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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 http://arxiv.org/abs/1203.0831). 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 $R$.

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2 Answers 2

up vote 6 down vote accepted

The paper http://www.math.uiuc.edu/~dan/cv.xhtml#binary 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.

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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.

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1  
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 at 19:56
    
You're right. I misunderstood the question. Should I erase this non-answer? –  Fernando Muro Apr 24 at 20:16

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