Q-construction

In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category C, the construction creates a topological space ${\displaystyle B^{+}C}$ so that ${\displaystyle \pi _{0}(B^{+}C)}$ is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for ${\displaystyle i=0,1,2}$, ${\displaystyle \pi _{i}(B^{+}C)}$ is the i-th K-group of R in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts

${\displaystyle K_{i}(C)=\pi _{i}(B^{+}C)}$

and call it the i-th K-group of C. Similarly, the i-th K-group of C with coefficients in a group G is defined as the homotopy group with coefficients:

${\displaystyle K_{i}(C;G)=\pi _{i}(B^{+}C;G)}$.

The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant algebraic K-theory as ${\displaystyle \pi _{*}}$ of ${\displaystyle B^{+}}$ of the category of equivariant sheaves on a scheme.

Waldhausen's S-construction generalizes the Q-construction in a stable sense; in fact, the former, which uses a more general Waldhausen category, produces a spectrum instead of a space. Grayson's binary complex also gives a construction of algebraic K-theory for exact categories.^[1] See also module spectrum#K-theory for a K-theory of a ring spectrum.