Implication Details

Claim: If a symmetric monoidal category is cartesian and is codistributive, then it is trivial.

Proof: In a codistributive symmetric monoidal category, for every object $X$ , the functor $X \otimes -$ preserves the terminal object $1$ . But since the symmetric monoidal structure is assumed to be cartesian, $1$ is the monoidal unit, so $1 = X \otimes 1 = X$ .

Show 9 symmetric monoidal categories using this implication

symmetric monoidal category of finitely generated abelian groups
symmetric monoidal category of abelian groups
cartesian symmetric monoidal category of small categories
symmetric monoidal category of finite-dimensional vector spaces
symmetric monoidal category of modules over a commutative ring
symmetric monoidal category of modules over an absolutely flat commutative ring
symmetric monoidal category of modules over a non-absolutely flat commutative ring
cartesian symmetric monoidal category of sets
cartesian symmetric monoidal category of topological spaces