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Stream: theory: categorical probability

Topic: Converse to de Finetti's theorem

Eigil Rischel (Nov 17 2025 at 00:16):

Let $\mathcal{C}$ be a positive Markov category, and let $A$ be an object. Let $\mathsf{Fin}_i$ be the category of finite sets and injections, and consider the diagram $\mathsf{Fin}_i^\mathrm{op} \to \mathcal{C}$ given by $F \mapsto A^{\otimes F}$. Suppose this diagram has a limit $P$ (that is, an object which classifies exchangable distributions on arbitrarily many copies of $A$), and that this limit is preserved by the tensor product. Then $P$ is a distribution object for $A$.

To see this, note that this limit is the (carrier of) the cofree (co)commutative comonoid on $A$. But for a positive Markov category, $\mathsf{CoMon}(\mathcal{C}) \cong \mathcal{C}_\mathrm{det}$, so this is the same universal property as being a distribution object.

This is a kind of converse to de Finetti's theorem - de Finetti says that the distribution object, if it exists (and $\mathcal{C}$ satisfies a few other conditions) classifies exchangable families. This says that if there is an object classifying exchangable families, it must be a distribution object. I'm thinking maybe the existence of this limit can be established in more general terms, which would be a different abstract path to the de Finetti theorem.

Has anyone seen this idea before?

(To see the claim about the cofree comonoid, it's not too hard to cook up the comonoid structure (using the fact that the tensor product preserves this limit) and show by hand that it's universal.)

Tobias Fritz (Nov 17 2025 at 08:36):

Eigil Rischel said:

To see this, note that this limit is the (carrier of) the cofree (co)commutative comonoid on $A$. But for a positive Markov category, $\mathsf{CoMon}(\mathcal{C}) \cong \mathcal{C}_\mathrm{det}$, so this is the same universal property as being a distribution object.

That's a really neat and clean way to put it! This formulation of the argument is new to me.

Tobias Fritz (Nov 17 2025 at 08:42):

But the statement itself isn't new. There are the following versions of it: Theorem 4.35 and Corollary 4.37 of Involutive Markov categories and the quantum de Finetti theorem give a version of that in the quantum case, while Theorem 2.23 in Empirical Measures and Strong Laws of Large Numbers in Categorical Probability provides a version for quasi-Markov categories (even without positivity).

Sam Staton (Nov 17 2025 at 09:10):

Nice question. As you may know, the cofree comm comonoid is important in linear logic, and Raphaelle Crubille had a very nice talk about the connection with de Finetti at TLLA, abstract here