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

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.