nLab restricted product

Idea

Given a set $S$ and a collection $\{K_s \stackrel{\iota_s}{\hookrightarrow} X_s\}_{s \in S}$ of morphisms in some category (typically monomorphisms), then the restricted product $\underset{s\in S}{\prod}^\prime X_s$ is vaguely the like the actual product $\underset{s\in S}{\prod} X_s$ of all the $X_s$, but subject to the restriction that for each element $(x_s)_{s \in S}$ all but a finite number of components $x_s$ are in the image of $\iota_s$.

Definition

Assume that the ambient category $\mathcal{C}$ has ordinary products. Write $\mathcal{P}_{fin} S$ for the poset of finite subsets of $S$ and consider the functor

$$X^K \colon \mathcal{P}_{fin} S \longrightarrow \mathcal{C}$$

which is given on a finite subset $U \subset S$ by the products

$$U \mapsto X^K_U \coloneqq \left(\underset{u \in U}{\prod} X_u\right) \times \left( \underset{s \in S-U}{\prod} K_s \right)$$

and which sends an inclusion $U_1 \hookrightarrow U_2$ of subsets to the evident morphism between these products whose components $f_s$ are the identity for $s \in S-(U_2-U_1)$ and are $\iota_s$ for $s \in U_2-U_1$.

Then the restricted product is the (filtered) colimit over this functor

$$\underset{s\in S}{\prod}^\prime X_s \coloneqq \underset{\underset{U \in \mathcal{P}_{fin}S}{\longrightarrow}}{\lim} X^K_U \,.$$

Examples

• A weak direct product is a restricted product whose trivial subobjects are specified global elements, hence where each $K_s = \ast$ are terminal objects.

• The ring of adeles of a global field $k$ is the restricted product of the formal completions $k_v$ at all its places $v$, with the restriction being that only finitely many components have norm greater than unity.

  Equivalently this is the restricted product for the inclusions $\iota_v \colon \mathcal{O}_v \hookrightarrow k_v$ of the ring of integers of $k_v$ into $k_v$.

• Under the function field analogy the above example is a special case of the following more general example: for $\Sigma$ an arithmetic curve then the restricted product of all the algebras of functions on all its punctured formal disks, with the restrictions being along the inclusions of the functions on the un-punctured formal disks, appears in the description of Cech cocycles with respect to covers of the curve by the complement of any finite points and the formal disks around these points (e.g Frenkel 05, section 3.2). This is discussed in detail at moduli stack of bundles – over curves.

References

• Wikipedia Restricted product

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)

• MO/96137/categorical-description-of-the-restricted-product-adeles