nLab
graded monoid

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

  • for each nNn \in \mathbf{N}, an object Φ n\Phi_n,
  • for each m,nNm,n \in \mathbf{N}, 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 June 12, 2021 at 23:41:57. See the history of this page for a list of all contributions to it.