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This question has been on my mind for a while. As I understand it, the Q-construction was the first definition for higher algebraic K-theory. Some details can be found here.


I have always wondered what train of thought led Quillen to come up with this definition. Does anyone know an interpretation of the Q-construction that makes it seem natural?

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For the development and underlying intuitions of the several K-theory constructions see Grayson's recent artcle: math.uiuc.edu/K-theory/1018 –  Peter Arndt Dec 10 '11 at 16:35

4 Answers 4

I've always liked the interpretation Quillen gave in his "On the group completion of a simplicial monoid" paper (Appendix Q in Friedlander-Mazur's "Filtrations on the homology of algebraic varieties"). Here is a somewhat revisionist version.

Associated to a monoidal category C, you can take its nerve NC, and the monoidal structure gives rise to a coherent multiplication (an A-space structure) on NC. (If you work a little harder you can actually convert it into a topological monoid.)

May showed in his paper "The geometry of iterated loop spaces" that an A-space structure on X is exactly the structure you need to produce a classifying space BX, and there is a natural map from X to the loop space Omega(BX) that is a map of A-spaces, and is a weak equivalence if and only if π0(X) was a group rather than a monoid using the A-monoid structure. In fact, Omega(BX) satisfies this property, and so you can think of it as a "homotopy theoretic" group completion of the coherent monoid X.

What Quillen showed was that you can recognize the homotopy theoretic group completion in the following way: the homotopy group completion of X has homology which is the localization of the homology ring of X by inverting the images of π0(X) in H0(X). Moreover, the connected component of the identity in the homotopy group completion is a connected H-space, so its fundamental group is abelian and acts trivially on the higher homotopy groups.

In particular, if X is the nerve of the category of finitely generated free modules over a ring R, then X is homotopy equivalent to a disjoint union of the classifying spaces BGLn(R), with monoidal structure induced by block sum. The monoid π0(R) is the natural numbers N, and so you can consider the map

X = coprod_(n∈N) BGL_n(R) → coprod_(n∈Z) BGL(R)

to a union of copies of the infinite classifying space. This map induces the localization of H*(X), so the space on the right has to have the same homology as the homotopy group completion, but the problem is that the connected component of the identity on the right (BGL(R)) doesn't have an abelian fundamental group that acts trivially on the higher homotopy groups, so this can't be the homotopy group completion yet.

So this leads to the plus-construction: to find the homotopy group completion you're supposed to take BGL(R) and produce a new space, which has to have the same homology as BGL(R), and which has an abelian fundamental group (plus stuff on higher homotopy groups). This is what the plus-construction does for you.

Quillen's Q-construction contains within it the symmetric monoidal nerve construction (you can consider just the special exact sequences that involve direct sum inclusions and projections), but it's got the added structure that it "breaks" exact sequences for you. I wish I could tell you how Quillen came up with this, but this is the best I can do.

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man. I wish underscores didn't try to italicize things. –  Tyler Lawson Oct 18 '09 at 12:41
they don't inside ` . Also, you can use <sub>` –  Ben Webster Oct 18 '09 at 14:15
Whoops, stupid lack of comment in previews. If you put things inside a pair of ``` they won't italicize. –  Ben Webster Oct 18 '09 at 14:16
@Tyler: I took the liberty of replacing symbols like oo, pi, and in with their Unicode analogs, if you don't mind. –  Dmitri Pavlov Dec 10 '11 at 14:20

Quillen's $Q$-construction naturally arises as a cleaned-up version of Segal's edgewise subdivision of Waldhausen's $s_\bullet$-construction.

Waldhausen's $s_\bullet$-construction gives a rather natural definition of the algebraic $K$-theory space of an exact category $C$. It is a simplicial set $s_\bullet C$, whose elements in degree $q\ge0$ are ``length $q$ filtrations'' $0 \rightarrowtail M_1 \rightarrowtail \dots \rightarrowtail M_q$ in $C$, plus some choices. The algebraic $K$-theory space $K(C)$ is defined as the loop space $\Omega |s_\bullet C|$ of the geometric realization of this simplicial set.

My understanding is that Quillen knew of this construction, but was unhappy with the passage from categories to simplicial sets. At some point he realized that even if $s_\bullet C$ is not the nerve of a category, its Segal subdivision $Sd(s_\bullet C)$ is (basically) the nerve of a category. That category is the $Q$-construction, $Q(C)$.

Since Segal subdivision does not change the homotopy type, the loop space $\Omega |Q(C)|$ of the classifying space of $Q(C)$ is as good a definition for $K(C)$ as the one I first stated.

Some people would write $BQ(C)$ where I write $|Q(C)|$. See Section 1.9 of

Waldhausen, Friedhelm, Algebraic $K$-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318--419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.

for details.

  • John
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I'd never heard that Quillen was aware of the S.-construction. On the other hand, it's very similar to Segal's construction of the Gamma-space associated to a category with sums, which Segal almost certainly learned from Quillen. So maybe in the end this isn't a huge surprise. But it does go a long way towards explaining where the Q-construction came from! –  Dan Ramras Sep 30 '10 at 19:52
Waldhausen's construction came after Quillen's. As it happens, in one of the few conversations I ever had with Quillen (this was in the mid-90s, in the course of a car ride from Cambridge to Providence) he asked me to tell him something about Waldhausen K-theory, and I remember telling him that the S. construction is related to his Q-construction by edgewise subdivision. –  Tom Goodwillie Dec 11 '11 at 16:26
I seem to recall that in Segal's paper about Gamma-spaces he modestly describes what he is doing as developing some of Quillen's ideas about algebraic K-theory. –  Tom Goodwillie Dec 11 '11 at 16:28

There is an altogether different motivation different from the ones discussed above that appears in a paper by Graeme Segal ("K-homology and algebraic K-theory," LNM 575 K-theory and Operator Algebras, Athens Georgia 1975, pp. 113–127).

The $Q$-construction there is motivated by considering self-adjoint Fredholm operators on Hilbert space.

More, precisely Segal shows that the homotopy type of the classifying space of Q-construction of the category of finite dimensional vector spaces over the reals or complex numbers is the same as that of the space $Saf(H)$ consisting of self-adjoint Fredholm operators on infinite dimensional Hilbert space $H$: $$ BQC \simeq Saf(H) . $$ A map $V\to V'$ in the $Q$-construction on the category $C =$ Vect of finite dimensional vector spaces is represented as a triple $(W_+,W_-;\alpha)$ in which $\alpha: W_+\oplus V\oplus W_- \to V'$ is an isomorphism of vector spaces.

According to Segal, the idea is supposed to be that a Fredholm operator is determined up to contractible choice by its kernel and cokernel, which are a pair of finite dimensional vector spaces. When a Fredholm operator is deformed continously, its kernel and cokernel can jump but only by adding isomorphic pieces to each.

In the self adjoint case, the operator is determined by its kernel up to contractible choice. The kernel then corresponds to an object of the $Q$-construction. When the operator is deformed, the kernel jumps in such a way that the part added to it is a direct sum of the part on which the operator was positive and a part on which it was negative. These correspond to the terms $W_\pm$ appearing above.

So the heuristic motivation in a nutshell is this: the objects of the $Q$-construction correspond to self-adjoint Fredholm operators and the morphisms correspond to deformations of such operators. The passage is given by taking operator kernels.

(note: I think Segal wants to consider the above $C$ as a topological category).

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Expanding on Tyler Lawson's comment on the Q-construction, I would say the following. The K0 functor involves two processes, a group completion (of the monoid structure given by direct sum) and an identification of all the extensions of any two objects. That is, K_0 of an exact category E is the free abelian group generated by the objects of E (group completion) together with the relations [B] = [A] + [C] for every exact sequence

A >---> B --->> C

Now, the higher K-theory is a sort of categorification, making both processes above remember higher homotopical data. Then we make the definition

Ki(E) = pii Omega BQE

Here Omega B corresponds to a homotopical group completion as explained by Tyler. Quillen's Q-construction changes the morphisms of the category E in a way that when group-completed, "the extensions become split" (giving the relation [B] = [A] + [C] above). Strictly speaking, the Q-construction does not make exact sequences split in QE, as QE has the same isomorphisms as E, only the non-isos change: a morphism A ---> B in QE correspond to an identification of A with a subquotient of B.

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