The K-theory of endomorphisms of spaces

TL;DR

This paper extends algebraic K-theory to endomorphisms of spaces, providing a non-linear analogue of Grayson's theorem and linking it to reduced K-theory of localized categories, generalizing previous results on nilpotent endomorphisms.

Contribution

It introduces a non-linear version of Grayson's theorem, relating the K-theory of endomorphisms over a space to localized categories, and generalizes Klein and Williams' description of nil-terms in A-theory.

Findings

Identifies the K-theory of endomorphisms over a space in terms of reduced K-theory.

Provides a non-linear analogue of a classical theorem in algebraic K-theory.

Generalizes previous results on nilpotent endomorphisms in A-theory.

Abstract

We prove a non-linear version of a theorem of Grayson which is an analogue of the Fundamental Theorem of Algebraic $K$-theory and identify the $K$-theory of the endomorphism category over a space $X$ in terms of reduced $K$-theory of a certain localisation of the category of $\NN$-spaces over $X$. In particular we generalise the result of Klein and Williams describing the nil-terms of $A$-theory in terms of $K$-theory of nilpotent endomorphisms.