of quasi-categories.) The category $\mathfrak{S}^{\mathbb{N}^{\op}}$ is closed symmetric monoidal with the Day convolution symmetric monoidal product due to Day [@Day] and it is a symmetric monoidal model category satisfying the pushout product axiom due to Propositions 2.2.15 and 2.2.16 in the Harvard thesis of Isaacson [@MR2713397]. We may therefore consider an object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ as a decreasingly filtered commutative monoid and equip the category $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ with the model structure created by the forgetful functor $$U: \Comm \mathfrak{S}^{\mathbb{N}^{\op}}\lra \mathfrak{S}^{\mathbb{N}^{\op}};$$ i.e., a map $f$ in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ is a fibration or, respectively, a weak equivalence if $U(f)$ is a fibration in $\mathfrak{S}^{\mathbb{N}^{\op}}$ or, respectively, $U(f)$ is a weak equivalence in $\mathfrak{S}^{\mathbb{N}^{\op}}$.

Due to S. Schwede and B. Shipley [@SS Thm 4.1], applying the forgetful functor to a cofibration with cofibrant source in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ produces a cofibration in $\mathfrak{S}^{\mathbb{N}^{\op}}$, so a cofibrant object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$ is a decreasingly filtered commutative monoid in $\mathfrak{S}$ whose underlying object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ is projective cofibrant. (Note that we need to cofibrantly replace the unit object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ and consider modules over the cofibrant unit to use the theorem of S. Schwede and B. Shipley [@SS].) A projective cofibrant object $I$ in $\mathfrak{S}^{\mathbb{N}^{\op}}$ has 