Category theory; homological algebra Theorem

Additivity theorem

Builds on Waldhausen category Waldhausen S-construction Homotopy equivalence

Statement

Let $\mathcal{C}$ be a Waldhausen category, and let $\mathcal{E} = S_2 \mathcal{C}$ be the Waldhausen category of cofibration sequences $A \rightarrowtail B \twoheadrightarrow C$ in $\mathcal{C}$ . The category $\mathcal{E}$ carries its standard cofibrations and weak equivalences. Let $F \colon \mathcal{E} \to \mathcal{C} \times \mathcal{C}$ be the functor $(A \rightarrowtail B \twoheadrightarrow C) \mapsto (A, C)$ . Then $F$ induces a homotopy equivalence $\abs{wS_\bullet \mathcal{E}} \xrightarrow{\ \simeq\ } \abs{wS_\bullet \mathcal{C}} \times \abs{wS_\bullet \mathcal{C}},$ where $wS_\bullet$ is the S-construction.

Remarks

A map back is induced by $(A, C) \mapsto (A \rightarrowtail A \vee C \twoheadrightarrow C)$ , where $A \vee C = A \cup_0 C$ is the coproduct. One composite is the identity of $\mathcal{C} \times \mathcal{C}$ . The work is to show that the other composite is homotopic to the identity on the realization, which Waldhausen did by a simplicial argument.

The main consequence: let $F' \rightarrowtail F \twoheadrightarrow F''$ be a cofibration sequence of exact functors $\mathcal{C} \to \mathcal{D}$ between Waldhausen categories. Then the maps that $F$ and $F' \vee F''$ induce on K-theory spaces are homotopic, so $F_* = F'_* + F''_*$ on $K_n$ for all $n \ge 0$ . In degree $0$ this is the relation $[F(X)] = [F'(X)] + [F''(X)]$ in $K_0(\mathcal{D})$ ; the theorem says the same additivity holds in all degrees. It is used, for instance, to compute the K-theory of filtered objects and of projective bundles.

Going deeper

For a student

Additivity is the basic tool of Waldhausen K-theory. Waldhausen derives from it his fibration theorem, and the fact that the iterated S-construction gives an $\Omega$ -spectrum. For exact categories with the isomorphisms as weak equivalences, it contains Quillen's additivity result for the Q-construction.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`2TW8ZQ0GNG`
URI	`m2://2TW8ZQ0GNG`
Kind	statement, theorem
Writing	W0 stub
MSC	18F25, 19D10
Status	This copy of the site has no build data. Run `m2 status 2TW8ZQ0GNG`.
About	Waldhausen S-construction
Uses	Waldhausen category, Waldhausen S-construction, Homotopy equivalence
Changed	3 November 2026