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    • I continued working my way through the lower realms of the Whitehead tower of the orthogonal group by creating special orthogonal group and, yes, orthogonal group.

      For the time being the material present there just keeps repeating the Whitehead-tower story.

      But I want more there, eventually: I have a query box at orthogonal group. The most general sensible-nonsense context to talk about the orthogonal group should be any lined topos.

      I am wondering if there is anything interesting to be said, from that perspective. Incidentally, I was prepared in this context to also have to create general linear group, only to find to my pleasant surprise that Zoran had already created that some time back. And in fact, Zoran discusses there an algebro-geometric perspective on GL(n) which, I think, is actually usefully thought of as the perspective of GL(n) in the lined topos of, at least, presheaves on  CRing^op .

      Presently I feel that I want eventually a discussion of all those seemingly boring old friends such as  \mathbb{Z} and  \mathbb{R} / \mathbb{Z} and  GL(n) etc. in lined toposes and smooth toposes. Inspired not the least by the wealth of cool structure that even just  \mathbb{Z} carries in cases such as the  \mathbb{B} -topos in Models for Smooth Infinitesimal Analysis.

    • created a page for Haynes Miller, since I just mentioned his name at string group as the one who coined that term.

      not much on the page so far. Curiously, I found only a German Wikipedia page for him

    • I've started listing differences between iTeX and LaTeX in the FAQ. That seemed the most logical place (I don't think we want a proliferation of places where users should look to find simple information) so either here or the HowTo seemed best. I chose the FAQ because the most likely time someone is going to look for this is when they notice something didn't look right.

      The issue is that whilst iTeX is meant to be close to LaTeX they are never going to be the same so it's worth listing known differences with their work-arounds.

      So far I've noted operator names, whitespace in \text, and some oddities on number handling.

    • Vishal Lama joined the Lab!

      on his page he promises to create Lab pages on some books on category theory and topos theory. Great, I am looking forward to it

    • I have typed into infinitesimal interval object a detailed description of the simplicial object inuced on a microlinear space from the infinitesimal interval in immediate analogy to the construction of the finite path simplicial object induced from an interval object (as discussed there).

      I also give the inclusion of the infinitesimal simplicial object into the finite one.

      All the proofs here are straightforward checking, which I think I have done rather carefully on paper, but not typed up. What I would appreciate, though, is if somebody gave me a sanity check on the definition of the infinitesimal simplicial object (which is typed in detail).

      In the very last section, which is the one that is still just a sketch, I am hoping to describe an isomorphism from my simplicial infinitesimal object to that considered by Anders Kock, which is currently described at infinitesimal singular simplicial complex in the case that the space X satisfies Kock's assumptions (it must be a "formal manifold").

      The construction I discuss at infinitesimal interval object is supposed to generalize Kock's construction to all microlinear spaces and motivated by having that canonical obvious inclusion into the finite version at interval object.

      The isomorphism should be evident: my construction evidently yields in degree k k-tuples of pairwise first oder neighbours if the space X admits that notion. But I want to sleep over this statement one more night...

    • Roger Witte asks a question at foundations that looks interesting but which I haven't really thought about yet.

    • I added the Lab itself to Online Resources, since that list is getting a lot of attention and may well be copied to other contexts.

    • 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} $.

      ?