Categorical Foundations for K-Theory

TL;DR

This paper develops a unified categorical framework for understanding the foundational structures used in algebraic K-theory, focusing on how objects are associated with structured categories to extract K-theoretic information.

Contribution

It introduces a conceptual framework that unifies various examples of objects in K-theory through categories of modules in monoidal fibred categories, clarifying the interaction between objects, categories, and morphisms.

Findings

Proposes a unified categorical approach to K-theory foundations.

Characterizes structured categories as modules in monoidal fibred categories.

Provides insights into the interaction of morphisms with categorical structures.

Abstract

Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...). One then applies to the category A_C a "$K$-theory machine", which provides an infinite loop space that is the $K$-theory K(C) of the object C. We study the first step of this process. What are the kinds of objects to be studied via $K$-theory? Given these types of objects, what structured categories should one associate to an object to obtain $K$-theoretic information about it? And how should the morphisms of these objects interact with this correspondence? We propose a unified, conceptual framework for a number of important examples of objects studied in $K$-theory. The structured categories associated to an object C are typically categories of modules in a monoidal (op-)fibred category. The modules considered are "locally trivial" with respect to a given class of trivial modules and a given Grothendieck topology on the object C's category.