The Clowder Project

The symmetric monoidal structure on the category ${\mathsf{Sets}}_{\ast }$ of Proposition 5.5.9.1.1 is uniquely determined by the following requirements:

1. 1. Two-Sided Preservation of Colimits. The tensor product
   $${\otimes }_{{\mathsf{Sets}}_{\ast }}:{\mathsf{Sets}}_{\ast }\times {\mathsf{Sets}}_{\ast }\to {\mathsf{Sets}}_{\ast }$$

   of ${\mathsf{Sets}}_{\ast }$ preserves colimits separately in each variable.

2. 2. The Unit Object Is $S^0$. We have ${\mathbb{1}}_{{\mathsf{Sets}}_{\ast }}\cong S^0$.

More precisely, the full subcategory of the category ${\mathcal{M}}_{{\mathbb{E}}_{\mathrm{\infty }}}\left({\mathsf{Sets}}_{\ast }\right)$ of spanned by the symmetric monoidal categories $(\phantom{{\lambda }^{{\mathsf{Sets}}_{\ast }}}{\mathsf{Sets}}_{\ast }$, ${\otimes }_{{\mathsf{Sets}}_{\ast }}$, ${\mathbb{1}}_{{\mathsf{Sets}}_{\ast }}$, ${\lambda }^{{\mathsf{Sets}}_{\ast }}$, ${\rho }^{{\mathsf{Sets}}_{\ast }}$, ${\sigma }^{{\mathsf{Sets}}_{\ast }})$ satisfying Item 1 and Item 2 is contractible.