We study elementary Tate objects in an exact category. We characterize the category of elementary Tate objects as the smallest subcategory of admissible Ind-Pro objects which contains the categories of admissible Ind-objects and admissible Pro-objects, and which is closed under extensions. We compare Beilinson’s approach to Tate modules to Drinfeld’s. We establish several properties of the Sato Grassmannian of an elementary Tate object in an idempotent complete exact category (e.g., it is a directed poset). We conclude with a brief treatment of n-Tate modules and n-dimensional adèles.

An appendix due to J. Šťovíček and J. Trlifaj identifies the category of flat Mittag-Leffler modules with the idempotent completion of the category of admissible Ind-objects in the category of finitely generated projective modules.