the right. The degeneracy maps are $$s_i= \id_R \smash ... \smash \id_R \smash \eta \smash \id_R \smash \dots \smash \id_R$$ where the unit map $\eta:S\rightarrow R$ from the sphere spectrum is in the $i$-th position.

Using the May filtration in each simplicial degree, we produce a filtration of simplicial objects in $\mathfrak{S}$ $$\xymatrix{
\vdots \ar[d] & \vdots \ar[d] & \vdots \ar[d] & \\
 \mathcal{M}^{X_0}_2(I_{\bullet}) \ar[r]\ar[d] & 
 \mathcal{M}^{X_1}_2(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
 \mathcal{M}^{X_2}_2(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
 \mathcal{M}^{X_0}_1(I_{\bullet}) \ar[r]\ar[d] & 
 \mathcal{M}^{X_1}_1(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r]\ar[d] &
 \mathcal{M}^{X_2}_1(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r]\ar[d] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] \\
 \mathcal{M}^{X_0}_0(I_{\bullet}) \ar[r] & 
 \mathcal{M}^{X_1}_0(I_{\bullet}) \ar@<1ex>[l] \ar@<-1ex>[l] \ar@<1ex>[r] \ar@<-1ex>[r] &
 \mathcal{M}^{X_2}_0(I_{\bullet}) \ar@<2ex>[l] \ar@<-2ex>[l] \ar[l] \ar@<2ex>[r] \ar@<-2ex>[r] \ar[r] &
 \dots \ar@<-3ex>[l] \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<3ex>[l] 
}$$ and when $X_{\bullet}=S^1_{\bullet}$ the bottom row is the simplicial object whose geometric realization is $THH(I_0)$.

\[def of thh-may ss\] If $I$ is a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, $S^1_{\bullet}$ is the minimal simplicial model for the circle, and $H_*$ is a connective generalized homology theory then the [*topological Hochschild-May spectral sequence*]{} is the spectral sequence obtained by applying $H_*$ to the tower of cofiber sequences $$\label{tower 