added to *inter-universal Teichmüller theory* a pointer to the recent note

- Yamashita,
*FAQ on ‘Inter-Universality’*(pdf)

(Though after reading I am not sure if that note helps so much.)

]]>Every now and then some spam gets through the filters on the forum. Toby and I can remove it from public view (it stays in the database so if it turns out that it wasn't spam it can easily be deleted). Also, I can set the user account which was used to "banned" to prevent it being used again. Other useful information, such as IPs and email addresses, is also logged. But to keep the place clean, we need to be aware of the spam occurring. I, in particular, don't read every post made here on the forum. So if you spot some spam, please alert someone. Use this discussion if you like.

]]>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.

]]>We are finalizing an article:

$\,$

$\,$

**Abstract.** *We extend the Chern character on K-theory, in its generalization to the Chern-Dold character on generalized cohomology theories, further to twisted non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L-∞ algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory, which we review in streamlined form, to a twisted non-abelian generalization of the de Rham theorem. We show that this non-abelian character reproduces the Chern-Weil homomorphism on non-abelian cohomology in degree 1, represented by principal bundles; and thus generalizes it to higher non-abelian cohomology, represented by higher bundles/higher gerbes. As a fundamental example we discuss the character map on twistorial Cohomotopy theory over 8-manifolds, which is a twisted non-abelian enhancement of the Chern-Dold character on topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory.*

$\,$

Comments are welcome. Please grab the latest version of the file from behind the above link.

]]>starting something, on the kind of theorems originating with

- Graeme Segal,
*The topology of spaces of rational functions*, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)

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

starting something – not done yet but I need to save

]]>am splitting off simplicial principal bundle from simplicial group

]]>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.

]]>I think we should expand this list of languages!

Anonymous

]]>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

- Ernesto Lupercio, Bernardo Uribe, Section 7.2 of:
*Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

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

- Marco Mackaay,
*A note on the holonomy of connections in twisted bundles*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62. (arXiv:math/0106019, numdam:CTGDC_2003__44_1_39_0)

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…

]]>starting some minimum

]]>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 $\infty$-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.

]]>