are the face and degeneracy maps of the simplicial finite set $X_{\bullet}$. Note that the coproduct in commutative ring spectra is the smash product and we are using that fact here. Similarly, the degeneracy map $$s_i: (X_{\bullet} \tilde{\otimes} R)_{n-1}\lra (X_{\bullet} \tilde{\otimes} R)_{n}$$ is given on each summand by the map $$R\{t\} \lra \bigwedge_{s\in X_n} R\{s\}$$ where $R\{t\}$ includes as the smash factor corresponding to $\sigma_i(t)\in X_n$.

In the case where $X_{\bullet}=\Delta[1]/\delta \Delta[1]=:S^1_{\bullet}$, the minimal simplicial model for the circle, $$S^1_{\bullet} \otimes R$$ is the geometric realization of the simplicial object in $\mathfrak{S}$ $$\label{}
S^1_{\bullet} \tilde{\otimes} R := \left \{  \xymatrix{ R  & \ar@<1ex>[l]\ar@<-1ex>[l] 
 R\wedge R &                         
 \ar@<2ex>[l]\ar@<-2ex>[l]  \ar[l]R\wedge R \wedge R& 
    \ar@<3ex>[l]\ar@<-3ex>[l]\ar@<1ex>[l]\ar@<-1ex>[l]  \dots   }  \right \}$$ with face and degeneracy maps given by the following formulas: the face maps are $$d_i=\left \{ \begin{array}{ll} 
 \id_{R} \smash ... \id_{R}  \smash \mu \smash \id_{R} \smash ... \smash \id_{R}  & \text{ if } i <n \\ 
(\mu \smash   \id_{R}   \smash \dots   \smash \id_{R}) \circ t_{n}  &  \text{ if } i=n\end{array} \right .$$ where the multiplication map $\mu:R\wedge R\rightarrow R$ is in the $i$-th position on the first line and $t_n: R^{\wedge n} \rightarrow R^{\wedge n}$ is the map that cyclicly permutes the factors to 