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1. • CommentRowNumber1.
   • CommentAuthorYemon Choi
   • CommentTimeOct 29th 2013
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   Author: Yemon Choi
   Format: MarkdownItexPopped my head round the door and made a couple of changes to [Banach algebra](http://ncatlab.org/nlab/show/Banach+algebra) The first change was to attempt a more lax position on what should constitute a Banach coalgebra: only looking at comonoids in the monoidal category of Banach spaces (geometric or topogical) with projective tensor product would rule out several important examples that have arisen in e.g. abstract harmonic analysis. The existence of different monoidal structures in the category of Banach spaces is a pain, but without it one would miss out on a rich world of examples. The second was to add, to the list of examples, the celebrated-in-my-world-and-possibly-no-others Arens products on the double dual of a Banach algebra. I've made a stab at linking them to the related concepts of tensorial strength and strong monad but would welcome feedback or improvements.

   Popped my head round the door and made a couple of changes to Banach algebra

   The first change was to attempt a more lax position on what should constitute a Banach coalgebra: only looking at comonoids in the monoidal category of Banach spaces (geometric or topogical) with projective tensor product would rule out several important examples that have arisen in e.g. abstract harmonic analysis. The existence of different monoidal structures in the category of Banach spaces is a pain, but without it one would miss out on a rich world of examples.

   The second was to add, to the list of examples, the celebrated-in-my-world-and-possibly-no-others Arens products on the double dual of a Banach algebra. I’ve made a stab at linking them to the related concepts of tensorial strength and strong monad but would welcome feedback or improvements.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 29th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Yemon! As for the Arens example, my own memory is that the double dual monad is strong (there is a canonical strength on any functor definable from the data of a symmetric monoidal closed category), but it is not a [[monoidal monad]]: the two ways available of defining an arrow $A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$, using tensorial strengths, disagree.

   Thanks, Yemon! As for the Arens example, my own memory is that the double dual monad is strong (there is a canonical strength on any functor definable from the data of a symmetric monoidal closed category), but it is not a monoidal monad: the two ways available of defining an arrow $A^{\ast\ast} \otimes B^{\ast\ast} \to (A \otimes B)^{\ast\ast}$, using tensorial strengths, disagree.

3. • CommentRowNumber3.
   • CommentAuthorYemon Choi
   • CommentTimeNov 2nd 2013
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   Author: Yemon Choi
   Format: MarkdownItexThanks for the clarification, Todd. I have added some more from the world of "Arens regularity". Eventually I hope to build up the links between this section and those on [[reflexive Banach space]].

   Thanks for the clarification, Todd.

   I have added some more from the world of “Arens regularity”. Eventually I hope to build up the links between this section and those on reflexive Banach space.

4. • CommentRowNumber4.
   • CommentAuthorYemon Choi
   • CommentTimeNov 4th 2013
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   Author: Yemon Choi
   Format: MarkdownItexRearranged some material, added a link to a talk given by Linton. Also corrected an erroneous claim (the word "compact" should have been "discrete").

   Rearranged some material, added a link to a talk given by Linton. Also corrected an erroneous claim (the word “compact” should have been “discrete”).

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 4th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexI added to the material on Arens products, linking to [[monoidal monad]] along the lines sketched in my earlier comment.

   I added to the material on Arens products, linking to monoidal monad along the lines sketched in my earlier comment.