of sequences} \xymatrix{ 
\vdots \ar[d] & \\
\left| \mathcal{M}_2^{S_{\bullet}}(I)\right| \ar[r]\ar[d] & \left| \mathcal{M}_2^{S_{\bullet}}(I)\right|/\left| \mathcal{M}_3^{S^{\bullet}}(I)\right| \\
\left| \mathcal{M}_1^{S_{\bullet}}(I)\right| \ar[r]\ar[d] & \left| \mathcal{M}_1^{S_{\bullet}}(I)\right|/\left| \mathcal{M}_2^{S_{\bullet}}(I)\right| \\
\left| \mathcal{M}_0^{S_{\bullet}}(I )\right| \ar[r] & \left| \mathcal{M}_0^{S_{\bullet}}(I )\right|/\left| \mathcal{M}_1^{S_{\bullet}}(I )\right|  .}$$ That is, it is the spectral sequence of the exact couple $$\xymatrix{ 
D^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right| \ar[rr] & & \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right| \ar[ld] \\
 & E^1_{*,*} \cong \bigoplus_{i,j} H_i \left| \mathcal{M}_j^{S_{\bullet}}(I )\right|/\left| \mathcal{M}_{j+1}^{S_{\bullet}}(I )\right| \ar[ul] & . }$$

We need to know that the map $$\left| \mathcal{M}_n^{S_{\bullet}}(I )\right| \lra \left| \mathcal{M}_{n-1}^{S_{\bullet}}(I )\right|$$ is cofibration, and this relies on a theorem of the author and A. Salch [@asReedy], which states that a map between two “good" simplicial objects in $\mathfrak{S}$ which is a flat cofibration at each simplicial level realizes to a cofibration. This is the one part of this paper that is model dependent and uses properties of the category of symmetric spectra in simplicial sets with the flat model structure. Otherwise, all the constructions are sufficiently general to work in any nice enough model category including the categories of $\Gamma$-spaces, symmetric spectra, $S$-modules, and orthogonal spectra.

Consequently, the sequence $$\dots \lra \left | \mathcal{M}_2^{S_{\bullet}}(I)\right| \lra \left| \mathcal{M}_{1}^{S_{\bullet}}(I)\right|\lra  \left| \mathcal{M}_{0}^{S_{\bullet}}(I)\right|$$ is 