Tate objects in stable $(\infty,1)$-categories

TL;DR

This paper develops a comprehensive theory of Tate objects within stable $( abla,1)$-categories, extending previous work from exact categories and including examples like spectra and derived algebraic structures.

Contribution

It introduces the theory of Tate objects in stable $( abla,1)$-categories, broadening the framework beyond exact categories and providing foundational properties.

Findings

Established main properties of Tate objects in the new setting

Included examples such as spectra and derived algebraic objects

Extended the applicability of Tate objects to structured infinite-dimensional spaces

Abstract

Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable $(\infty,1)$-categories, while the literature only treats with exact categories. We will prove the main properties expected from Tate objects. This new setting includes several useful examples: Tate objects in the category of spectra for instance, or in the derived category of a derived algebraic object -- which can be thought as structured infinite dimensional vector bundle in derived setting.