In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups Ki(C) associated to it. If C is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on C a structure of an exact category, or of a Waldhausen category, or of a dg-category, or possibly some other variants. Thus, there are several constructions of those groups, corresponding to various kinds of structures put on C. Traditionally, the K-theory of C is defined to be the result of a suitable construction, but in some contexts there are more conceptual definitions. For instance, the K-theory is a 'universal additive invariant' of dg-categories and small stable ∞-categories. The motivation for this notion comes from algebraic K-theory of rings. For a ring R Daniel Quillen in Quillen (1973) introduced two equivalent ways to find the higher K-theory. The plus construction expresses Ki(R) in terms of R directly, but it's hard to prove properties of the result, including basic ones like functoriality. The other way is to consider the exact category of projective modules over R and to set Ki(R) to be the K-theory of that category, defined using the Q-construction. This approach proved to be more useful, and could be applied to other exact categories as well. Later Friedhelm Waldhausen in Waldhausen (1985) extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces.
K-theory of Waldhausen categories In algebra, the S-construction is a construction in algebraic K-theory that produces a model that can be used to define higher K-groups. It is due to Friedhelm Waldhausen and concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be thought of as analogous to a monomorphism, and a category with cofibrations is one in which, roughly speaking, monomorphisms are stable under pushouts. According to Waldhausen, the "S" was chosen to stand for Graeme B. Segal. Unlike the Q-construction, which produces a topological space, the S-construction produces a simplicial set.
Details The arrow category A r ( C ) {\displaystyle Ar(C)} of a category C is a category whose objects are morphisms in C and whose morphisms are squares in C. Let a finite ordered set [ n ] = { 0 < 1 < 2 < ⋯ < n } {\displaystyle [n]=\{0<1<2<\cdots <n\}} be viewed as a category in the usual way. Let C be a category with cofibrations and let S n C {\displaystyle S_{n}C} be a category whose objects are functors f : A r [ n ] → C {\displaystyle f:Ar[n]\to C} such that, for i ≤ j ≤ k {\displaystyle i\leq j\leq k} , f ( i = i ) = ∗ {\displaystyle f(i=i)=*} , f ( i ≤ j ) → f ( i ≤ k ) {\displaystyle f(i\leq j)\to f(i\leq k)} is a cofibration, and f ( j ≤ k ) {\displaystyle f(j\leq k)} is the pushout of f ( i ≤ j ) → f ( i ≤ k ) {\displaystyle f(i\leq j)\to f(i\leq k)} and f ( i ≤ j ) → f ( j = j ) = ∗ {\displaystyle f(i\leq j)\to f(j=j)=*} . The category S n C {\displaystyle S_{n}C} defined in this manner is itself a category with cofibrations. One can therefore iterate the construction, forming the sequence S ( m ) C = S ⋯ S C {\displaystyle S^{(m)}C=S\cdots SC} . This sequence is a spectrum called the K-theory spectrum of C.
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