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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### The $K$-theory of endomorphisms of spaces

Pages: 325 – 338

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n1.a17

#### Author

#### 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 $\mathbb{N}$-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.

#### Keywords

$K$-theory of endomorphisms, algebraic $K$-theory of spaces, non-linear projective line.

#### 2010 Mathematics Subject Classification

19D10, 19D35, 55N15

Published 31 May 2016