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Aug
12
asked Is the ideal of functions vanishing at a set complementable in $C(X)$?
Aug
12
revised Identities that connect antipode with multiplication and comultiplication
added 35 characters in body
Aug
12
comment Identities that connect antipode with multiplication and comultiplication
As far as I understand, Sweedler considers Hopf algebras in the category of vector spaces. Is the same true for all (braded?) monoidal categories?
Aug
12
revised Identities that connect antipode with multiplication and comultiplication
deleted 3 characters in body
Aug
12
asked Identities that connect antipode with multiplication and comultiplication
Aug
5
accepted Is antipode unique for bialgebras in arbitrary monoidal categories?
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
They are visual, that's important. It will take me some time to look at the explanations. Thank you, Darij!
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
Yes, I expected something like this from the very beginning. But, Darij, it's not correct to write $f\otimes g\otimes h$. You should write either $(f\otimes g)\otimes h$ or $f\otimes (g\otimes h)$, and this is a supplementary headache here. If you don't mind, I think, I'll accept the answer by Evan after getting accustomed to his style. :)
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
I did not see such a proof. Do you mean this trick with string diagrams?
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
@darij grinberg: This becomes obvious only if the assocoativity of convolution in $Hom(B^c,B^a)$ is already proved. This fact is not obvious -- in the book by Dascalescu, Nastasescu and Raianu, for example, they use the Sweedler notations for proving it. But the Sweedler notations, I would say, is a risky way in arbitrary monoidal categories. Chari and Pressley mention this fact, but without references. These were two sources for me. I saw before the pictures like those that Evan Jenkins gave in his answer, but I did not understand them and I did not know that they can be used here.
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
Dima, thank you!
Aug
5
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
Todd and UwF, thank you!
Aug
4
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
@UwF: I guessed also. :) But this style of explanation - "I drop a hint, you guess!" - is a bit annoying when you need a reference for a paper. Are there texts where this technique is described more or less intelligible? :)
Aug
4
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
Yes, I guessed. However, a text with explanations would be highly appreciated.
Aug
4
comment Is antipode unique for bialgebras in arbitrary monoidal categories?
Evan, I don't understand these pictures. Do you know a text where they are explained?
Aug
4
revised Is antipode unique for bialgebras in arbitrary monoidal categories?
added 3 characters in body
Aug
4
asked Is antipode unique for bialgebras in arbitrary monoidal categories?
Jul
2
awarded  Curious
Jun
5
comment Characterization of $l_p$ up to a linear isometry
Ah, I see... You mean that the unit ball (of the norm) must be a polytope, and in addition it must have $2n$ facets. Is this true? Anyway, this looks too complicated... Is there a way to describe the situation without the additional assumption that the inut ball is a polytope?
Jun
5
comment Characterization of $l_p$ up to a linear isometry
Mark, if we take this definition of facet, then for an absorbing bounded balanced convex set $B$ in ${\mathbb R}^3$ having 6 facets is not equivalent to being a unit ball for a space isometrically isomorphic to $\ell_\infty^3$. You can take $B=\{x\in {\mathbb R}^3: \ \max_i|x_i|\le 1\quad \&\quad x_1^2+x_2^2+x_3^2\le 2\}$ (the intersection of a cube with the edge 2 and the usual ball of radius $\sqrt{2}$) -- this set has 6 facets but it is not a cube.