nLab
Waldhausen S-construction

Contents

Idea

The general procedure in K-theory is to assign an algebraic K-theory spectrum K(C)\mathbf{K}(C) to a stable (∞,1)-category CC.

In practice these stable (,1)(\infty,1)-categories are usually presented by homotopical categories called Waldhausen categories.

The Waldhausen S-construction on a Waldhausen category CC' produces a simplicial set equivalent to the K-theory spectrum (see below) of the simplicial localization CC of CC': it is a concrete algorithm for computing K-theory spectra.

Definition

Recall from the definition at K-theory that the K-theory spectrum K(C)K(C) of the (∞,1)-category CC is the diagonal simplicial set on the bisimplicial set Core(Func(Δ n,C))Core(Func(\Delta^n,C)) of sequences of morphisms in CC and equivalences between these (the core of the Segal space induced by CC).

The Waldhausen S-construction mimics precisely this: for CC' a Waldhausen category for every integer nn define a simplicial set S nCS_n C' to be the nerve of the category whose

  • objects are sequences 0A 0,1A 0,n0 \hookrightarrow A_{0,1} \hookrightarrow \cdots \hookrightarrow A_{0,n} of Waldhausen cofibrations;

    • together with choices of quotients A ij=A 0,j/A 0,iA_{i j} = A_{0, j}/ A_{0,i}, i.e. cofibration sequences A 0,iA 0,jA ijA_{0,i} \to A_{0,j} \to A_{i j}
  • morphisms are collections of morphisms {A i,jB i,j}\{A_{i,j} \to B_{i,j}\} that commute with all diagrams in sight.

Then one finds that the realization of the bisimplicial set S CS_\bullet C' with respect to one variable is itself naturally a topological Waldhausen category. Therefore the above construction can be repeated to yield a sequence of topological categories S (n)CS^{(n)}_\bullet C'. The corresponding sequence of thick topological realizations is a spectrum

K(C) n=|S (n)C| \mathbf{K}(C)_n = |S^{(n)}_\bullet C'|

this is the S-construction of the Waldhausen K-theory spectrum of CC'.

(… roughly at least, need to polish this, see link below meanwhile…)

References

The Waldhausen S-construction is recalled for instance in section 1 of

  • Paul D. Mitchener, Symmetric Waldhausen K-theory spectra of topological categories (pdf)

A combinatorial construction of symmetries due Nadler has a relation to the S-construction in a special case:

  • David Nadler, Cyclic symmetries of A nA_n-quiver representations, arxiv/1306.0070

This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen’s S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an immediate solution.

Revised on May 29, 2014 02:53:01 by Urs Schreiber (88.128.80.76)