Tensor Product of Commutative Monoids

The tensor product $M\otimes N$ of two Commutative Monoids $M,N$ is the quotient of the Free Commutative Monoid on $M\times N$ by the Equivalence Relation generated by

• $(a_{1},b)+(a_{2},b)\sim(a_{1}+a_{2},b)$

• $(a,b_{1})+(a,b_{2})\sim(a,b_{1}+b_{2})$

• $(0,b)\sim 0$

• $(a,0)\sim 0$

As is usual with tensor products, the tensor product of commutative monoids classifies the Bilinear Morphism of Monoids, insofar the bilinear morphisms $X\times Y\to Z$ are equivalent to Monoid Homomorphisms $X\otimes Y\to Z$.

Properties

• The tensor product is Associative up to Isomorphism.

• The tensor product is Symmetric.

  We can construct a map $\sigma:M\otimes N\to N\otimes M$ as $\sigma(m,n)=(n,m)$; this clearly respects the quotient. Moreover, it is also an involution.

• The unit of the tensor product is the additive monoid on the Natural Numbers.

  For the left unitor, observe that we have a map $\lambda:\mathbb{N}\otimes M\to M$ that sends each $(n,a)$ to $n\cdot a$, using the fact that every commutative monoid is a $\mathbb{N}$-Module over a Semiring.

  Moreover, $\lambda$ is Invertible, with inverse $\lambda^{-1}(a)=(1,a)$. Moreover, we can get a right unitor from the composite $\lambda\circ\sigma$.

• Thus, the tensor product of commutative monoids endows the Category of Commutative Monoids with the structure of a Symmetric Monoidal Category.

References

• https://ncatlab.org/nlab/show/tensor+product+of+commutative+monoids