nLab infinitary tensor product

Idea

In a category with all products, one can take finite products of objects, and hence one has a monoidal category. One can however also take infinite products. In some situations, such as in categorical probability, one is interested in taking an infinitary version of a tensor product.

In measure-theoretic probability theory, such a construction is provided by Kolmogorov's extension theorem, and the same idea can be used in general:

• In a category with products, an infinite product can be expressed as a cofiltered limit of finite products.
• Similarly, one can define an infinitary tensor product as a cofiltered limit, if it exists, of finite tensor products.

Definition

Let $(C,\otimes,I)$ be a symmetric monoidal category equipped with a distinguished map $del_X:X\to I$ for each object $I$. (This happens for example if $C$ is cartesian semicartesian, or if it has a Markov or copy-discard structure.)

Let $J$ be a possibly infinite set, and let $(X_i)_{i\in J}$ be a $J$-indexed collection of objects of $C$.

For every finite subset $F\subseteq J$, denote by

$$\bigotimes_{i\in F} X_i$$

the tensor product of the family $(X_i)_{i\in F}$. (Note that $F$ is not ordered, but the ordering of the terms in the tensor product does not matter up to coherent isomorphism, since $C$ is symmetric monoidal.)

Consider now finite subsets $F\subseteq G$ of $J$. We have a canonical “projection” morphism defined as follows. First of all, for $i\in G$, let

$$Y_i = \begin{cases} X_i & i\in F ; \\ I & i\notin F. \end{cases}$$

Similarly, let $e_i:X_i\to Y_i$ be

$$e_i = \begin{cases} id_{X_i} & i\in F ; \\ del_{X_i} & i\notin F. \end{cases}$$

The map $\pi_{G,F}$ is given by the tensor product of the $e_i$ as follows, where the isomorphism on the right is given by the unitors of the monoidal category.

Let now $Fin(J)$ be the poset of finite subsets of $G$. The union of finite subsets is finite, so $Fin(J)$ is a directed set, hence a filtered category. The functor $Fin(J)^op\to C$ mapping the inclusion $F\subseteq G$ to $\pi_{G,F}$ defined above is hence a cofiltered diagram.

We say that an object is an infintary tensor product of the family $(X_i)_{i\in J}$ in $C$, and denote it by $X_J$, if

• It is a cofiltered limit of the functor $Fin(J)^op\to C$ described above, and moreover
• For every object $A$, the functor $A\otimes-:C\to C$ preserves this limit.

Examples

• In a cartesian monoidal category, every infinite product, if it exists, is an infinitary tensor product.

• More generally, consider a cartesian monoidal category $\mathcal{C}$ and a monoidal monad (or commutative monad) $T$. Recall that the Kleisli category of a monoidal monad is canonically a monoidal category. Suppose now that an infinite product in $\mathcal{C}$ exists, and express it as a cofiltered limit of finite products. The left adjoint $\mathcal{C}\to\mathcal{C}_T$ preserves this limit if and only if the endofunctor $T$ does (see here). So, given a category with infinite products and a monoidal monad on it which preserves cofiltered limits, its Kleisli category has infinitary tensor products.

• In a Markov category, Kolmogorov products are particular infinitary products compatible with the copy-discard structure. In particular, the category BorelStoch has all countable infinitary tensor products.

• The Giry monad on standard Borel spaces is an instance of both of the examples above, by the Kolmogorov extension theorem.

• Reversing all arrows, the infinite tensor product of rings is defined as the filtered colimit of all the finitary tensor products. (The canonical arrows $I\to X$ are the units of the rings.)

See also

• monoidal category, cartesian monoidal category
• filtered category, filtered limit
• Markov category, Kolmogorov extension theorem

References

• George Janelidze and Ross Street, Real sets, Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, pp. 23-49. [doi:10.1515/tmj-2017-0101]

• Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)