Homology, Homotopy and Applications

Volume 19 (2017)

Number 2

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

Pages: 373 – 395

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a18


Benjamin Hennion (Max-Planck Institut für Mathematik, Bonn, Germany)


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.


Tate object, higher category, $K$-theory

2010 Mathematics Subject Classification

18F25, 18G55

Received 5 August 2016

Received revised 4 April 2017

Published 29 November 2017