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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
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} $.
?
started synthetic differential geometry - contents and added it as floating table of contents to the relevant entries
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
started integration axiom, but incomplete for the moment
at schreiber:integration of oo-Lie algebroid valued differential forms I am thinking about the higher version of this
Looks like maybe Todd is right at cyclic order that the cycle category cannot be defined in that way.
Just learnt about detexify from the Secret Blogging Seminar and thought it worth adding to the online resources page.
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".
I found another term for a tight relation.
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.
started synthetic differential supergeometry
so far just to record the (three!) references
created entries for the following people
Riccardo D'Auria, Pietro Fre, Leonardo Castellani, Hisham Sati
not much there, yet, but these pages serve a purpose as listing the pages that link to them, which is useful I think
Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.
Added the Yoneda-embedding way to talk about group objects and hence supergroups.
A list of all recently edited entries can be seen at Recently Revised. But that list tends to contain lots of minor changes: it's not easy to spot the important ones. So, if you feel people's attention should be drawn to some changes you make, then please mention them here. This way the rest of us can spot them, so we can learn what you know — and maybe make further improvements!
[[wikilink]] to the page or pages that you edited.Questions? Ask them here!
created String Lie 2-algebra
uploaded course notes by Bryant to exterior differential system (linked to from the Reference section)
Added references to blob homology.
Here are archives of the changes before 2009 October:
See also all changes.