# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• Added in the formula giving the associated crossed complex from the Moore complex of the simplicial group(oid).

• Page created, but author did not leave any comments.

• Because of the algebraic Kan complex entry I had a look at the simplicial T-complex page. I am not sure that the current page is quite right in its wording. It is a bit the age old problem of structure or properties. In the algebraic Kan complex, the filler choice function is part of the structure. In a T-complex the thin elements form part of the structure but then properties of the thin elements show that there is a unique choice function taking thin values. They then satisfy some equational conditions.

My thought would be that there should be a bit more precision on the differences between them. For instance I think it is true (but I would need to prove it in detail) that any simplicial T-complex gave an algebraic Kan complex, yielding an ’inclusion functor’ from SimpT to Alg Kan. That functor should have a left adjoint which kills off the Whitehead products etc, (that need not be trivial for an algebraic Kan complex but are for a simplicial T-complex). I do not see how to construct this explicitly but am sure there must be a simple way of imposing conditions on an alg. Kan complex and looking at ’varieties’ in that category. (I have not read Thomas’s thesis and he may have done something related to this already.) In other words, can one impose equations on alg. Kan complexes, in this way. The present definition is more or less the free algebras case (?).

Before altering the simp. T-complex page, I thought it worth asking this question of ’varieties’ as the answer (if it is known) would influence how best to do the edit.

• Thomas's guest post at cafe and his paper should maybe be reflected in entry infinity-category and other places in nlab where various "models" fro infinity categories are listed, as it should have a very important role in my opinion, but still better experts should do carefully these changes. I might give a slightly uninformed interepretation of the role of this work in comparison to the experts like Mike.

• some minimum, for the moment just so as to record the recent results by the FIRE-2 computer simulation

• added to Grothendieck construction a section Adjoints to the Grothendieck construction

There I talk about the left adjoint to the Grothendieck construction the way it is traditionally written in the literature, and then make a remark on how one can look at this from a slightly different perspective, which then is the perspective that seamlessly leads over to Lurie's realization of the (oo,1)-Grothendieck construction.

There is a CLAIM there which is maybe not entirely obvious, but straightforward to check. I'll provide the proof later.

• I have given Grothendieck construction for model categories its own entry, in order to have a place for recording references. In particular I added pointer to the original references (Roig 94, Stanculescu 12)

(There used to be two places in the entry Grothendieck construction where an attempt was made to list the literature on the model category version, but they didn’t coincide and were both inclomplete. So I have replaced them with pointers to the new entry.)

• started a bare minimum at Poisson-Lie T-duality, for the moment just so as to have a place to record the two original references

• Hello,

I noticed DFT page has not been updated in a while and I added a couple of sections: some sketchy introductory material (analogy between Kaluza-Klein and DFT) and a little insight about a more rigorous geometrical formulation of DFT.

It is still quite sketchy but I would be happy to refine it.

PS: this is my first edit, I hope I played by the rules. And thank you all for this wiki

Luigi

• I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here

• a minimum, for the moment just so as to record some references on $Pin(2)$-equivariant homotopy theory (as kindly pointed out by David Roberts)

• Added in the usual group presentation of the dihedral group $D_{2n}$ plus a warning that this group is also denoted $D_n$ by some authors (including myself!!!)

• Page created, but author did not leave any comments.

Anonymous

• added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:

On general grounds, since orbifolds $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$

into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$

for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$

This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of global equivariant homotopy theory, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.

Concretely, in global equivariant homotopy theory the plain orbit category $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$

More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.

At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)

• added a line on $Pin_\pm(n)$, and added pointer to the example of Pin(2)

• started a stubby nPOV-description at the beginning of BV-BRST formalism

somebody please stop me, though, because I urgently need to be doing something else... :-)

• I gave the entry logical relation an Idea-section, blindly stolen from a pdf by Ghani that I found on the web. Please improve, I still don’t know what a “logical relation” in this sense actually is.

Also, I cross-linked with polymorphism. I hope its right that “parametricity” may redirect there?

• brief category:people-entry for hyperlinking references at

• trying to collect references on the state-of-the-art of computer simiulations on cosmic structure formations. Will try to expand as I find more…

• Page created, but author did not leave any comments.

• started a minimum at M-wave

(I was after the kind of statement as cited by Chu-Isono there, but have added now a minimum of the background literature, too).

• added references for higher curvature corrections in 11d supergravity