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Mike Pierce
Mike Pierce
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For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have natural transformations $\eta \colon 1_\mathcal{C} \Rightarrow RL$ and $\varepsilon \colon LR \Rightarrow 1_\mathcal{D}$, called the unit and counit respectively, that satisfy the triangle identities.

On the other hand, a Hopf algebra $H$ is an associative and coassociative bialgebra equipped with a multiplication $\nabla$, a comultiplication $\Delta$, an antipode $S$, and unit and counit maps $\eta$ and $\varepsilon$ such that $\nabla \circ (1_H \otimes S) \circ \Delta = \Delta \circ (S \otimes 1_H) \circ \nabla = \eta\circ \varepsilon$.

How are these two different definitions of unit/counit related? I mean, I know they must be the same idea when viewed in the right context, but I haven't figured it out yet, and neither nLab nor WikipediaWikipedia spells it out. I'm sure I could figure this out myself eventually and type up an nice answer (and will if no one else feels like typing it up), but I'll bet someone on this site already knows, the relationship well and can provide some useful insight too.

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have natural transformations $\eta \colon 1_\mathcal{C} \Rightarrow RL$ and $\varepsilon \colon LR \Rightarrow 1_\mathcal{D}$, called the unit and counit respectively, that satisfy the triangle identities.

On the other hand, a Hopf algebra $H$ is an associative and coassociative bialgebra equipped with a multiplication $\nabla$, a comultiplication $\Delta$, an antipode $S$, and unit and counit maps $\eta$ and $\varepsilon$ such that $\nabla \circ (1_H \otimes S) \circ \Delta = \Delta \circ (S \otimes 1_H) \circ \nabla = \eta\circ \varepsilon$.

How are these two different definitions of unit/counit related? I mean, I know they must be the same idea when viewed in the right context, but I haven't figured it out yet, and neither nLab nor Wikipedia spells it out. I'm sure I could figure this out myself eventually and type up an nice answer, but I'll bet someone on this site already knows, and can provide some useful insight too.

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have natural transformations $\eta \colon 1_\mathcal{C} \Rightarrow RL$ and $\varepsilon \colon LR \Rightarrow 1_\mathcal{D}$, called the unit and counit respectively, that satisfy the triangle identities.

On the other hand, a Hopf algebra $H$ is an associative and coassociative bialgebra equipped with a multiplication $\nabla$, a comultiplication $\Delta$, an antipode $S$, and unit and counit maps $\eta$ and $\varepsilon$ such that $\nabla \circ (1_H \otimes S) \circ \Delta = \Delta \circ (S \otimes 1_H) \circ \nabla = \eta\circ \varepsilon$.

How are these two different definitions of unit/counit related? I mean, I know they must be the same idea when viewed in the right context, but I haven't figured it out yet, and neither nLab nor Wikipedia spells it out. I'm sure I could figure this out myself eventually and type up an nice answer (and will if no one else feels like typing it up), but I'll bet someone on this site already knows the relationship well and can provide some useful insight too.

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Mike Pierce
Mike Pierce
  • 1.2k
  • 1
  • 9
  • 26

How are the unit/counit of a Hopf algebra and of an categorical adjunction related?

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have natural transformations $\eta \colon 1_\mathcal{C} \Rightarrow RL$ and $\varepsilon \colon LR \Rightarrow 1_\mathcal{D}$, called the unit and counit respectively, that satisfy the triangle identities.

On the other hand, a Hopf algebra $H$ is an associative and coassociative bialgebra equipped with a multiplication $\nabla$, a comultiplication $\Delta$, an antipode $S$, and unit and counit maps $\eta$ and $\varepsilon$ such that $\nabla \circ (1_H \otimes S) \circ \Delta = \Delta \circ (S \otimes 1_H) \circ \nabla = \eta\circ \varepsilon$.

How are these two different definitions of unit/counit related? I mean, I know they must be the same idea when viewed in the right context, but I haven't figured it out yet, and neither nLab nor Wikipedia spells it out. I'm sure I could figure this out myself eventually and type up an nice answer, but I'll bet someone on this site already knows, and can provide some useful insight too.