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This paper contains a construction of generators and partial relations for $K_1$ of a simplicial Waldhausen category where cofiber sequences split up to weak equivalence. The main application of these generators and relations is to produce generators for $K_1$ of a (simplicial) assembler.
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is
We provide a more economical refined version of Evrards categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred functor which sa
In this note we show that Waldhausens K-theory functor from Waldhausen categories to spaces has a universal property: It is the target of the universal global Euler characteristic, in other words, the additivization of the functor sending a Waldhause
Suppose k is a field of characteristic 2, and $n,mgeq 4$ powers of 2. Then the $A_infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_infty$-structure on $H^*(C_ntimes
We show that the Waldhausen trace map $mathrm{Tr}_X colon A(X) to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to emph{weak} homotopy with respect to transfer maps in algebraic