2012/07/27 by Fiore, Thomas M., Pieper, Malte
#18A99 (Secondary) #19D10 #55N15 #55U10 (Primary) #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1207.6613
We use a simplicial product version of Quillen's Theorem A to prove classical Waldhausen Additivity of wS., which says that the "subobject" and "quotient" functors of cofiber sequences induce a weak equivalence wS.E(A,C,B)--> wS.A x wS.B . A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra K(A)vK(B)--> K(E(A,C,B)). This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature.