Parametrized K-Theory

TL;DR

This paper introduces a general framework for defining K-theory of objects via categories of locally trivial objects, focusing on the first step of the traditional two-step K-theory process and exploring conditions for exact structures.

Contribution

It develops a unified approach to the first step of K-theory construction using categories of locally trivial objects and examines conditions for their exactness.

Findings

Established conditions for exact structures on categories of locally trivial objects

Defined K-theories for schemes and morphisms between them

Provided a framework connecting local triviality with K-theory construction

Abstract

In nature, one observes that a K-theory of an object is defined in two steps. First a "structured" category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a general framework that deals with the first step of this process. The K-theory of an object is defined via a category of "locally trivial" objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.