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Stream: learning: questions

Topic: Pcoh and biproducts

Alejandro Díaz-Caro (Mar 21 2024 at 21:24):

A quick question: Does the category of Probabilistic Coherent Spaces (Pcoh) have biproducts? I was checking Danos and Ehrhard's paper and I am not completelly sure. There is certainly a Cartesian product, noted as the additive conjunction $\&$, and it is said that $\oplus_{i} X_{i} = (\&_{i} X_i^{\bot})^{\bot}$. But I am not sure if that implies that the coproduct is the same as the Cartesian product.

Morgan Rogers (he/him) (Mar 22 2024 at 09:29):

It seems like they're different: the $\oplus$ notation should not be interpreted as a biproduct unless they explicitly say so.

Damiano Mazza (Mar 22 2024 at 09:52):

Yes, they are different. For example, if $\mathbf 1$ is the space $(\{\ast\},[0,1])$ (the monoidal unit), then both $\mathbf 1\mathrel{\&}\mathbf 1$ and $1\oplus\mathbf 1$ have web $\{0,1\}$, but $\mathsf P(\mathbf 1\mathrel{\&}\mathbf 1)=[0,1]\times[0,1]$ (the unit square), whereas $\mathsf P(\mathbf 1\oplus\mathbf 1)$ is the set of all pairs $(\alpha,\beta)\in[0,1]\times[0,1]$ such that $\alpha+\beta\leq 1$ (the "unit triangle").

Damiano Mazza (Mar 22 2024 at 09:55):

Notice that the type $\mathbf 1\oplus\mathbf 1$ is the type of Booleans, and in fact the space $\mathbf 1\oplus\mathbf 1$ reflects this intuition in the probabilistic case: an element of $\mathsf P(\mathbf 1\oplus\mathbf 1)$ is a sub-probability distribution on $\{0,1\}$, i.e., a probabilistic Boolean. On the other hand, since $\mathbf 1$ is the space of sub-probability distributions on the singleton (intuitively representing the probability that a program of unit type terminates), $\mathbf 1\mathrel{\&}\mathbb 1$ is just the space of pairs of such things.

Alejandro Díaz-Caro (Mar 22 2024 at 10:23):

Thank you both!
Damiano: your explanation is exactly what I was looking for. Now I think I understand the intuition much better.