nLab graded monoid

Contents

Contents

Definition

Let MM be a commutative monoid. A MM-graded monoid Φ\Phi in a symmetric monoidal category 𝒱\mathcal{V} with unit object II is the data of

  • for each mMm \in M, an object Φ m\Phi_m,
  • for each m,nMm,n \in M, a morphism
    Φ mΦ nΦ m+n \Phi_m \otimes \Phi_n \to \Phi_{m+n}
  • a morphism
    IΦ 0 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 𝒱\mathcal{V}.

Examples

See also

Last revised on August 13, 2022 at 15:15:51. See the history of this page for a list of all contributions to it.