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
is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for
(The notation "+" is meant to suggest the construction adds more to the classifying space BC.)
One puts 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: The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context.
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.
Let QC be the category whose objects are the same as those of C and morphisms from X to Y are isomorphism classes of diagrams
such that the first arrow is an admissible epi and the second admissible mono and two diagrams are isomorphic if they differ only at the middle and there is an isomorphism between them.
is the classifying space of the category QC (geometric realization of the nerve).
As it turns out, it is uniquely defined up to homotopy equivalence (so the notation is justified.)
is the category of finitely generated projective modules over R. One can easily show this map (called transfer) agrees with one defined in Milnor's Introduction to algebraic K-theory.
[2] The construction is also compatible with the suspension of a ring (cf.
A theorem of Daniel Quillen states that, when C is the category of finitely generated projective modules over a ring R,
is the i-th K-group of R in the classical sense for
Weibel 2013) relies on an intermediate homotopy equivalence.
If S is a symmetric monoidal category in which every morphism is an isomorphism, one constructs (cf.
that generalizes the Grothendieck group construction of a monoid.
Let C be an exact category in which every exact sequence splits, e.g., the category of finitely generated projective modules, and put
, the subcategory of C with the same class of objects but with morphisms that are isomorphisms in C. Then there is a "natural" homotopy equivalence:[3] The equivalence is constructed as follows.
Let E be the category whose objects are short exact sequences in C and whose morphisms are isomorphism classes of diagrams between them.
, which is a subcategory, consists of exact sequences whose third term is X.
We now take C to be the category of finitely generated projective modules over a ring R and shows that
The image actually lies in the identity component of
be the full subcategory of S consisting of modules isomorphic to
be the component containing R. Then, by a theorem of Quillen, Thus, a class on the left is of the form
is an isomorphism for any local coefficient system L on X, then Proof: The homotopy type of
Thus, we can replace the hypothesis by one that Y is simply connected and
(Coincidentally, by reversing argument, one can say this implies
says: An inspection of this spectral sequence gives the desired result.