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    • I added the case of Set-enriched category theory to the example section of Cauchy complete category (thanks to David Corfield for fixing my LaTeX errors), and inserted the definition at Karoubi envelope. There is an issue of choosing how to split idempotents which someone like Toby might want to say something about.

    • created page for Johan Louis Dupont, cited at simplicial deRham complex

      (given that at that entry I am trying to merge some of Dupont's work with some of that of Anders Kock, it is curious that JL Dupont and Anders Kock are decade-long colleagues in Aarhus, as Anders Kock kindly reminds me a minute ago)

    • To the entry on regular category I added a brief note describing an application of this idea and the calculus of relations to a paper of Knop. For the future I will try to flesh this note out as well as add a page on tensor categories.

      By the way, does the definition of a tensor category have to include linearity? It seems that the definitions vary depending on where one looks (e.g. whether the monoidal structure is an additive functor). Thanks.

    • I started a section

      dependence on the underlying site at model structure on simplicial presheaves.

      So far this quotes a result from Jardine's lectures and then looks a bit at an example.

      At that example I would really like to conclude that the Quillen adjunction discussed there is actually a Quillen equivalence. But I have to interrupt now to make a telephone call... :-)

    • I started an entry simplicial deRham complex

      on differential forms on simplicial manifolds.

      In parts this is for me to collect some standard references and definitions (still very incomplete on that aspect, help is appreciated -- is there a good reference by Dupont that is online available?)

      and in parts this is to discuss the deeper abstract-nonsense origin of this concept.

      I am thinking that

      • with differential forms understood in the synthetic context as just the image under Dold-Kan of the cosimplicial algebra of functions on the simplicial object of infinitesimal simplices in some space

      • it follows that the simplicial deRham complex of a simplicial object is just the image under Dold-Kan of the cosimplicial algebra of functions on the realization of the bisimplicial object of infinitesimal simplices in the given simplicial space.

      This looks like it is prretty obvious, once one stares at the coend-formula, but precisely that makes me feel a bit nervous. Maybe i am being too sloppy here. Would appreciate you eyeballing this.

    • Began entry with that name.

    • I wrote Poincare group as an entree to the project of carrying on in nLab the blog discussion on unitary representations of the Poincare group. I'm not a specialist of course, so I ask the experts to please examine for accuracy.

    • I expanded and polished the discussion of the abstract definition of of G-principal oo-bundles in an arbitrary (oo,1)-topos at principal infinity-bundle.

      Parts of this could/should eventually be moved/copied to action and action groupoid, but I won't do that now.

      I'd be interested in comments. One would expect that for the case that the ambient (oo,1)-topos is Top this style of definition should be well known in the literature, but I am not sure if it is. In fact, the examples listed further below in the entry, (the construction by Quillen and the Stasheff-Wirth construction) seems to indicate that this very simple very general nonsense picture has not been conceived as such before. Could that be true?

    • I've removed the request for help link from the main contents. It didn't get used much (though I got answers to my questions there!). Since we have yet to actually delete a page, rather than just blank the request for help page I've put a pointer to where one can ask questions (pretty similar to that on the FAQ).

    • I created a page for S-Sch as a notation for S-schemes to refer to in another post. Zoran pointed out that the notation is nonstandard (I do not know why I thought it was normal) and changed the title to Sch/S. I thus changed the first sentence to read Sch/S instead.

    • I added a description of the degenerate affine Hecke algebra to the Hecke algebra page as one of the many variants.

      I added the categorical generalization of Schur's lemma to that page.

      I wrote a short stub on the additive envelope of a category, which Mike Shulman has expanded.

      I mentioned the generalization of the Morse lemma to Hilbert manifolds.

      I added the generalization of Hilbert's basis theorem to the case of where the ground ring is noetherian (not necessarily a field).

      I wrote a short page on the Eilenberg swindle.

    • I see that Akil Mathew has worked on a bunch of entries. Great! We should try to contact him and ask hom to record his changes here.

    • I added Alex's recent lecture notes to cobordism hypothesis and in that process polished some typesetting there slightly.

      Then I was pleased to note that Noah Snyder joined us and worked on fusion category. I created a page for him.

    • I don't think that the (non-full) essential image of an arbitrary functor is well-defined.

    • I added a fairly long (but still immensely incomplete) examples section to smooth topos.

      I mention the "well adapted models" and say a few words about the point of it. Then I have a sectoin on how and in which sense algebraic geometry over a field takes place in a smooth topos. here the model is described easily, but I spend some lines on how to think of this. In the last example sections I have some remarks on non-preservation of limits in included subcategories of tame objects, but all that deserves further expansion of course.

    • 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

    • 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} $.

      ?

    • 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

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