On fundamental theorems of algebraic K-theory

Systems of diagram categories and K-theory. II

“…We refer the reader to Staffeldt's work [21]. The author knows two proofs of that theorem for Waldhausen's categories: by Waldhausen [25] and by McCarthy [15]. We shall follow Waldhausen's proof.…”

“…In this case, one can think of the K-theory of D in terms of this spectrum. This does not affect the K-groups, because: [15] proof of additivity -in fact, both have the same complexity -to dérivateurs (or systems of diagram categories).…”

“…A weaker version of additivity has been shown in [6]. One could try to adapt Waldhausen's [25] or McCarthy's [15] proof of additivity in this context to show the strong form of additivity. At the first glance, the machinery of dérivateurs seems to have some of the necessary ingredients.…”

On the Karoubi Filtration of a Category

“…One of the basic tools is the additivity theorem [12, Theorem 1.4.2 and Proposition 1.3.2], see also [6].…”

On the computation of stabilized tensor functors and the relative algebraic 𝐾-theory of dual numbers

“…We will say that F is a p-product functor if F preserves products in a range 0 ≤ i ≤ p (that is, π i (ρ) is an isomorphism for all 0 ≤ i ≤ p). By the proof of additivity found in [12], for any F…”