## The additivity theorem in $$K$$-theory.(English)Zbl 1077.19001

The authors give new versions, in the style of Quillen’s Theorem A, of F. Waldhausen’s proof [Lect. Notes Math. 1126, 318–419 (1985; Zbl 0579.18006)] of the additivity theorem for algebraic $$K$$-theory. This is carried out by showing some results that allow to convert a theorem B style proof into a theorem A style proof. The first of these results is Theorem Â showing conditions such that, for any functors of small categories $$f:C\rightarrow D$$ and $$g:C\rightarrow E$$, the induced functor $$(f,g):C\rightarrow D\times E$$ is a homotopy equivalence. This theorem is then rewritten for maps of simplicial sets obtaining the simplicial set versions Theorem Â* and Theorem Â$$'$$.

### MSC:

 19D06 $$Q$$- and plus-constructions 55U10 Simplicial sets and complexes in algebraic topology

Zbl 0579.18006
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