the property that $I_i$ is cofibrant for all $i$ and $f_i:I_i\rightarrow I_{i-1}$ is a cofibration for all $i$ (see [@thhmay Lem. 2.1.5]).

A decreasingly filtered commutative monoid in $\mathfrak{S}$ is a *cofibrant decreasingly flltered commutative monoid* in $\mathfrak{S}$ if it is cofibrant in the model structure specified above on $\Comm\mathfrak{S}^{\mathbb{N}^{\op}}$.

Naturally, one would like to define the associated graded commutative monoid in $\mathfrak{S}$ of a decreasingly filtered commutative monoid in $\mathfrak{S}$. The maps $\rho_{i,j}$ are the structure necessary to make sense of such an associated graded commutative monoid in $\mathfrak{S}$.

\[assoc grd\]Let $I$ be a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$. We define $E_0^*I$, the [*associated graded commutative monoid of $I$*]{}, to be the graded commutative monoid object in $\mathfrak{S}$ defined as follows:

-   additively, as an object in $\mathfrak{S}$, $$E_0^*I \cong \coprod_{n\in\mathbb{N}} I_n/I_{n+1},$$

-   The unit map $S \rightarrow E_0^*I$ is the composite $$S\stackrel{\eta}{\longrightarrow} I_0 \rightarrow I_0/I_1\hookrightarrow E_0^*I .$$

-   The multiplication on $E_0^*I$ is given as follows. Since the smash product commutes with colimits, hence with coproducts in $\mathfrak{S}$, to specify a map $$E_0^*I \wedge E_0^*I \rightarrow E_0^*I$$ it suffices to specify a component map $$\nabla_{i,j}: I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow E_0^*I$$ for every $i,j\in\mathbb{N}$. We define such a map 