S-CONSTRUCTION // COFIBER SEQUENCES // COFINAL SUBCATEGORIES // SATURATION
Waldhausen's S-construction builds algebraic K-theory from categories with cofibrations and weak equivalences (Waldhausen categories). Given a Waldhausen category C, the simplicial set S•C has S_n C = sequences of cofibrations:
* → A_1 ↪ A_2 ↪ ... ↪ A_n
with chosen quotients A_{ij} = A_j/A_i. The K-theory space is K(C) = Ω|wS•C|.
A cofibration sequence in C is A ↪ B ↠ B/A where the arrow A ↪ B is a cofibration (satisfies certain extension/pushout axioms) and B/A is the cofiber (quotient).
cof(A,B) : A ↪ B ↠ C with C ≃ B/A
A full subcategory D ⊆ C is cofinal if for every X ∈ C, there exists Y ∈ D with X ↪ Y. Cofinal inclusions induce K-theory equivalences on K_0.
The saturation axiom says: if f, g are composable morphisms and two of f, g, gf are weak equivalences, then so is the third. This 2-of-3 property ensures the localization behaves well.