The Waldhausen S-construction on a Waldhausen category produces a simplicial set equivalent to the K-theory spectrum (see below) of the simplicial localization of : it is a concrete algorithm for computing K-theory spectra.
Recall from the definition at K-theory that the K-theory spectrum of the (∞,1)-category is the diagonal simplicial set on the bisimplicial set of sequences of morphisms in and equivalences between these (the core of the Segal space induced by ).
objects are sequences of Waldhausen cofibrations;
morphisms are collections of morphisms that commute with all diagrams in sight.
Then one finds that the realization of the bisimplicial set 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 . The corresponding sequence of thick topological realizations is a spectrum
this is the S-construction of the Waldhausen K-theory spectrum of .
(… roughly at least, need to polish this, see link below meanwhile…)
The Waldhausen S-construction is recalled for instance in section 1 of
A combinatorial construction of symmetries due 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.