This paper generalizes the normally ordered tensor product from Tate vector spaces to Tate objects over arbitrary exact categories. We show how to lift bi-right exact monoidal structures, duality functors, and construct external Homs. We list some applications: (1) Adeles of a flag can be written as ordered tensor products; (2) Intersection numbers can be interpreted via these tensor products; (3) Pontryagin duality uniquely extends to n-Tate objects in locally compact abelian groups.
Keywords: Tate vector space, Tate object, normally ordered product, higher adeles,higher local fields
2010 MSC: 14A22, 18B30
Theory and Applications of Categories, Vol. 33, 2018, No. 13, pp 296-349.