Algebraic Kasparov K-theory. I

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.Comment: This is the final revised versio

The homotopy coniveau tower

We examine the "homotopy coniveau tower" for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky's slice tower for $S^1$-spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel-Voevodsky stable homotopy category, and we identify this $P^1$-stable homotopy coniveau tower with Voevodsky's slice filtration for $P^1$-spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general $P^1$-spectrum the structure of a module over motivic cohomology. This recovers and extends recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral sequence.Comment: A revised and extended version of an earlier paper, which is on the K-theory serve

A C-system defined by a universe category

This is a major update of the previous version. The methods of the paper are now fully constructive and the style is "formalization ready" with the emphasis on the possibility of formalization both in type theory and in constructive set theory without the axiom of choice. This is the third paper in a series started in 1406.7413. In it we construct a C-system $CC({\cal C},p)$ starting from a category $\cal C$ together with a morphism $p:\widetilde{U}\rightarrow U$, a choice of pull-back squares based on $p$ for all morphisms to $U$ and a choice of a final object of $\cal C$. Such a quadruple is called a universe category. We then define universe category functors and construct homomorphisms of C-systems $CC({\cal C},p)$ defined by universe category functors. As a corollary of this construction and its properties we show that the C-systems corresponding to different choices of pull-backs and final objects are constructively isomorphic. In the second part of the paper we provide for any C-system CC three constructions of pairs $(({\cal C},p),H)$ where $({\cal C},p)$ is a universe category and $H:CC\rightarrow CC({\cal C},p)$ is an isomorphism. In the third part we define, using the constructions of the previous parts, for any category $C$ with a final object and fiber products a C-system $CC(C)$ and an equivalence $(J^*,J_*):C \rightarrow CC$