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nLab > Latest Changes: graded monoid

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  1.  

  1. Added the description of the unit morphism, as it was not present before.

    Anonymous

    diff, v3, current

  2. Removed unnecessary binding for variable n in the unit morphism definition.

    Anonymous

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 29th 2022

    Link to graded comonoid created

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 2nd 2022

    Generalized the definition for grading in a commutative monoid.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 15th 2022

    Added some relations with monoids and defined graded monoid with trivial unit.

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorncfavier
    • CommentTimeApr 20th 2023

    Mentioned that graded monoids are monoidal functors

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJul 28th 2023

    This page defines a graded monoid to be connected if

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

    I don’t understand what the second condition is for, or even what it means. In a non-strict monoidal category, the source and target of η⊗∇ 0,p\eta \otimes \nabla_{0,p} and ∇ n,0⊗η\nabla_{n,0} \otimes \eta are not equal, so they can’t be identities. They could be equal to unit coherence isomorphisms, but surely that follows from the first condition and the unit axioms of any graded monoid?

    • CommentRowNumber8.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2023
    • (edited Jul 28th 2023)
    • Corrected the definition of connected graded monoid.
    • Deleted a non-interesting example
    • Addded four examples of connected ℕ\mathbb{N}-graded monoid: tensor powers, symmetric powers, divided powers and exterior powers, defining them in monoidal categories, symmetric monoidal categories or symmetric monoidal categories enriched over abelian groups.
    • Mentioned that if the symmetric monoidal category is enriched over ℚ +\mathbb{Q}^+-modules, then the examples of symmetric and divided powers are isomorphic.

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 28th 2023

    Thanks. But aren’t those two triangles just the “unit axioms” that hold in any graded monoid?

    • CommentRowNumber10.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2023

    Reorganized, corrected and improved a few things.

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2023
    • (edited Jul 28th 2023)

    Re 9: Yes, sorry there is nothing more to require that η\eta is an isomorphism. I deleted this.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJul 28th 2023

    Thanks! (I know I could have fixed it too, but I wasn’t 100% sure that I wasn’t missing something.)

1 to 12 of 12

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