Given a set and a collection of morphisms in some category (typically monomorphisms), then the restricted product is vaguely the like the actual product of all the , but subject to the restriction that for each element all but a finite number of components are in the image of .
Assume that the ambient category has ordinary products. Write for the poset of finite subsets of and consider the functor
which is given on a finite subset by the products
and which sends an inclusion of subsets to the evident morphism between these products whose components are the identity for and are for .
Then the restricted product is the (filtered) colimit over this functor
A weak direct product is a restricted product whose trivial subobjects are specified global elements, hence where each are terminal objects.
The ring of adeles of a global field is the restricted product of the formal completions at all its places , with the restriction being that only finitely many components have norm greater than unity.
Equivalently this is the restricted product for the inclusions of the ring of integers of into .
Under the function field analogy the above example is a special case of the following more general example: for 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.
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
Last revised on January 14, 2021 at 09:59:51. See the history of this page for a list of all contributions to it.