Locally Compact Objects in Exact Categories

TL;DR

This paper explores two categories of locally compact objects in exact categories, establishing their properties, relations, and their roles in K-theory, with a focus on the Beilinson and Kato constructions.

Contribution

It identifies and compares two constructions of locally compact objects in exact categories, proving that the Beilinson category is an exact category suitable for K-theoretical applications.

Findings

The Beilinson category lim A is an exact category.

The Kato category k(A) is related to lim A and studied comparatively.

Categories of countably indexed ind/pro-objects can be described as localizations.

Abstract

We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories Ind_{aleph_0}(C), Pro_{aleph_0}(C) of countably indexed ind/pro-objects over any category C can be described as localizations of categories of diagrams over C.