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    • pairing — pretty simple, but not to be confused with the product

    • started infinitesimal neighbour and began creating a circle of entries surrounding this:

      infinitesimal path infinity-groupoid in a smooth topos; path infinity-groupoid in a smooth topos; simplex in a lined topos

      This is heading in the direction of giving a full discussion of X^{\Delta^n_{inf}} for X a microlinear space, mentioned presently already at differential forms in synthetic differential geometry. I though i could just point to the literature for that, but not quite, apparently. Anders Kock discusses this for X a "formal manifold", an object with a cover by Kock-Lawvere vector spaces. But it should work a bit more generally using microlinear spaces, as indicated in the appendix of Models for Smooth Infinitesimal Analysis. There is an obvious general-nonsense definition wich I discuss, but I need yet to insert discussion of that and how this reproduces Kock's definition (but I think it does).

      It has been an esteemed insight for me that the best way to think of all these constructions of "combinatorial differential forms" (still have to expand the discussion of those at differential forms in synthetic differential geometry) is by organizing them into their natural simplicial structures and then noticing that the model category structure available in the background allows us to think of the resulting simplicial objects in the topos as interna oo-groupoids. I think this must clearly the nLab way of thinking about this, so I created entries with the respective titles.

      You may have noticed that on my personal web I am developing the further step that goes from (infinitesimal) path oo-groupoids of objects in a 1-topos to (infinitesimal) path oo-groupoids of objects in a "smooth (oo,1)-topos". I don't want to impose that fully (oo,1)-material on the main nLab as yet, before this is further developed, but the bits now added that just have oo-groupoids of paths in a 1-topos object is straightforward enough to warrent discussion here. i think.

      While working on this, I also did various minor edits on the synthetic differential geometry context cluster, such as

      splitting off lined topos from smooth topos

      rewriting the introduction at Models for Smooth Infinitesimal Analysis (the previous remarks are by now better explained in the corresponding sub-entries and the new summary is supposed to get the main message of the book across better). Also created section headers there for each of the single models, which I hope I'll eventually describe there in a bit more detail each. Those toposes \mathcal{N} and \mathcal{B} they have there are mighty cool, I think, giving not only a well-adapted model for SDG but on top of that an implementation of nonstandard analysis, and of distribution theory. I am thinking that the toposophers among my co-laborants might enjoy looking at their smooth natural number object in \mathcal{N} a bit more. It's fun and seems to be much more relevant than seems to be widely appreciated.

      Notice that at simplex in a lined topos I am asking for a reference. It's this standard construction of simplices as collpsed cylinders on lower dim simplicies. I don't think I should re-invent that wheel. What's the canonical reference for this general construction?

      Finally please notice that all entries mentioned above are more or less stubby for the moment and need more work. But I thought it was about time to drop a latest-changes alert here now, before waiting longer.

    • 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 \mathcal{N} and \mathcal{B} 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.

    • Spent all day with being distracted from this single thing that I planned to finish this morning: now at least a rough sketch is done

      at infinitesimal interval object in the last section with the long section name I mean to define the "infinitesimal singular simplicial complex" in a new way.

      Anders Kock defines this guy for "formal manifols", roughly, for spaces that have an atlas by vector spaces. There the simple definition applies recalled at infinitesimal singular simplicial complex.

      But there should be a definition for arbitrary microlinear spaces, And it should be such that it is almost manfestly the inifnitesimal version of the path oo-groupoid construction described at interval object. This is what I am aiming to describe here.

      One crucial thing is that we want that morphisms out of the objects in degree k of the infinitesimal singular simplicial complex that vanish on degenerate k-simplices are automatically fiberwise skew-linear. Seeing this in the construction that I am presenting there seems to be different to the way Anders Kock describes it in the other setup. This is the main thing I need to check again when i am more awake.

    • I created homotopy - contents and added it as a floating table of contents to relevant entries.

      This was motivated from the creation of infinitesimal interval object and the desire to provide a kind of map that shows how that concept sits in the greater scheme of things. The homotopy - contents was supposed to be a step in that diretion.

      I really meant to expand at infinitesimal interval object on something I already meant to provide yesterday, but then that table of contents kept distracting me, and the fact that I came across mapping cone while editing it and felt compelled to improved that entry first, which I did

    • I removed my recent material at simplex in a lined topos and instead inserted this now, expanded, at

      interval object

      where it belongs. There is now a section there that discusses how interval objects gives rise to cubical and simplicial path oo-categories.

      The proposition I state there I have carefully checked. Should be correct. But haven't typed the proof, it doesn't lend itself to being typed (straightforward but tedious, as one says).

      But if it is indeed correct, this must be standard well-known stuff. Does anyone have a reference?!

      I also restructured and edited the rest of the entry a bit, trying to make it a bit nicer. THis entry deserves more attention, it is a pivotal entry.

      Tomorrw when I am more awake I'll remove simplex in a lined topos and redirect links to it suitably to interval oject.

    • I moved the instructions on making diagrams from FAQ to HowTo, which seemed a better fit, and added a comment about including images as another method. I also made the individual questions at FAQ into ### headers, rather than numbered lists, so that they would show up in the automatic table of contents.

    • created stub for smooth loci

      (should it be "smooth locus" instead?)

    • sty addition to generalized smooth algebra: remark on terminology added and section on "internal definition" added.

      planning to polish thinmgs later

    • created entry on Israel Gelfand with the material that John posted to the blog.

      turns out the "Timeline" entry was already requesting it

    • created microlinear space

      One thing I might be mixed up above:

      in the literature I have seen it seems to say that

      $ X^D x_X X^D \simeq X^{D(2)}$

      with

      $ D(2) = { (x_1,x_2) \in R \times R | x_i x_j = 0} $.

      But shouldn't it be

      $ D(2)' = { (x_1,x_2) \in R \times R | x_i^2 = 0} $.

      ?

    • quick addition of "formal infinitesimal spaces" and Weil algebras to infinitesimal space

      but am planning to polish this entry further later, it is a bit of a mess at the moment

    • A little more detail at natural isomorphism, including when one can speak of the functor satisfying certain conditions.

    • I worked on synthetic differential geometry:

      I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

      Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.

    • created smooth topos on the axioms on toposes used in synthetic differential geometry.

    • By chance I came across an old CatTheory mailing list post by e. Dubuc, where he complains about how is work on SDG is not sufficiently recognized and asks people to speak of the "Dubuc topos".

      I added a remark about this to synthetic differential geometry in the section on "Well adapted models".

    • A few more sections at A Survey of Elliptic Cohomology - elliptic curves on

      • topol. invariants of the moduli stack of ell. curves

      • the compactified version

      • the definition of Gromov-Witten invariants

      • an example.

      As before, this is raw material which I am thinking lends itself to be turned into entries.