graded monoid

A **graded monoid** $\Phi$ in a symmetric monoidal category $\mathcal{V}$ with unit object $I$ is the data of

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

- 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**$\Sigma = (\Sigma_n)_n$.

Last revised on June 12, 2021 at 23:41:57. See the history of this page for a list of all contributions to it.