Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Hoel et al on emergence

David Corfield (Oct 24 2025 at 11:44):

Erik Hoel has a series of papers with coauthors looking to understand emergence. Seeing the most recent one come out the other day reminded me that I was contemplating a category-theoretic understanding of his approach a while ago.

Central to his position is the idea of seeing how near we are to situations where a cause necessarily and sufficiently brings about its effect. So Comolatti and Hoel (Causal emergence is widespread across measures of causation) explore emergence in terms of differences in causal strength (CS) calculated at different levels, where CS is measuring the control one has on effects (E) via causes (C). If this were perfect, there would be an isomorphism $C \to E$.

In general, the process is a probabilistic mapping, a matrix with non-negative entries, each row summing to $1.$ A cause is sufficient for an effect if $P(e|c) = 1$, and necessary if $P(e|\neg c) = 0$. Departures over the set of causes and effects correspond to indeterminacy and degeneracy, respectively. A good coarse-graining of a system will display high determinacy and low degeneracy.

I had thought to tie this to

• @Caterina Puca, @Amar Hadzihasanovic, @Fabrizio Romano Genovese, @Bob Coecke, Obstructions to Compositionality,

which looks to provide measures of departure from isomorphism via mono and split epi characteristics. They end with a comment on generalizing beyond $\mathbf{Set}$:

Most importantly, we hope to have opened a new avenue in “formal compositionality theory”. The greatest challenge will be to graduate from proof-of-concept examples to ones that reveal more interesting structure, perhaps in non-$\mathbf{Set}$-like categories where a split epi or mono is not simply a surjective or injective map. We have been looking at case studies of this sort, which nevertheless have manageable combinatorics permitting an exhaustive study of their homotopy posets, and we hope to discuss them in future work.

I was thinking then that we might consider departures from isomorphism in the Markov category of finite sets and probability distributions. I think we have that: if each effect has a sufficient cause, then the morphism is a split epi. If each cause is necessary to its effects, then it's mono.

Does that sound right?

Matteo Capucci (he/him) (Oct 25 2025 at 09:03):

David Corfield said:

I think we have that: if each effect has a sufficient cause, then the morphism is a split epi. If each cause is necessary to its effects, then it's mono.

How does that work? What is a split epi in the Markov category of finite sets and probability distributions [aka $\bf FinStoch$]?

David Corfield (Oct 25 2025 at 09:11):

I think a split epi from $A$ to $B$ should be such that for every $b \in B$ there's an $a$ such that $P(b|a)=1$. Then there's a suitable deterministic map backwards.

David Corfield (Oct 27 2025 at 13:06):

Does something like this crop up in category-theoretic probability theory, where (proximity to) being mono/split epi in a Markov category has the flavour of necessary/sufficient causation? (Hoping the likes of @Tobias Fritz, @Paolo Perrone, @Sam Staton might be able to answer.)

Paolo Perrone (Oct 27 2025 at 13:42):

I don't quite know the 'causation' meaning of what follows, but maybe someone finds it suggestive.

• Most Markov categories of interest for probability are positive, which corresponds roughly to the fact that there are no negative probabilities. FinStoch, Stoch and all those are positive.
• In a positive Markov category, if we have a retraction $f:X\to Y$, $g:Y\to X$ with $g\circ f = 1_X$, then $g$ is $f$-almost surely deterministic. So, in a certain sense, split epis are 'almost' deterministic.
• Concretely, in (say) FinStoch, this means that, up to probability zero:
• • $g$ is a deterministic function, which partitions $Y$ into 'cells', namely its fibers $g^{-1}(x)$, indexed by the $x\in X$.
• • $f$, seen as a map from points to distributions, maps each $x\in X$ to a probability distribution supported on its fiber $g^{-1}(x)$.
• • This way, when we apply $g$ after $f$, all the 'mass' goes back to $x$.
• Outside of FinStoch, the same intuition holds but things are a little more involved, a kernel like $f$ is sometimes called a proper kernel.

I don't know how to relate this to causality, but I hope it helps!

David Corfield (Oct 27 2025 at 15:00):

Thanks, Paolo! It seems that many people were after some way of measuring a process as causally efficacious, where the ideal is for a specific effect to be brought about uniquely and deterministically by a specific cause:

image.png

(Causal emergence is widespread across measures of causation, Fig. 1)

For their version of this measure, the latest Hoel paper (with Jansma), Engineering Emergence, looks at how it varies under coarse-graining a Markov chain. Greater determinacy and less degeneracy may emerge after coarse-graining from the microscale.

Tobias Fritz (Oct 28 2025 at 05:40):

David Corfield said:

I think a split epi from $A$ to $B$ should be such that for every $b \in B$ there's an $a$ such that $P(b|a)=1$. Then there's a suitable deterministic map backwards.

Just to confirm this, yes, those are precisely the split epis. You've already explained why the condition is sufficient, and the fact that it's necessary follows from Paolo's comment on almost sure determinism.

The split monos are those $P : A \to B$ for which the distributions $P(-|a)$ have disjoint support as $a$ varies.

Tobias Fritz (Oct 28 2025 at 05:44):

David Corfield said:

So Comolatti and Hoel (Causal emergence is widespread across measures of causation) explore emergence in terms of differences in causal strength (CS) calculated at different levels, where CS is measuring the control one has on effects (E) via causes (C). If this were perfect, there would be an isomorphism $C \to E$.

In general, the process is a probabilistic mapping, a matrix with non-negative entries, each row summing to $1.$ A cause is sufficient for an effect if $P(e|c) = 1$, and necessary if $P(e|\neg c) = 0$.

Is this in the setting of a Bayesian network where the DAG is simply $C \to E$? Or are we merely given a stochastic matrix $P : C \to E$ without a distribution on $C$? In the latter case, what does $P(e|\neg c)$ mean if $c$ has more than two elements? Perhaps the condition formally reads as $P(e|c') = 0$ for all $c' \neq c$?

Tobias Fritz (Oct 28 2025 at 05:57):

Assuming that my interpretation of necessary is correct, it looks to me like your idea that "split epi = each effect has a sufficient cause" and "split mono = each cause is necessary to its effects" works out very nicely. Interesting!

David Corfield (Oct 28 2025 at 08:46):

Thanks, Tobias! As to the setting for this, in the most recent paper

• Abel Jansma and Erik Hoel, Engineering Emergence

we see them considering a transition probability matrix, each node considered as a cause probabilistically realising its effects. Then starting out with a distribution over nodes (chosen to be uniform), they can calculate a specificity (= $1-$ degeneracy) measure and a determinacy measure for the network, $T$, and so a value measuring how near the transition probability matrix is to being a permutation matrix, ($CP = 1$):

$CP = \text{determinacy}_T + \text{specificity}_T - 1$

I was wondering whether, if we put things in a good category-theoretic setting, then there would be a natural measure of the departure from isomorphism between causes and effects.

Jansma and Hoel then consider all the possible coarse-grainings of the network of a transition probability matrix to see how this $CP$ value changes, how the causality appears at different scales.

Tobias Fritz (Oct 28 2025 at 09:24):

So in other words, one can more directly write $CP = \mathrm{determinacy} - \mathrm{degeneracy}$? In this form it reminds me of the index of a linear map. It's not a perfect analogy because of CP = 1 for isomorphisms while index = 0 for invertible linear maps, but perhaps one can make the analogy more precise by using different conventions?

David Corfield (Oct 28 2025 at 10:00):

Tobias Fritz said:

So in other words, one can more directly write $CP = \mathrm{determinacy} - \mathrm{degeneracy}$?

Yes, I was wondering why they did that ("For simplicity’s sake").

I reckon there could be something useful in the paper I mentioned above, Puca et al. Obstructions to Compositionality, and their measure of divergence from isomorphism.

I had a brief exchange with @Caterina Puca on X a while ago. I was wondering about comparison of their work with the cohomological approach of Elie Adam's thesis Systems, Generativity and Interactional Effects

image.png
image.png

and their

image.png

She supported the idea that you can view emergence as failure of compositionality.

Tobias Fritz (Oct 28 2025 at 10:00):

On the other hand, you may not want a measure of causal strength to behave like an index, because an index can be negative, and having index 0 is necessary but not sufficient for being an isomorphism.

David Corfield (Oct 28 2025 at 11:53):

One might hope for Jansma-Hoel meets Baez-Courser on Coarse-Graining Open Markov Processes. Let's see if @John Baez can help me out.

Is there some kind of measure on open Markov processes which generalizes (and maybe modifies) the quantities that Jansma-Hoel are assigning to closed Markov processes (failure to be a deterministic permutation) so that we can get a handle on what makes for a "good" coarse-graining? Some kind of lax double functor from $\mathbb{M}\mathbf{ark}$ which detects how close a Markov process is to being deterministic and non-degenerate.

John Baez (Oct 28 2025 at 16:36):

I don't know the Jansma-Hoel metric (or whatever it is). For finite n, there's a unique nice topology on n x n stochastic matrices, coming from the Euclidean topology on n x n real matrices. Any decent metric should give that topology. But some metrics might be well-motivated.

John Baez (Oct 28 2025 at 16:37):

In the infinite-dimensional case things get more tricky.