nLab graded monoid

Idea

A graded monoid is like a monoid but where elements of the monoid possess some degree and such that when you multiply, you sum the degrees. It can be defined in any monoidal category.

Definition and properties

Let $M$ be a commutative monoid (in the category $\mathbf{Sets}$). A $M$-graded monoid $\Phi$ in a 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
  $$\nabla_{m,n}: \Phi_m \otimes \Phi_n \to \Phi_{m+n}$$
• 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.

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

Examples

The following are all examples of connected $\mathbb{N}$-graded monoids.

• In the symmetric monoidal category of groups with the cartesian product, an example of $\mathbb{N}$-graded monoid is the trivial one $1 = (1)_n$

• In the same category, another example is the graded monoid of symmetric groups $\Sigma = (\Sigma_n)_n$.

• In any monoidal category, defining $\Phi_n := A^{\otimes n}$, $\nabla_{m,n}:A^{\otimes m} \otimes A^{\otimes n} \rightarrow A^{\otimes (m+n)}$ given by associators and $\eta = 1_{I}$, we obtain the connected $\mathbb{N}$-graded monoid of tensor powers.

• In any symmetric monoidal category, defining $S^n(A)$ equal to the coequalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$ and defining the multiplication $S^n(A) \otimes S^p(A) \rightarrow S^{n+p}(A)$ by using the universal property of the coequalizer, we obtain the connected commutative $\mathbb{N}$-graded monoid of symmetric powers.

• In any symmetric monoidal category, defining $\Gamma^n(A)$ equal to the equalizer of the $n!$ permutations $A^{\otimes n} \rightarrow A^{\otimes n}$ and defining the multiplication $\Gamma^n(A) \otimes \Gamma^p(A) \rightarrow \Gamma^{n+p}(A)$ by using the universal property of the equalizer, we obtain the connected commutative $\mathbb{N}$-graded monoid of divided powers.

• In any symmetric monoidal category enriched over the category of abelian groups, defining $\Lambda^n(A)$ equal to the coequalizer of the $n!$ signed permutations $sgn(\sigma).\sigma:A^{\otimes n} \rightarrow A^{\otimes}$ where $sgn(\sigma)$ is the signature of the permutation $\sigma$ and defining the multiplication $\Lambda^n(A) \otimes \Lambda^p(A) \rightarrow \Lambda^{n+p}(A)$ by using the universal property of the equalizer, we obtain the connected graded-commutative (ie. such that $\sigma;\nabla_{p,n} = sgn(\sigma).\sigma:\Phi_{n} \otimes \Phi_{p} \rightarrow \Phi_{n+p}$) $\mathbb{N}$-graded monoid of exterior powers. Replacing the coequalizer by an equalizer always provides an isomorphic $\mathbb{N}$-graded monoid.

The connected commutative $\mathbb{N}$-graded monoid of symmetric powers and of divided powers are not isomorphic in general, but they are if the symmetric monoidal category is enriched over $\mathbb{Q}^{+}$-modules, for instance in the category of vector spaces over a field of characteristic $0$ or in the category of sets and relations (where $S^{n}(X) \cong \Gamma^{n}(X)$ is equal to the set of all multisets of $n$ elements of $X$).

See also

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