized how poorly it was written. Let me rephrase the algebra vs geometry part: Forget about homological mirror symmetry, the "algebra is dual to geometry" philosophy has many incarnations, e.g., Boolean rings vs Stone spaces, unital commutative C*-algebras vs compact Hausdorff spaces, or finitely generated reduced k-algebras vs affine k-varieties.

Now naively algebraic things behave like sets and functions. So while we think about generalizing the notion of functions, how does this manifest on the geometry side? One such example is the convolution product in Boore-Moore homology. Chriss&Ginzberg's "Representation Theory and Complex Geometry", section 2.7 explains precisely how the convolution product is a homology version of how things in $Rel$ compose. Now this story is more on the geometry side rather than the alegrba side, so I want to say something like in the dual case this is like the 2-category $Ring$, but this is very imprecise and just a vague idea.

Also I should mention I was totally unaware of this kind of viewpoint of generalized maps until reading through some recent very interesting discussions of profunctors and linear algebra in the Zulip channel, which connected some dots for me, since @John Baez described the 2-category $Alg$ around a year ago, thanks!