Waldhausen's Additivity Theorem

The additivity theorem is the fundamental structural result for Waldhausen K-theory. It states that the K-theory of the category of cofibration sequences splits as the product of the K-theories of the "fiber" and "cofiber." This result underpins essentially all computations in Waldhausen K-theory.

Statement

Theorem7.3Waldhausen's Additivity Theorem

Let $\mathcal{C}$ be a Waldhausen category and $E(\mathcal{C}) = S_2(\mathcal{C})$ the category of cofibration sequences in $\mathcal{C}$ (objects are $A \rightarrowtail B \twoheadrightarrow C$ with $C = B/A$ ). The "source and quotient" functor:

$(s, q): E(\mathcal{C}) \to \mathcal{C} \times \mathcal{C}, \quad (A \rightarrowtail B \twoheadrightarrow C) \mapsto (A, C)$

induces a homotopy equivalence on K-theory spaces:

$K(E(\mathcal{C})) \xrightarrow{\;\simeq\;} K(\mathcal{C}) \times K(\mathcal{C}).$

Equivalently, $K(s) + K(q) = K(\operatorname{id}): K(E(\mathcal{C})) \to K(\mathcal{C})$ , meaning:

$[B] = [A] + [C] \text{ in } K_0(\mathcal{C}) \quad \text{for every } A \rightarrowtail B \twoheadrightarrow C.$

Proof

The proof uses the S-construction and a "swallowing" argument.

Step 1: Setup. We work with the bisimplicial set $wS_p S_q \mathcal{C}$ and its realization. The additivity theorem is equivalent to showing that the two functors

$s, q: S_2 \mathcal{C} \rightrightarrows \mathcal{C}$

(source and quotient of the cofibration sequence) satisfy $|wS_\bullet s| + |wS_\bullet q| \simeq |wS_\bullet \operatorname{total}|$ at the level of K-theory spaces.

Step 2: The "generic cofibration" argument. Consider the functor $F: \mathcal{C} \to E(\mathcal{C})$ defined by $A \mapsto (A \rightarrowtail A \twoheadrightarrow *)$ . Then $(s, q) \circ F = (\operatorname{id}, 0)$ . Similarly, $G: \mathcal{C} \to E(\mathcal{C})$ via $C \mapsto (* \rightarrowtail C \twoheadrightarrow C)$ gives $(s, q) \circ G = (0, \operatorname{id})$ .

The functor $\operatorname{total}: E(\mathcal{C}) \to \mathcal{C}$ , $(A \rightarrowtail B \twoheadrightarrow C) \mapsto B$ , satisfies $\operatorname{total} \circ F = \operatorname{id}$ and $\operatorname{total} \circ G = \operatorname{id}$ .

Step 3: Key lemma (Waldhausen's "swallowing" lemma). Define two functors $F', G': \mathcal{C} \to E(\mathcal{C})$ that are naturally weakly equivalent to $F$ and $G$ but have the property that $F'$ and $G'$ together "generate" $E(\mathcal{C})$ .

Consider the filtration on $S_n E(\mathcal{C})$ . An object is an $n$ -step filtration of cofibration sequences:

$(*) = (A_0 \to B_0 \to C_0) \rightarrowtail (A_1 \to B_1 \to C_1) \rightarrowtail \cdots \rightarrowtail (A_n \to B_n \to C_n)$

This can be decomposed into the filtration on the $A$ -components and the $C$ -components independently, using the cofibration axioms and the fact that cofibrations in $E(\mathcal{C})$ are cofibrations on each component.

Step 4: Realization argument. The bisimplicial identity $wS_\bullet E(\mathcal{C}) \simeq wS_\bullet \mathcal{C} \times wS_\bullet \mathcal{C}$ follows from the decomposition in Step 3 and the "realization lemma": a map of bisimplicial sets that is a weak equivalence on each row (or column) is a weak equivalence on realizations.

More precisely, the functor $(s, q)$ on $S_n$ -level gives:

$wS_n E(\mathcal{C}) \to wS_n \mathcal{C} \times wS_n \mathcal{C}$

which is a homotopy equivalence for each $n$ (by induction on $n$ using the cofibration structure). Taking geometric realization over $n$ preserves the equivalence.

Step 5: Passage to loop spaces. Since $K(\mathcal{C}) = \Omega|wS_\bullet \mathcal{C}|$ and $K(E(\mathcal{C})) = \Omega|wS_\bullet E(\mathcal{C})|$ , the equivalence on $|wS_\bullet|$ gives $K(E(\mathcal{C})) \simeq K(\mathcal{C}) \times K(\mathcal{C})$ . $\square$

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Consequences

ExampleEuler characteristic

The additivity theorem implies that K-theory is an "additive invariant": for any exact sequence $A \rightarrowtail B \twoheadrightarrow C$ , we have $[B] = [A] + [C]$ in $K_0$ . This is the algebraic version of the Euler characteristic formula:

$\chi(B) = \chi(A) + \chi(C)$

for short exact sequences of complexes. More generally, for a bounded complex $C^\bullet$ with homology $H^i(C^\bullet)$ :

$[C^\bullet] = \sum_i (-1)^i [C^i] = \sum_i (-1)^i [H^i(C^\bullet)] \in K_0$

(assuming all terms are in the Waldhausen category).

RemarkFibration theorem

The additivity theorem implies Waldhausen's fibration theorem: if $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ is a "short exact sequence" of Waldhausen categories (meaning the functor from $\mathcal{A}$ to the "fiber" of $\mathcal{B} \to \mathcal{C}$ is a K-theory equivalence), then there is a homotopy fiber sequence:

$K(\mathcal{A}) \to K(\mathcal{B}) \to K(\mathcal{C}).$

This gives long exact sequences in K-groups:

$\cdots \to K_{n+1}(\mathcal{C}) \xrightarrow{\partial} K_n(\mathcal{A}) \to K_n(\mathcal{B}) \to K_n(\mathcal{C}) \to \cdots$

The Thomason--Trobaugh localization sequence is a special case, applied to $\operatorname{Perf}_Z(X) \to \operatorname{Perf}(X) \to \operatorname{Perf}(U)$ .

ExampleSplitting and cofinality

The additivity theorem implies:

Eilenberg swindle: If $\mathcal{C}$ has countable coproducts, then $K(\mathcal{C}) \simeq *$ (contractible), because $[A] = [A \oplus B] - [B]$ and $[A \oplus B] = [A] + [B]$ , giving $[A] = 0$ for all $A$ .
Cofinality theorem: If $\mathcal{A} \subseteq \mathcal{B}$ is cofinal (every $B \in \mathcal{B}$ is a summand of some $A \in \mathcal{A}$ ), then $K_n(\mathcal{A}) \cong K_n(\mathcal{B})$ for $n \geq 1$ and $K_0(\mathcal{A}) \hookrightarrow K_0(\mathcal{B})$ with cokernel determined by the "missing" summands.
Agreement principle: For two Waldhausen category structures on the same category that have the same weak equivalences, the K-theories agree. This is because additivity forces the K-theory to depend only on the "homological" structure.