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1h
awarded  Yearling
1h
awarded  Excavator
2h
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
@ZhenLin, great, thanks! Do you want to make this into an answer? (If you don't have the time I can do that as well)
2h
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
Oh, maybe I understand what you're getting at: While $Y$ and $Y$ are the same objects, $(Y,\epsilon,\eta)$ and $(Y,e,h)$ are different duals. So the whole dual data is unique, but not the underlying object?
2h
comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
@ZhenLin, what does "compatible with all that data" mean, other than "define the new $\epsilon$ and $\eta$ by composition" and "the new dual satisfies the snake identities"? As I see it now, my $(Y, e, h)$ is another dual and $f$ is a unique iso compatible with the data.
2h
revised A coherence question in rigid monoidal categories
Clarified spelling and grammar
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comment Why are duals in a rigid/autonomous category unique up to unique isomorphism?
@ZhenLin, yes, that's why I prove the snake identities for the new dualities. Or is there any more structure that the new duality must be compatible with?
3h
comment A coherence question in rigid monoidal categories
Are you implicitly assuming a particular isomorphism $\hat{\hat Y} \cong Y$? So maybe you're actually considering a pivotal category? It's nonstandard to define a rigid category as a monoidal symmetrical closed category with duals. Usually, only the monoidal structure is needed to define duals.
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suggested suggested edit on A coherence question in rigid monoidal categories
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asked Why are duals in a rigid/autonomous category unique up to unique isomorphism?
Jul
8
answered $[0, 1)$ and $S^1$ not homeomorphic?
Jul
2
awarded  Curious
Apr
21
comment Underlying functor of tensor product in a closed and symmetric monoidal category.
I wonder how many people have actually read this question completely. Can you shorten it somehow? (If necessary, restrict the audience to more experienced people by shortening)
Apr
21
answered Is there a category of categories?
Apr
18
answered Arrow between endofunctors over a symmetric monoidal category.
Apr
18
awarded  Citizen Patrol
Mar
23
comment Are there homomorphisms of group algebras that don't come from a group homomorphism?
Ah, so already $\mathbb{C}[\mathbb{Z}_4] \cong \mathbb{C}[\mathbb{Z_2} \times \mathbb{Z}_2]$? I wonder whether these kind of isomorphisms are also equivalences of representation categories.
Mar
23
accepted Are there homomorphisms of group algebras that don't come from a group homomorphism?
Mar
23
comment Are there homomorphisms of group algebras that don't come from a group homomorphism?
That sounds interesting! You say "it is well known", but when I search for dihedral group, group algebra and quaternions, I can't find anything about this algebra isomorphism.
Mar
23
comment Are there homomorphisms of group algebras that don't come from a group homomorphism?
@Dustan, it seems I'm not familiar enough. How does the isomorphism you mean look like?