nLab Kolmogorov product

Idea

A Kolmogorov product is a way of forming infinite tensor products in a semicartesian monoidal category which is not necessarily cartesian. In a cartesian monoidal category it coincides with the usual infinite cartesian product (if it exists).

If the category in question is a poset, Kolmogorov products correspond to a finitary complete monoid? structure (up to reversing the arrows).

Definition

In what follows, let $(C,\otimes 1)$ be a symmetric semicartesian monoidal category.

Let $X_1$ and $X_2$ be objects of $C$. The unique map $!:X_1\to 1$ induces a map sometimes called the projection (or marginal? in the probability literature). This projection is natural in $X_2$, and there is a commutative diagram

More generally, let $\{X_1, \dots, X_n\}$ be a finite set of objects of $C$. All the possible projections of their product onto the factors form a finite Boolean lattice, or more precisely, a functor $B \to C$ from a finite Boolean lattice $B$ (considered as a thin category) to $C$. We call this lattice the lattice of projections.

Even more generally, let now $\{X_i\}_{i\in I}$ be a set of objects of $C$, possibly infinite. We can form the lattice of all the finite products of the $X_i$ and their projections, with the arrows in the form

$$\bigotimes_{i\in F} X_i \to \bigotimes_{j\in S} X_j ,$$

where $F$ is a finite subset of $I$, and $S\subseteq F$. This is again a lattice (it may be infinite, but it is closed under finite joins and meets), which we call the lattice of finite projections. In particular, this is a cofiltered diagram. We call the Kolmogorov product of the set $\{X_i\}_{i\in I}$ the cofiltered limit of this diagram, if it exists, and if it is preserved by the functor $-\otimes Y$ for every object $Y$ of $C$. We denote it by

$$\bigotimes_{i\in I} X_i .$$

Properties

• The Kolmogorov product of a finite set of objects is just their $n$-ary tensor product.

• In a cartesian monoidal category, the Kolmogorov product coincides with the possibly infinite cartesian product.

Examples

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See also

• filtered category, filtered limit
• finitary complete monoid?
• monoidal category, cartesian monoidal category, semicartesian monoidal category
• Kolmogorov extension theorem

References

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• Tobias Fritz and Eigil Fjeldgren Rischel, The zero-one laws of Kolmogorov and Hewitt–Savage in categorical probability, 2019. (arXiv:1912.02769)