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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 8th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownI tried to prettify the entry [[infinitesimal object]]: I expanded and restructured the "Idea" section. I tried to emphasize the point that Lawvere's axioms are the right general point of view and that the wealth of constructions in algebraic geoemtry is, from this abstract nonsense point of view, to be regarded as taking place in a _model_ for these axioms. I cite Anders Kocks's latest book for the most simple minded version of how algebraic geometry is a model for sdg, but I think this goes through for more sophisticated versions, too. It would be nice to discuss this eventually elsewhere in some entry on "algebraic geometry as models for smooth toposes". I have also tried to subsume the approach of nonsstandard analysis as yet another special case of Lawvere's general axioms, by referring to Moerdijk-Reyes' topos <latex>\mathcal{N}</latex> and <latex>\mathcal{B}</latex> in which "objects of invertible infinitesimals/infinities" exist and model nonstandard analysis. This, too, would deserve being expanded on further, but I am thinking for the nLab this abstract-nonsense-first perspective is the right one. Then I inserted some links to the now separate [[infinitesiaml interval object]] that I am still working on. I also changed the ideosyncratic terminology "infinitesimal k-cube" and "infinitsimal k-disk" to "cartesian product of inf. intervals" and "k-dimensional infinitesimal interval". Anders Kock calls the latter a "monad", following Leibniz, but I am hesitating to overload [[monad]] this way, given that Kock's use of it doesn't seem to be wide spread.

   I tried to prettify the entry infinitesimal object:

   I expanded and restructured the "Idea" section. I tried to emphasize the point that Lawvere's axioms are the right general point of view and that the wealth of constructions in algebraic geoemtry is, from this abstract nonsense point of view, to be regarded as taking place in a model for these axioms. I cite Anders Kocks's latest book for the most simple minded version of how algebraic geometry is a model for sdg, but I think this goes through for more sophisticated versions, too. It would be nice to discuss this eventually elsewhere in some entry on "algebraic geometry as models for smooth toposes".

   I have also tried to subsume the approach of nonsstandard analysis as yet another special case of Lawvere's general axioms, by referring to Moerdijk-Reyes' topos and in which "objects of invertible infinitesimals/infinities" exist and model nonstandard analysis. This, too, would deserve being expanded on further, but I am thinking for the nLab this abstract-nonsense-first perspective is the right one.

   Then I inserted some links to the now separate infinitesiaml interval object that I am still working on. I also changed the ideosyncratic terminology "infinitesimal k-cube" and "infinitsimal k-disk" to "cartesian product of inf. intervals" and "k-dimensional infinitesimal interval". Anders Kock calls the latter a "monad", following Leibniz, but I am hesitating to overload monad this way, given that Kock's use of it doesn't seem to be wide spread.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeOct 8th 2009
   • PermaLink
   Author: TobyBartels
   Format: MarkdownFixed link: [[infinitesimal interval object]] Personally, I *like* infinitesimal cubes and discs; that terminology makes a lot of sense to me.

   Fixed link: infinitesimal interval object

   Personally, I like infinitesimal cubes and discs; that terminology makes a lot of sense to me.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 9th 2009
   • PermaLink
   Author: Urs
   Format: Markdown<blockquote> I like infinitesimal cubes and discs; that terminology makes a lot of sense to me. </blockquote> Yes, but I became worried that these were the wrong terms the way I used them. But maybe we should at least keep a remark that D(n) for instance might be thought of as the infinitesimal n-disk. I was just getting a bit worried about how to justify that language rigorously. But its certainly the right intuition.

   This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> I like infinitesimal cubes and discs; that terminology makes a lot of sense to me. </blockquote> <p>Yes, but I became worried that these were the wrong terms the way I used them. But maybe we should at least keep a remark that D(n) for instance might be thought of as the infinitesimal n-disk. I was just getting a bit worried about how to justify that language rigorously. But its certainly the right intuition.</p> </div>
4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 9th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownI don't believe that SDG can be said to "subsume" NSA; I added some comments to [[infinitesimal object]] regarding why.

   I don't believe that SDG can be said to "subsume" NSA; I added some comments to infinitesimal object regarding why.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 9th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownThanks, Mike!

   Thanks, Mike!