Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $X$).