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added to inter-universal Teichmüller theory a pointer to the recent note
(Though after reading I am not sure if that note helps so much.)
I reorganized linearly distributive category by moving the long block of history down to the bottom, adding an “Idea” section and a description of how -autonomous categories give rise to linearly distributive ones and linearly distributive ones give rise to polycategories. I also cross-linked the page better with polycategory and star-autonomous category.
I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?
added statement of and references to the weak equivalence with the Fulton-MacPherson operad (here)
added the same statement also at Fulton-MacPherson operad
Added to fix a dead link.
am splitting off simplicial principal bundle from simplicial group
Made some edits and additions at simplicial complex.
starting something, on the kind of theorems originating with
Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)
There is an empty page entitled Towards an enumerative geometry of the moduli space of curves.
a bare subsection with a list of references, to be !include-ed into the References-section of relevant enties, such as at triangulation and at triangulation theorem, for ease of synchronizing
brief category:people-entry for hyperlinking references at triangulation theorem and Riemann surface
brief category:people-entry for hyperlinking references at triangulation theorem
brief category:people-entry for hyperlinking references at topological manifold
brief category:people-entry for hyperlinking references at triangulation theorem and 3-manifold
brief category:people-entry for hyperlinking references at triangulation theorem
On the occasion of Brandenburg 14 I have – finally – created an entry 2-algebraic geometry.
We had almost created that a few times before, only that we never did. Maybe the closest we came in the section Derived algebraic geometry – Relation to noncommutative geometry.
I have tried to do some minimum cross-linking, with 2-ring, etc. But one might want to do more.
brief category:people-entry for hyperlinking references at infinity loop space and E-infinity ring spectrum
am slowly starting to add some genuine content to twisted K-theory
brief category:people-entry for hyperlinking references at higher twisted K-theory
Created:
The Swiss cheese operad is an analogue of the little disks operad, where disks are replaced by half-disks, which contain both ordinary disks in their interior, as well as half-disks positioned at the flat boundary.
This structure can be organized into an operad in the category of modules over the little disks operad.
Alexander Voronov, The Swiss-Cheese Operad, arXiv:math/9807037.
Najib Idrissi, Swiss-Cheese operad and Drinfeld center, arXiv:1507.06844.
I gave regular cardinal its own page.
Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.
Looks better to me, but let me know what you think.
Created:
An operad is a monoid in the monoidal category of symmetric sequences equipped with the substitution product.
A module over an operad is just a right module over this monoid.
Right modules are very different from left modules, the latter are essentially algebras over an operad.
V. A. Smirnov. ON THE COCHAIN COMPLEX OF TOPOLOGICAL SPACES. Mathematics of the USSR-Sbornik 43:1 (1982), 133–144. doi.
Martin Markl, Models for operads, Comm. Algebra 24 (1996), no. 4, 1471–1500. arXiv:hep-th/9411208v1.
I have added pointer to
Their Prop. 7.2.2 is verbatim the characterization that BCMMS made the definition of “bundle gerbe module” a month and a half later (except that LU focus on open covers instead of more general surjective submersions, but that’s not an actual restriction and in any case not the core of the definition).
Also added pointer to
which essentially recovers Lupercio & Uribe’s Def. 7.2.1.
From the arXiv timestamps I gather that it must have been an intense couple of weeks for all these auhtors in spring 2001. But Lupercio & Uribe came out first, by a fair margin. And in equivariant generality, right away…
I have been working on the entry twisted bundle.
Apart from more literature, etc. I have started typing something like a first-principles discussion: first a general abstract definition from twisted cohomology in any cohesive -topos, then unwinding this in special cases to obtain the traditional cocycle formulas found in the literature.
Needs more polishing here and there, but I have to pause now.
added pointer to
brief category:people-entry for hyperlinking references at Gromov-Witten theory
added pointer to:
added this pointer:
added pointer to:
I have added pointer to the arXiv copy to the item
Added some references at Morse theory.
brief category:people-entry for hyperlinking references at birational geometry
brief category:people-entry for hyperlinking references at Hilbert scheme and birational geometry
Have added a bunch of references to this entry.
Question: What precisely can one say about the relation between the topological space underlying the Hilbert scheme of points of and/or , and the Fulton-MacPherson compactification of the corresponding configuration spaces of points?
There is commentary in just this direction on p. 189 of:
but it remains unclear to me what exactly the statement is, in the end.
brief category:people-entry for hyperlinking references at homotopy of rational maps
added to Eckmann-Hilton argument the formal proposition formulated in any 2-category.
BTW, doesn’t anyone have a gif with the nice picture proof?
brief category:people-entry for hyperlinking references at homotopy of rational maps
brief category:people-entry for hyperlinking references at homotopy of rational maps
I have expanded the list of references at differential K-theory with a few comments thrown in.
added publication data to:
Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on -algebras in the topos language really nicely…).
I have to rush to the dinner now. But to remind myself, I have added cross-links between Boolean topos and measurable space and for the moment pointed to
for more. Simon Henry’s thesis will be out soon.
Have to rush now…
created over-topos
added a list of examples to Lie groupoid
added a sentence to the Idea-section at Kan complex
this is a bare list of (commented) references, to be !include-ed as a subsection into the References-section of relevant entries (notably at model structure on simplicial sest and at classical model structure on simplicial sets, from whose joint content this list here is compiled), for ease of synchronizing across entries
I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.
For the record, I found the inductive characterisation in Cisinski’s book [Les préfaisceaux comme modèles des types d’homotopie, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [Notes on simplicial homotopy theory], but they take it as a definition.