nLab graded monoid

Definition

Let $M$ be a commutative monoid. A $M$-graded monoid $\Phi$ in a symmetric monoidal category $\mathcal{V}$ with unit object $I$ is the data of

• for each $m \in M$, an object $\Phi_m$,
• for each $m,n \in M$, a morphism
  $$\Phi_m \otimes \Phi_n \to \Phi_{m+n}$$
• a morphism
  $$I \to \Phi_0$$

  such that the obvious associativity and unit axioms hold.

Thus, a graded monoid is in particular a graded object. In fact, a graded monoid is just a monoid in the monoidal category of graded objects of $\mathcal{V}$.

Examples

• In the symmetric monoidal category of groups with the cartesian product, two examples of $\mathbb{N}$-graded monoids are the trivial one $1 = (1)_n$ and the graded monoid of symmetric groups $\Sigma = (\Sigma_n)_n$.

See also

• monoid object
• graded object
• graded comonoid
• graded bimonoid