MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Algebraic K-theory of an exact category $\mathcal{C}$ is a certain universal non-connective spectrum $K(\mathcal{C})$. In particular, objects of $\mathcal{C}$ give elements of $K_0(\mathcal{C})$.

There are models for the delooping of $K(\mathcal{C})$, i.e. spectra $X(\mathcal{C})$ such that $\Omega X(\mathcal{C})\cong K(\mathcal{C})$. For instance, one can take $X(\mathcal{C})$ to be the K-theory of Tate objects in $\mathcal{C}$ (see This, in particular, explains that Tate objects of $\mathcal{C}$ give rise to elements of $K_{-1}(\mathcal{C})$.

Are there models for the looping of K-theory? More precisely, is there an exact category $\mathcal{D}$ constructed from $\mathcal{C}$ such that $K(\mathcal{D})\cong \Omega K(\mathcal{C})$?

I am mainly interested in the case when $\mathcal{C}$ is the category of finitely-generated projective modules over a ring $R$.

share|cite|improve this question
up vote 9 down vote accepted

The paper comes close to answering your question, but instead of yielding an exact category $\mathcal D$ as requested, it yields a split pair of exact categories. That's just as good, I think.

share|cite|improve this answer

Yes. Apart from the different natural constructions of loop spaces for arbitrary simplicial sets. You have a specific construction of Gillet and Grayson published in:

MR0909784 Reviewed Gillet, Henri; Grayson, Daniel R. The loop space of the Q-construction. Illinois J. Math. 31 (1987), no. 4, 574–597. (Reviewer: A. J. Berrick) 18F25 (19D06)

with erratum:

MR2007234 Reviewed Gillet, Henri; Grayson, Daniel R. Erratum to: "The loop space of the Q-construction'' [Illinois J. Math. 31 (1987), no. 4, 574–597; MR0909784 (89h:18012)]. Illinois J. Math. 47 (2003), no. 3, 745–748. 18F25 (19D06)

Simplices are pairs of filtered objects $(A_0\subset\cdots\subset A_n,B_0\subset\cdots\subset B_n)$ with the same subquotients $A_i/A_j=B_i/B_j$. Simplicial operators are defined as in the nerve of a category.

I'd like to remark that, although the same construction makes sense for Waldhausen categories, it doesn't produce a looping in that case, in general.

share|cite|improve this answer
I think this answers a slightly different question. The Gillet--Grayson construction yields a simplicial set $G.\mathcal{C}$ with a natural homotopy equivalence $|G.\mathcal{C}| \simeq \Omega K(\mathcal{C})$, but not another exact category $\mathcal{D}$ such that $K(\mathcal{D}) \simeq \Omega K(\mathcal{C})$. – Tom Harris Apr 24 '14 at 19:56
You're right. I misunderstood the question. Should I erase this non-answer? – Fernando Muro Apr 24 '14 at 20:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.