Timeline for Shuffle Hopf algebra: how to prove its properties in a slick way?

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Jul 21, 2012 at 22:05	comment	added	Duchamp Gérard H. E.		Thanks and to finish the proof in the general case, one can proceed as follows : Let $V$ be a $k$-module (free or not) and $X=x_i,i∈I$ (finite or not) a generating family of $V$ with $\gamma : F=k^{(I)}\rightarrow V$ the canonical map. Then, the tensor extension $$ T(\gamma) : k\langle X\rangle \rightarrow T(V) $$ is a onto mapping between of algebra with unit and co-algebra with counit between shuffle algebras and concatenation co-algebras. From this, one gets without effort the bi-algebra structure of $T(V)$. Antipode is derived as previously for the proof of $log_*$.
Jul 21, 2012 at 17:39	comment	added	darij grinberg		Thanks. This is a nice variation of the proof from YBL's post. (I think $K\left\langle \left\langle X \right\rangle \right\rangle$ is a Hopf algebra in an appropriate category of topological vector spaces with some kind of completed tensor products, but I don't know this category. Or is there a good reason why there is no such category?)
Jul 21, 2012 at 16:37	history	edited	Duchamp Gérard H. E.	CC BY-SA 3.0	added 84 characters in body; edited body; added 10 characters in body
Jul 21, 2012 at 16:27	history	answered	Duchamp Gérard H. E.	CC BY-SA 3.0