A graded monoid $\Phi$ in a symmetric monoidal category $\mathcal{V}$ is the data of

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

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 graded monoids are the trivial one $1 = (1)_n$ and the graded monoid of symmetric groups $\Sigma = (\Sigma_n)_n$.