Can any suspension spectrum be realized as Waldhausen K-theory?

Question

If we consider the category of finite, pointed sets and declare cofibrations to be inclusions and weak equivalences to be bijections, we get a Waldhausen category whose $K$-theory spectrum is the sphere spectrum.

Given a pointed space $X$, is there a nice description (analogous to the one above) of a Waldhausen category whose $K$-theory spectrum is the suspension spectrum of $X$?

My first guess is something like finite, pointed sets equipped with a pointed map to $X$, but I don't really know enough about $K$-theory to prove that this guess is correct (if it even is....)

Answer 1

(1) Given a simplicial monoid $G$ let $R^0(*, G)$ be the Waldhausen category of pointed finite free $G$-simplicial sets weakly equivalent to $(\coprod^k G)_+$, for varying $k\ge0$. This is the special case $n=0$ of the notation from sections 2.1 and 2.2 of Waldhausen's ''Algebraic $K$-theory of spaces''. He obtains a homotopy equivalence $$ |hR^0_k(*, G)| \simeq BH^0_k(G) = B(\Sigma_k \ltimes |G|^k) , $$ so by Segal's theorem $$ \Omega |h N_\bullet R^0(*, G)| \simeq Z \times colim_k |hR^0_k(*, G)|^+ $$ and the Barratt-Priddy-Quillen-Segal theorem $$ Z \times colim_k B(\Sigma_k \ltimes |G|^k)^+ \simeq Q(B|G|_+) $$ you know that the ``direct sum'' $K$-theory $\Omega |h N_\bullet R^0(*, G)|$ of $R^0(*, G)$ is a model for $Q(B|G|)_+)$. I think the natural map $$ |hN_\bullet R^0(*, G)| \to |hS_\bullet R^0(*, G)| $$ is an equivalence, since all cofibrations are split in this case, but I don't have a reference at hand.

For a space $X$, let $R^0(X)$ be the Waldhausen category of finite retractive spaces over $X$, homotopy equivalent to $X$ disjoint union finitely many points (or $0$-cells, if you like). If $X \simeq B|G|$, there is a homotopy equivalence $$ |hS_\bullet R^0(X)| \simeq |hS_\bullet R^0(*, G)| , $$ so $R^0(X)$ should do the trick for you, also if $X$ is not connected.

(2) For simplicial sets $X \colon [q] \mapsto X_q$ there is a different construction, in an unpublished preprint of Igusa-Waldhausen (my copy is from 1991), of a simplicial Waldhausen category $C_0(X)$ that in degree $q$ is given by a non-obvious Waldhausen category of finite sets over $X_q$. Their Corollary 1.5 is an equivalence $$ \Omega |iS_\bullet C_0(X)| \simeq Q(|X|_+) . $$