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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeOct 2nd 2012
   • (edited Oct 2nd 2012)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI noted that Thomas Holder has an excellent post on the [Café:]( http://golem.ph.utexas.edu/category/2012/09/where_do_ultraproducts_come_fr.html#c042242). Is this worth editing into the [[ultraproduct]] entry in the Lab? If so is it easier to do it with the source or just a copy or paste? (I do not have access to the source I think.) There are connections in his list that I would love to see followed up within the model theoretic sections, but I do not feel knowledgeable about that area so am reluctant to attempt it.

   I noted that Thomas Holder has an excellent post on the Café:.

   Is this worth editing into the ultraproduct entry in the Lab? If so is it easier to do it with the source or just a copy or paste? (I do not have access to the source I think.)

   There are connections in his list that I would love to see followed up within the model theoretic sections, but I do not feel knowledgeable about that area so am reluctant to attempt it.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 11th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexFor some reason Tim's post seemed to direct me to a post other than one by Thomas Holder. I'll try again: [here](https://golem.ph.utexas.edu/category/2012/09/where_do_ultraproducts_come_fr.html#c042243). The entry [[ultraproduct]] had received more than a dozen edits (some of them spam) with still not much to show for it, so I thought I'd try to help it along. I'm taking a little break for the moment. The extract attributed to Barr bothers me, especially the bit about categorical ultraproducts: as a standalone quote this seems to me somewhat misleading, or a misguided emphasis. Sure, you don't take reduced products of fields *directly*. But so what? When people speak about taking ultraproducts of fields (as they do), it's easy to make categorical sense of it by taking ultraproducts of their underlying *structures* (in the sense of model theory), and then asserting that they have the *properties* expected of models. Categorical ultraproducts (as filtered colimits of systems of products) should thus be viewed primarily as functors on categories of structures, which tend to be certain cocomplete quasitoposes; secondarily they do induce functors between full subcategories of models. (Maybe Barr says something like this; I didn't go to the source of the quotation yet to see the greater context.)

   For some reason Tim’s post seemed to direct me to a post other than one by Thomas Holder. I’ll try again: here.

   The entry ultraproduct had received more than a dozen edits (some of them spam) with still not much to show for it, so I thought I’d try to help it along. I’m taking a little break for the moment.

   The extract attributed to Barr bothers me, especially the bit about categorical ultraproducts: as a standalone quote this seems to me somewhat misleading, or a misguided emphasis. Sure, you don’t take reduced products of fields directly. But so what? When people speak about taking ultraproducts of fields (as they do), it’s easy to make categorical sense of it by taking ultraproducts of their underlying structures (in the sense of model theory), and then asserting that they have the properties expected of models. Categorical ultraproducts (as filtered colimits of systems of products) should thus be viewed primarily as functors on categories of structures, which tend to be certain cocomplete quasitoposes; secondarily they do induce functors between full subcategories of models. (Maybe Barr says something like this; I didn’t go to the source of the quotation yet to see the greater context.)

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 12th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSorry, what I said about quasitoposes was just incredibly dumb. But categories of structures of signatures are complete and cocomplete, which is what is relevant here.

   Sorry, what I said about quasitoposes was just incredibly dumb. But categories of structures of signatures are complete and cocomplete, which is what is relevant here.

4. • CommentRowNumber4.
   • CommentAuthorThomas Holder
   • CommentTimeJul 12th 2015
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItexBack then I had flooded Tom's post mostly with references of which I've added the most useful to [[ultraproduct]]. Feel free to dispose of the mentioning of Keisler-Shelah or Los; eventually these should get more suitable places.

   Back then I had flooded Tom’s post mostly with references of which I’ve added the most useful to ultraproduct. Feel free to dispose of the mentioning of Keisler-Shelah or Los; eventually these should get more suitable places.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 12th 2015
   • (edited Jul 12th 2015)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexNice additions, Thomas! (I was planning to get to Los, so I think it should stay definitely. Probably also Keisler-Shelah.)

   Nice additions, Thomas! (I was planning to get to Los, so I think it should stay definitely. Probably also Keisler-Shelah.)

6. • CommentRowNumber6.
   • CommentAuthorjesse
   • CommentTimeMay 25th 2017
   • PermaLink
   Author: jesse
   Format: MarkdownItexI've added a brief remark on how ultrapowers of [[Banach spaces]] turn approximate eigenvalues of a bounded operator into eigenvalues of (the ultrapower of) that operator.

   I’ve added a brief remark on how ultrapowers of Banach spaces turn approximate eigenvalues of a bounded operator into eigenvalues of (the ultrapower of) that operator.