nLab
Kolmogorov product

Contents

Contents

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,1)(C,\otimes 1) be a symmetric semicartesian monoidal category.

Let X 1X_1 and X 2X_2 be objects of CC. The unique map !:X 11!:X_1\to 1 induces a map

sometimes called the projection (or marginal? in the probability literature). This projection is natural in X 2X_2, and there is a commutative diagram

More generally, let {X 1,,X n}\{X_1, \dots, X_n\} be a finite set of objects of CC. All the possible projections of their product onto the factors form a finite Boolean lattice, or more precisely, a functor BCB \to C from a finite Boolean lattice BB (considered as a thin category) to CC.

We call this lattice the lattice of projections.

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

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

where FF is a finite subset of II, and SFS\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} iI\{X_i\}_{i\in I} the cofiltered limit of this diagram, if it exists, and if it is preserved by the functor Y-\otimes Y for every object YY of CC. We denote it by

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

Properties

Examples

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See also

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)

Last revised on February 8, 2020 at 08:55:54. See the history of this page for a list of all contributions to it.