Waldhausen K-theory of spaces via comodules

Waldhausen K-theory of Spaces via Comodules

Let X be a simplicial set. We construct a novel adjunction between the categories RX of retractive spaces over X and ComodX+ of X+comodules, then apply recent work on left-induced model category structures [5], [16] to establish the existence of a left proper, simplicial model category structure on ComodX+ with respect to which the adjunction is a Quillen equivalence after localization with res...

K-theory of a Waldhausen Category as a Symmetric Spectrum

If C is a Waldhausen category (i.e., a “category with cofibrations and weak equivalences”), it is known that one can define its K-theory K(C) as a connective symmetric Ω-spectrum. The goal of these notes is to explain the construction in such a way that it can be not only understood, but also (hopefully) remembered by a non-expert. We do not claim that our exposition shows the process by which ...

The 1-type of a Waldhausen K-theory Spectrum

We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.

Model theory of comodules

Relative Homological Algebra, Waldhausen K-theory, and Quasi-frobenius Conditions

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated K-theory groups are “stable algebraic G-theory,” which in degree zero form a certain stable representation group. We prove both some existence and nonexistence results about such Waldhausen category structure...