# Q-construction

## From Wikipedia, the free encyclopedia

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 $B^{+}C$ so that $\pi _{0}(B^{+}C)$ is the Grothendieck group of *C* and, when *C* is the category of finitely generated projective modules over a ring *R*, for $i=0,1,2$, $\pi _{i}(B^{+}C)$ 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

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

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:

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

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 $\pi _{*}$ of $B^{+}$ 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.