nLab graded monoid

Definition

Let $M$ be a commutative monoid (in the category $\mathbf{Sets}$). 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
  $$\nabla_{n,m}: \Phi_m \otimes \Phi_n \to \Phi_{m+n}$$
• a morphism
  $$\eta: I \to \Phi_0$$

  such that the obvious associativity and unit axioms hold.

Thus, a $M$-graded monoid is in particular a $M$-graded object. In fact, a $M$-graded monoid is just a monoid in the monoidal category of $M$-graded objects of $\mathcal{V}$, hence a lax monoidal functor $M \to \mathcal{V}$ by this proposition (where $M$ is viewed as a discrete monoidal category).

We say that a $M$-graded monoid is connected if

• $\eta$ is an isomorphism,
• $\eta \otimes \nabla_{0,p}$ and $\nabla_{n,0} \otimes \eta$ are equal to the identity.

Properties

• If $\Phi$ is a $M$-graded monoid in $\mathcal{V}$, then $(\Phi_{0},\nabla_{0,0},\eta)$ is a monoid in $\mathcal{V}$.

• If $\mathcal{V}$ is a symmetric monoidal category which is also a enriched over pointed sets, and if $(A,\nabla,\eta)$ is a monoid in $\mathcal{V}$, then we get a $\mathbb{N}$-graded monoid $\Phi$ by defining:

  • $\Phi_{m} := A$ for every $m$
  • $\nabla_{0,0} := \nabla$
  • $\nabla_{m,n} := 0$ if $m \neq 0$ or $n \neq 0$

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 $\Sigma = (\Sigma_n)_n$.

The two examples above in the category of groups are connected (by considering that $\Sigma_{0} = 1$).

See also

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