nLab graded monoid




Let MM be a commutative monoid (in the category Sets\mathbf{Sets}). 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
    n,m:Φ mΦ nΦ m+n\nabla_{n,m}: \Phi_m \otimes \Phi_n \to \Phi_{m+n}
  • a morphism
    η:IΦ 0 \eta: I \to \Phi_0

    such that the obvious associativity and unit axioms hold.

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

We say that a MM-graded monoid is connected if

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


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

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

    • Φ m:=A\Phi_{m} := A for every mm
    • 0,0:=\nabla_{0,0} := \nabla
    • m,n:=0\nabla_{m,n} := 0 if m0m \neq 0 or n0n \neq 0


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

See also

Last revised on April 20, 2023 at 17:49:23. See the history of this page for a list of all contributions to it.