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

Tate objects 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 deals with exact categories. We will prove the main properties expected from Tate objects. In particular, we show that the $\mathrm{K}$-theory of Tate objects is a delooping of that of the original category. This gives us a procedure to transport invariants from finite dimensional objects to Tate objects, hence providing interesting invariants.

Keywords

Tate object, higher category, $K$-theory

2010 Mathematics Subject Classification

18F25, 18G55

Full Text (PDF format)

Received 5 August 2016

Received revised 4 April 2017

Published 29 November 2017