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

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.