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1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeMay 8th 2010
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   Author: Eric
   Format: MarkdownItexTodd is helping me understand [[opposite categories]] beginning with $FinSet^{op}$ [here](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=1229&Focus=10072#Comment_10072). This discussion helped prompt some improvement of the page [[opposite category]]. When I look at that page now, I see the statement: >The idea of [[noncommutative geometry]] is essentially to define a category of _spaces_ as the opposite category of a category of algebras. This reminded me of a remark I made in the "Forward" to a paper I wrote back in 2002, so I've now itexified that "Foreward" here: [[Noncommutative Geometry and Stochastic Calculus]] By the way, this also suggests that the category $FinSet$ is the category of _spaces_ opposite to the category of finite [[Boolean algebras]] in the sense of [[space and quantity]].

   Todd is helping me understand opposite categories beginning with $FinSet^{op}$ here.

   This discussion helped prompt some improvement of the page opposite category. When I look at that page now, I see the statement:

   The idea of noncommutative geometry is essentially to define a category of spaces as the opposite category of a category of algebras.

   This reminded me of a remark I made in the “Forward” to a paper I wrote back in 2002, so I’ve now itexified that “Foreward” here:

   Noncommutative Geometry and Stochastic Calculus

   By the way, this also suggests that the category $FinSet$ is the category of spaces opposite to the category of finite Boolean algebras in the sense of space and quantity.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 8th 2010
   • (edited May 8th 2010)
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   Author: Todd_Trimble
   Format: MarkdownItexYes, that's quite right: $FinSet$ *is* a category of spaces opposite to... But they are not the most interesting spaces in the world, being finite sets with the discrete topologies. Nevertheless, your comment is quite *a propos*. What we are doing at "opposite of finite sets" is a 'baby' version of Stone duality, which posits a general duality between general Boolean algebras (not just finite ones) and so-called 'Stone spaces'. This was perhaps the first theorem of its kind, which went on to be extrapolated to Gelfand-Naimark duality (on maximal ideal spectra for C*-algebras), and thence to the more sophisticated noncommutative geometries. I had actually begun a series of posts (over on my all-but-defunct blog with Vishal Lama) on Stone duality... but maybe you shouldn't read that right away, Eric, as we're carrying on this quasi-Socratic (or Moore method) experiment. :-)

   Yes, that’s quite right: $FinSet$ is a category of spaces opposite to… But they are not the most interesting spaces in the world, being finite sets with the discrete topologies.

   Nevertheless, your comment is quite a propos. What we are doing at “opposite of finite sets” is a ’baby’ version of Stone duality, which posits a general duality between general Boolean algebras (not just finite ones) and so-called ’Stone spaces’. This was perhaps the first theorem of its kind, which went on to be extrapolated to Gelfand-Naimark duality (on maximal ideal spectra for C*-algebras), and thence to the more sophisticated noncommutative geometries.

   I had actually begun a series of posts (over on my all-but-defunct blog with Vishal Lama) on Stone duality… but maybe you shouldn’t read that right away, Eric, as we’re carrying on this quasi-Socratic (or Moore method) experiment. :-)

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 8th 2010
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   Author: Mike Shulman
   Format: MarkdownItexIt's a very interesting question to me of whether the category of (perhaps finite) sets should be regarded as a category of *spaces* or as a category of *algebras*. On the one hand, you can think of a set as a space with a discrete topology. But on the other hand, you can think of it as an algebra with no operations. Of course whichever position one takes will suggest that $Set^{op}$ is the other sort of category, and I think there are mathematical consequences of that. For instance, would you expect $Set^{op}$ to have a small [[dense subcategory]]? If $Set$ is a category of spaces, so that $Set^{op}$ is a category of algebras, then I would see nothing wrong with its having a small dense subcategory; but if $Set$ is a category of algebras, so that $Set^{op}$ is a category of spaces, then I would expect it probably not to. But it turns out that $Set^{op}$ has a small dense subcategory if and only if there does *not* exist a proper class of measurable cardinals. So whether or not you believe in measurable cardinals could be seen as taking a stand on whether $Set$ is a category of spaces or a category of algebras. (-:

   It’s a very interesting question to me of whether the category of (perhaps finite) sets should be regarded as a category of spaces or as a category of algebras. On the one hand, you can think of a set as a space with a discrete topology. But on the other hand, you can think of it as an algebra with no operations.

   Of course whichever position one takes will suggest that $Set^{op}$ is the other sort of category, and I think there are mathematical consequences of that. For instance, would you expect $Set^{op}$ to have a small dense subcategory? If $Set$ is a category of spaces, so that $Set^{op}$ is a category of algebras, then I would see nothing wrong with its having a small dense subcategory; but if $Set$ is a category of algebras, so that $Set^{op}$ is a category of spaces, then I would expect it probably not to. But it turns out that $Set^{op}$ has a small dense subcategory if and only if there does not exist a proper class of measurable cardinals. So whether or not you believe in measurable cardinals could be seen as taking a stand on whether $Set$ is a category of spaces or a category of algebras. (-:

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 8th 2010
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   Author: Todd_Trimble
   Format: MarkdownItex> But it turns out that $Set^{op}$ has a small dense subcategory if and only if there does not exist a proper class of measurable cardinals. Huh! That's very interesting. Where can I read about that? (If it's in one of your papers on set theory and category theory, then I'll be a little bit embarrassed that I didn't pick up on that, but go ahead and say...)

   But it turns out that $Set^{op}$ has a small dense subcategory if and only if there does not exist a proper class of measurable cardinals.

   Huh! That’s very interesting. Where can I read about that? (If it’s in one of your papers on set theory and category theory, then I’ll be a little bit embarrassed that I didn’t pick up on that, but go ahead and say…)

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeMay 8th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> Where can I read about that? It's Theorem A.5 in Adamek&Rosicky, "Locally presentable and accessible categories." I think it's easier to understand if you first prove that the full subcategory on $\mathbb{N}$ is dense in $Set^{op}$ if and only if there exist *no* measurable cardinals. For any set $A$, the canonical colimit over this subcategory in $Set^{op}$ turns out to be essentially the set of countably additive measures on $A$, hence isomorphic to $A$ iff $A$ admits no countably additive measure. Then the extension to higher cardinals is fairly straightforward.

   Where can I read about that?

   It’s Theorem A.5 in Adamek&Rosicky, “Locally presentable and accessible categories.” I think it’s easier to understand if you first prove that the full subcategory on $\mathbb{N}$ is dense in $Set^{op}$ if and only if there exist no measurable cardinals. For any set $A$, the canonical colimit over this subcategory in $Set^{op}$ turns out to be essentially the set of countably additive measures on $A$, hence isomorphic to $A$ iff $A$ admits no countably additive measure. Then the extension to higher cardinals is fairly straightforward.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 8th 2010
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Mike! It's very strange to me that I haven't looked into their appendix before, because it's very well-written and a lot of stuff of interest to me. One thing I like about it is that it is written from a structuralist point of view, which I find immensely easier to follow than more materialist accounts.

   Thanks, Mike! It’s very strange to me that I haven’t looked into their appendix before, because it’s very well-written and a lot of stuff of interest to me. One thing I like about it is that it is written from a structuralist point of view, which I find immensely easier to follow than more materialist accounts.