so that for $x\in \mathbb{N}^S$ $$|x| = \Sigma_{s\in S} x(s).$$ We then define a sub-poset of $\mathbb{N}^S$ by $$\mathcal{D}_n^S : = \{ x\in \mathbb{N}^S ; |x|\ge n \}.$$ Let $I$ be a cofibrant decreasingly filtered commutative monoid. Define a functor $$\mathcal{F}^S(I) : (\mathbb{N}^{S})^{\op} \lra \mathfrak{S}$$ on objects by $\mathcal{F}^S(I)(x)=\bigwedge_{s\in S} I_{x(s)}$ for $x\in \mathbb{N}^{S}$ and on morphisms in the apparent way. We can precompose the functor with the inclusion to produce a functor $$\xymatrix{\mathcal{F}_n^S(I):(\mathcal{D}_n^S)^{\op} \ar[r] &  (\mathbb{N}^{S})^{\op} \ar[r]^{\mathcal{F}^S(I)} & \mathfrak{S} .}$$ We then define the May filtration associated to the finite set $S$ to be the collection of objects $$\mathcal{M}_n^S(I) := \colim \mathcal{F}_n^S(I)$$ in $\mathfrak{S}$ for $n\in \mathbb{N}$ along with maps $$\mathcal{M}_n^S(I) \lra \mathcal{M}_{n-1}^S(I)$$ given by precomposing with the inclusion $(\mathcal{D}_n^S)^{\op}\lra (\mathcal{D}_{n-1}^S)^{\op}$ and taking colimits.

Let $R$ be an object in $\Comm \mathfrak{S}$. We define the tensor product $$X_{\bullet} \otimes R$$ to be the realization of the simplicial object in $\mathfrak{S}$ $$X_{\bullet} \tilde{\otimes} R$$ where $(X_{\bullet} \tilde{\otimes} R)_n=\bigwedge_{s\in X_n} R\{s\}$ with face maps $$d_i :(X_{\bullet} \tilde{\otimes} R)_n\lra (X_{\bullet} \tilde{\otimes} R)_{n-1}$$ given on each summand of the coproduct by the map $$R\{s\}\lra \bigwedge_{t\in X_{n-1}} R\{t\}$$ which includes $R\{s\}$ into the summand corresponding to $\delta_i(s)\in X_{n-1}$. Here the maps $\delta_i$ and $\sigma_i$ 