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Stream: learning: questions

Topic: Kleisli charts(/lenses)?

Harrison Grodin (Apr 28 2026 at 17:34):

A chart from $(A^-, A^+)$ to $(B^-, B^+)$ is typically represented as a pair of functions $f : A^+ \to B^+$ and $f_\flat : A^+ \times A^- \to B^-$. Equivalently, using exponentials, it's a single function $(f, f_\flat) : A^+ \to (B^+ \times (A^- \Rightarrow B^-))$.

I've seen @David Jaz Myers (and likely others) generalize charts such that the backwards map is "effectful" using some monad $M$, i.e. $f_\flat : A^+ \times A^- \to M(B^-)$. I've recently encountered a situation, though, that instead looks like a combined map $A^+ \to M(B^+ \times (A^- \Rightarrow M(B^-)))$, where the "forward part" is also effectful, and the "reverse part" happens under that same monad. (This map can't literally be broken down using the universal property of the product, since $M$ need not preserve products or anything.) For a concrete example, the $M$ I have in mind is the finitely-supported distributions (ie, free convex set) monad.

Has anyone seen this structure before? It feels like a sort of "Kleisli chart"...

Martti Karvonen (Apr 28 2026 at 18:20):

I know very little about lenses generally speaking, but can't help but chime in as I'm interested in answers as well. This is because I'm quite familiar with what I suspect to be an instance of this, and in my case the monad is also given by finitely supported distributions (possibly over other semirings too). Namely, in the sheaf-theoretic approach to contextuality the relevant maps of scenarios have both a forward and backward compoinent, making them more or less lenses, and in general we want both of them to be stochastic. See e.g. here where we define the relevant maps by first defining the deterministic maps and then applying the distribution functor to the old homsets (i.e., we freely encrich the starting category in convex sets).

Jules Hedges (Apr 28 2026 at 21:30):

You can do this for any symmetric monoidal category: a morphism $(A^-, A+) \to (B^-, B^+)$ is an element of the coend $\int^M \mathcal C (A^+, M \otimes B^+) \times \mathcal C (M \otimes A^-, B^-)$, aka an equivalence class of triples. (If $\otimes$ is a cartesian product this is equivalent to the basic definition of charts). This is a standard way to generalise lenses (the reference for that is Riley, Categories of optics) and it can be usefully generalised more than that too (see Profunctor optics: A categorical update). I never saw anybody write down that it works for charts too, but it's foklore for sure

Jules Hedges (Apr 28 2026 at 21:33):

This is almost but not exactly equivalent to the direct definition you wrote (I was confused about this point for years). For lenses I went into excruciating detail about it in this blog post
https://julesh.com/posts/2024-04-12-enriched-closed-lenses.html
which will definitely go exactly the same for charts

Jules Hedges (Apr 28 2026 at 21:49):

(I just discovered a formula in that post that didn't survive the migation of my website, I'll fix that tomorrow)

Harrison Grodin (Apr 28 2026 at 21:53):

Ohh right, this is exactly what I was looking for, thank you!! I now remember reading this post and finding it very lovely a few years ago, but couldn't quite recall where I'd seen this discussion. :) Many thanks!

Bruno Gavranović (Apr 28 2026 at 23:27):

In addition to what was said already, perhaps its useful to point out another "extreme" direction you can take this to. Using lens terminology, you can distinguish between

• the forward category $\mathcal{C}$ together with an action of some monidal category $\mathcal{M}$ on it
• the backwards category ${D}$, and the action of some monoidal category $\mathcal{M'}$ on it
• the "connecting tissue" of a lax monoidal functor $\mathcal{M} \times \mathcal{M'} \to \mathbf{Set}$ that mediates the state between the forward and the backward pass.

Of course, all of this works for charts too, which is why I'm mentioning it. While not directly stated at the level of the charts, I unpacked all of this in my thesis (Section 4.3 Weighted Optics), and the trick of flipping the element of the coend that Jules mentioned should apply here too, without a problem.

I found it useful in providing a unifying story of both probability-shaped processes, and also gradient-based ones in a way where you can get a handle on the type of process taking part in each specific component of the lens/chart. It sounds like this might be what you're after here

Jules Hedges (Apr 29 2026 at 10:22):

If anyone wants a small puzzle: In the case that $\mathcal C$ is cartesian, $\mathbf{Lens} (\mathcal C)$ and $\mathbf{Chart} (\mathcal C)$ are both fibred over $\mathcal C$ and are fibrewise opposites. In the monoidal case neither of them are fibred over $\mathcal C$, there should be some 2-categorical sense in which they are probably fibrewise opposite over $\mathbf{Copara} (\mathcal C)$, but this is much less obvious and afaik not written down anywhere

Jules Hedges (Apr 29 2026 at 10:24):

The main reason that's important is because you want them to be fibrewise opposites in order to form the monoidal version of Myers' double category of formal commutative squares where horizontal 1-cells are lenses and vertical 1-cells are charts