## On the normally ordered tensor product and duality for Tate objects

### Summary

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) Adèles 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.

14A22, 18B30

### Keywords/Phrases

Tate vector space, Tate object, normally ordered product, higher Adèles, higher local fields