Categorical Foundations for K-Theory

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.

💡 Research Summary

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The thesis addresses a foundational gap in algebraic K‑theory: while the modern K‑theory machinery (Quillen‑exact, Waldhausen, or symmetric monoidal categories) is well‑understood, there has been no systematic description of how to associate a suitable structured category to an arbitrary object (such as a space, a scheme, or a ring) before feeding it into the K‑theory machine. The author proposes a unified categorical framework that simultaneously captures the “object‑to‑category” step and respects morphisms between objects.

The central construction is that of a fibred site: a Grothendieck fibration (P\colon E\to B) equipped with a Grothendieck topology on the base category (B). A covering function assigns to each object (B) a family of morphisms ({f_i\colon B_i\to B}). Within this setting the notion of locally trivial objects is introduced. One first fixes a class of “trivial” objects in the fibres (for example free modules, trivial bundles, or the unit object of a monoidal structure). An object (E) over (B) is locally trivial if there exists a covering ({f_i}) such that each pull‑back (f_i^{*}E) is isomorphic to a trivial object. This abstracts the familiar ideas of locally free sheaves, vector bundles, torsors, etc., and places them in a highly general, fibre‑wise context.

The thesis then develops the theory of monoidal fibred categories, showing a 2‑equivalence with the corresponding indexed monoidal categories. Inside a monoidal fibred category one can define monoids and modules fibre‑wise; the total category of modules naturally forms a pair of composable fibrations \