The category of Hopf monoids in a braided monoidal category. #
TODO #
- Show that in a Cartesian monoidal category Hopf monoids are exactly group objects.
- Show that
Hopf (ModuleCat R) ā HopfAlgCat R
.
A Hopf monoid in a braided category C
is a bimonoid object in C
equipped with an antipode.
- mul_assoc : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight mul X) mul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X X X).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X mul) mul)
- comul_assoc : CategoryTheory.CategoryStruct.comp comul (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X comul) = CategoryTheory.CategoryStruct.comp comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight comul X) (CategoryTheory.MonoidalCategoryStruct.associator X X X).hom)
- mul_comul : CategoryTheory.CategoryStruct.comp MonObj.mul ComonObj.comul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom ComonObj.comul ComonObj.comul) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorμ X X X X) (CategoryTheory.MonoidalCategoryStruct.tensorHom MonObj.mul MonObj.mul))
The antipode is an endomorphism of the underlying object of the Hopf monoid.
Instances
Alias of HopfObj
.
A Hopf monoid in a braided category C
is a bimonoid object in C
equipped with an antipode.
Equations
Instances For
The antipode is an endomorphism of the underlying object of the Hopf monoid.
Equations
- HopfObj.termš® = Lean.ParserDescr.node `HopfObj.termš® 1024 (Lean.ParserDescr.symbol "š®")
Instances For
The antipode is an endomorphism of the underlying object of the Hopf monoid.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A Hopf monoid in a braided category C
is a bimonoid object in C
equipped with an antipode.
- X : C
The underlying object in the ambient monoidal category
Instances For
Alias of Hopf
.
A Hopf monoid in a braided category C
is a bimonoid object in C
equipped with an antipode.
Instances For
A Hopf monoid is a bimonoid.
Instances For
Alias of Hopf.toBimon
.
A Hopf monoid is a bimonoid.
Equations
Instances For
Morphisms of Hopf monoids are just morphisms of the underlying bimonoids. In fact they automatically intertwine the antipodes, proved below.
Morphisms of Hopf monoids intertwine the antipodes.
The antipode is an antihomomorphism with respect to both the monoid and comonoid structures. #
Auxiliary calculation for antipode_comul
.
This calculation calls for some ASCII art out of This Week's Finds.
| |
n n
| \ / |
| / |
| / \ |
| | S S
| | \ /
| | /
| | / \
\ / \ /
v v
\ /
v
|
We move the left antipode up through the crossing,
the right antipode down through the crossing,
the right multiplication down across the strand,
reassociate the comultiplications,
then use antipode_right
then antipode_left
to simplify.
Auxiliary calculation for mul_antipode
.
|
n
/ \
| n
| / \
| S S
| \ /
n /
/ \ / \
| / |
\ / \ /
v v
| |
We move the leftmost multiplication up, so we can reassociate.
We then move the rightmost comultiplication under the strand,
and simplify using antipode_right
.
In a commutative Hopf algebra, the antipode squares to the identity.