nLab Waldhausen S-construction




The Waldhausen S S_\bullet-construction is a procedure that produces algebraic K-theory (as an infinite loop space or connective spectrum) from a category or (infinity,1)-category equipped with a Waldhausen structure?.

The construction can take as input any of the following:


For Waldhausen categories

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…)


For Waldhausen categories

  • F. Waldhausen, Algebraic K-theory of spaces, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, pdf.

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

  • R. W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, 1990, 247-435.

or in section 1 of

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

For Waldhausen (infinity,1)-categories and stable (infinity,1)-categories


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

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.

Interpretation in terms of a categorified Dold-Kan correspondence is discussed in

Last revised on October 25, 2017 at 13:49:37. See the history of this page for a list of all contributions to it.