again a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, which we will call $\left | \mathcal{M}^{S_{\bullet}}(I) \right |$, and therefore, we can define $E_0^*\left | \mathcal{M}^{S_{\bullet}}(I) \right | $. The input of the THH-May spectral sequence as defined is $H_*\left (E_0^*\left| \mathcal{M}^{S_{\bullet}}(I)\right | \right )$. The main theorem of [@thhmay] produces a more computable $E_2$-page in a more general setting.

Let $X_{\bullet}$ be a simplicial finite set and $I$ be a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, then there is a weak equivalence in the category $\Comm \mathfrak{S}$ $$E_0^*\left| \mathcal{M}^{X_{\bullet}}(I)\right | \simeq  X_{\bullet} \otimes E_0^* I.$$

See [@thhmay] for a detailed proof.

We therefore produce a spectral sequence $$\label{thh map ss} E^2_{s,t} = H_{s,t}(THH(E_0^*(I)))\Rightarrow H_s (THH(I_0))$$ with differential $$d_r : E^r_{s,t}  \lra E^r_{s-1,t+r}$$ for any connective generalized homology theory $H_*$. The spectral sequence strongly converges as long as $\pi_k(I_i)\cong 0$ for $k<i$ and the differentials satisfy a Leibniz rule [@thhmay].

\[ thh-may ss with coeff\] Given a cofibrant decreasingly filtered commutative monoid $I$ and a cofibrant $I$-module $M$ in the category $\mathfrak{S}^{\mathbb{N}^{\op}}$, we can also construct a THH-May spectral sequence with coefficients and prove an analogous fundamental theorem of the May filtration with coefficients. This produces a spectral sequence 