Puzzle 226:

Guess 1: Rel (the category where objects are sets and morphisms are relations): my guess is that this can be viewed as the subcategory of Feas where the objects are discrete posets (only morphisms are the identities). But one could also probably show it's a monoidal category more directly: the cartesian product would be a tensor; for example, {1}xA is naturally isomorphic to A, and cartesian product is associative (up to isomorphism).

Guess 2: the category of finite dimensional vector spaces, equipped with the tensor product (from which I suppose guess 1 would follow). Reasons for guess: tensor product is associative, maps to a vector space where the basis is formed of a cartesian product, and has the word "tensor" in it.

Guess 3: the category of sets with conditional probability distributions as morphisms, i.e. (A->B) is the set of conditional distributions over B given A. The unital set is unit, and the tensor would be effectively cartesian product again.

Guess 1: Rel (the category where objects are sets and morphisms are relations): my guess is that this can be viewed as the subcategory of Feas where the objects are discrete posets (only morphisms are the identities). But one could also probably show it's a monoidal category more directly: the cartesian product would be a tensor; for example, {1}xA is naturally isomorphic to A, and cartesian product is associative (up to isomorphism).

Guess 2: the category of finite dimensional vector spaces, equipped with the tensor product (from which I suppose guess 1 would follow). Reasons for guess: tensor product is associative, maps to a vector space where the basis is formed of a cartesian product, and has the word "tensor" in it.

Guess 3: the category of sets with conditional probability distributions as morphisms, i.e. (A->B) is the set of conditional distributions over B given A. The unital set is unit, and the tensor would be effectively cartesian product again.