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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have started a table of contents [[measure theory - contents]] and started adding it as a floating toc to the relevant entries

   I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 4th 2012
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   Author: zskoda
   Format: MarkdownItexQuery archived from [[measure space]]: > [[Eric]]: Is there some nice "arrow theoretic" way to state the above? It seems to be screaming to be a functor or an internalization or something. > [[Eric]]: The "products" $\bigcup$ and $\sum$ look like they should be "products" in two different categories. Is that silly? > [[John Baez]]: $\bigcup$ is less like a "product" than a "sum" --- also known as a [[coproduct]]. The collection of subsets of $X$ is a [[poset]], which is a kind of category, and the union of a bunch of subsets can be seen as their coproduct in this category. Unfortunately I don't see a great way to understand the sum of real numbers as a coproduct! So, I can't quite do what I think you're hoping for. > _Toby_: Well, they are still operations that make both $\Sigma$ and $\mathbf{R}^+$ into [[monoidal categories]]; since they\'re also posets, this makes them [[monoidal poset]]s. We can talk about countable additivity using [[transfinite composition]]. I doubt that this adds much to the theory of measure spaces, but it points the way to some of the generalisation below, as well to possibilities for categorification.

   Query archived from measure space:

   Eric: Is there some nice “arrow theoretic” way to state the above? It seems to be screaming to be a functor or an internalization or something.

   Eric: The “products” $\bigcup$ and $\sum$ look like they should be “products” in two different categories. Is that silly?

   John Baez: $\bigcup$ is less like a “product” than a “sum” — also known as a coproduct. The collection of subsets of $X$ is a poset, which is a kind of category, and the union of a bunch of subsets can be seen as their coproduct in this category. Unfortunately I don’t see a great way to understand the sum of real numbers as a coproduct! So, I can’t quite do what I think you’re hoping for.

   Toby: Well, they are still operations that make both $\Sigma$ and $\mathbf{R}^+$ into monoidal categories; since they're also posets, this makes them monoidal posets. We can talk about countable additivity using transfinite composition. I doubt that this adds much to the theory of measure spaces, but it points the way to some of the generalisation below, as well to possibilities for categorification.