University of David
I shall not say much about Quillen’s refoundation of algebraic K-theory here, as so much has been written about it elsewhere. As he explained it in 1969–1970, one key starting point was the calculation of the homology of BGL1(Fp), and another was when he noticed that the known Pontrjagin ring of the union of the classifying spaces of the symmetric groups essentially coincided with the also-known Pontrjagin ring of Ú1S1, the infinite loop space of the infinite sphere. This led him to the idea that from a category with a suitable operation of “sum”—such as the category of finite sets under disjoint union, or of modules over a ring under the direct sum—one can obtain a cohomology theory if, instead of forming the Grothendieck group from the semigroup of isomorphismclasses, one constructs in the homotopy category the group completion of the topological semigroup which is the space of the category. The famous “plus construction”, which he used in his 1970 ICM talk, is a nice way to realize the group completion concretely; it came from a suggestion of Sullivan, but I do not think it was the basic idea. Throughout his year in Princeton, Quillen was making lightning progress understanding the homotopy theory of categories, which he had not much thought about before. He realized that he must find a homotopy version of the more general construction of Grothendieck groups in which the relations come from exact sequences rather than just from direct sums, and eventually he settled on the “Q-construction” as his preferred method of defining the space. The culmination of this work was the definitive treatment he wrote for the 1972 Seattle conference on algebraic K-theory. He published only one paper on algebraic K-theory after that: his proof in 1976 of Serre’s conjecture that projective modules over polynomial rings are free. This came fromreflecting deeply onwhatwas already known about the question—especially the work of Horrocks—and seeing that, when brewed lovingly in the way Grothendieck advocated for opening nuts, the result fell out.