The n-Category Café

If my understanding is correct, the notion of a convex linear combination of morphisms is a (well-motivated) “notation trick”. With a different choice of notation it might be less confusing.

To make sense of it in a very wordy way, note that there is a forgetful functor:

acting on objects by $F[(X,p)] = X$, and a morphism $f \colon (X, p) \to (Y, q)$ in $\mathbf{FinMeas}$ by forgetting the information about $p$ and $q$ and just remembering the function $X \to Y$, which we denote $F(f)$.

As a remark: remember that a morphism in FinMeas is simply a morphism in $\mathbf{FinSet}$ that satisfies some condition based on the extra data of the measures. Thus, the forgetful functor above roughly preserves data, but forgets information about condition (which wouldn’t make sense in FinSet anyway).

With that in mind, for any two morphisms $f \colon (X, p) \to (X', p')$ and $g \colon (Y,q) \to (Y', q')$ the convex linear combination: $\lambda f + (1 - \lambda) g$ for any $\lambda \in [0,1]$ is given by the data of its underlying function:

and it is guaranteed to satisfy the appropriate condition that makes it into a morphism from $(X \coprod Y, \lambda p + (1-\lambda) q)$ to $(X' \coprod Y', \lambda p' + (1-\lambda)q')$. Note that the value of $\lambda$ doesn’t tell us anything about the underlying functions in $\mathbf{FinSet}$, but tells us about info about what objects we are considering in $\mathbf{FinMeas}$.

That might clarify things a bit more.

Allan wrote:

It seems a little weird to be working with actual functions between finite sets, rather than matrices where each row is a probability measure.

The latter are called ‘stochastic maps’ and they don’t always decrease entropy. But measure-preserving functions — actual functions between finite sets — always decrease entropy. As I point out, the latter fact is an example of the ‘data processing inequality’. And my slides give this slogan for what’s going on:

Deterministic processing of random data always decreases entropy!

Allan wrote:

Is this what’s happening when you talk about “convex combinations of functions”? I couldn’t quite parse that slide.

Here’s what the slide says:

We can define convex linear combinations of objects in $FinProb$. For any $0 \le \lambda \le 1$, let

be the disjoint union of $X$ and $Y$, with the probability distribution given by $\lambda p$ on $X$ and $(1-\lambda)q$ on $Y$.

We can also define convex linear combinations of morphisms.

give

This is simply the function that equals $f$ on $X$ and $g$ on $Y$.

As I emphasized in my actual talk, the big scary-looking equation (3) is really no big deal: it’s just saying the source and target of a function between finite sets, and its name! And the function itself is very simple: it’s just the function that equals $f$ on the set $X$ and $g$ on the set $Y$.

The point is that taking convex linear combinations is functorial on $FinProb$: we can do it both to objects and morphisms, and the usual rules of a functor apply. So what I’m really saying is the $FinProb$ is a category internal to the category of convex spaces.

I do. But if you go here, you should be able to see the slides and a video both for my talk and Tai-Danae Bradley’s related talk “Operads and entropy”.