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

• am splitting this off from holographic QCD, since the latter is getting too crowded. Prompted by today’s

• following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

• Added some selected writings. Could add more but will stop for the moment.

• The stub for “associative” bialgebroid. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at Hopf algebroids.

• I have been trying to polish weak homotopy equivalence by adding formal Definition/Proposition-environements. Also expanded the Idea-section and edited here and there.

The following remark used to be in the entry, but I can’t see right now how it makes sense. If I am mixed up, please clarify and I’ll re-insert it into the entry:

It is tempting to try to restate the definition as “$f$ induces an isomorphism $f_*: \pi_n(X,x) \to \pi_n(Y,f(x))$ for all $x \in X$ and $n \geq 0$,” but this is not literally correct; such a definition would be vacuously satisfied whenever $X$ is empty, without regard to what $Y$ might be. If you really want to go this way, therefore, you still must add a clause for $\Pi_{-1}$ (the truth value that states whether a space is inhabited), so the definition is no shorter.

Then, there used to be the following discussion box, which hereby I am moving from there to here. I have added a brief remark on how weak homotopy equivalences are homotopy equivalences after resolution. But maybe it deserves to be further expanded.

[ begin forwarded discussion ]

+–{.query} Is there any reason for calling these ’weak’ homotopy equivalences rather than merely homotopy equivalences? —Toby

Mike: By “these” I assume you mean weak homotopy equivalences of simplicial sets, categories, etc. My answer is yes. One reason is that in some cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

Toby: Interesting. I would have guessed that any weak homotopy equivalence could be strengthened to a homotopy equivalence in this sense, but maybe not.

Tim: I think the initial paragraph is somehow back to front from a philosophical point of view, as well as a historical one. Homotopy theory grew out of studying spaces up to homotopy equivalence or rather from studying paths in spaces (and integrating along them). This leads to some invariants such as homology and the fundamental group. Weak homotopy type (and it might be interesting to find out when this term was first used) is the result and then around the 1950s with the development of Whitehead’s approach (CW complexes etc.) the distinction became more interesting between the two concepts.

I like to think of ’weak homotopy equivalence’ as being ’observational’, i.e. $f$ is a w.h.e if when we look at it through the observations that we can make of it, it looks to be an ’equivalence’. It is ’top down’. ’Homotopy equivalence’ is more ’constructive’ and ’bottom up’. The idea of simple homotopy theory takes this to a more extreme case, (which is related to Toby’s query and to the advent of K-theory).

With the constructive logical side of the nLab becoming important is there some point in looking at this ’constructive’ homotopy theory as a counter balance to the model category approach which can tend to be very demanding on the set theory it calls on?

On a niggly point, the homotopy group of a space is only defined if the space is non-empty so one of the statements in this entry is pedantically a bit dodgy!

Toby: I would say that it has a homotopy group at every point, and this is true even if it is empty. You can only pretend that it has a homotopy group, period, if it's inhabited and path-connected.

Anyway, how do you like the introduction now? You could add a more extensive History section too, if you want.

Tim: It looks fine. I would add some more punctuation but I’m a punctuation fanatic!!!

With all these entries I suspect that in a few months time we will feel they need some tender loving care, a bit of Bonsai pruning!! For the moment lets get on to more interesting things.

Do you think some light treatment of simple homotopy theory might be useful,say at a historical level? =–

[ end forwarded discussion ]

• Started this, largely to disambiguate.

• brief category:people-entry for hyperlinking references

• I noticed that an entry with this title was missing. Created a bare minimum here, for the moment just so that the link works.

• brief category:people-entry for hyperlinking references

• Just started and I’m called away, but I’ll save anyway.

• I've split module over a monad off from algebra for an endofunctor. It still needs work, notably the definition of tensor product of bimodules, but it's late and I'm tired.

Also I added a remark to internal category about internal cats as monads in the bicategory of spans. I'm leading up to talking about internal profunctors and 2-sided fibrations, which Mike has been helping me understand at the café.

• added a pointer for the higher dimensional case

• M. H. A. Newman, Theorem 7 in: The Engulfing Theorem for Topological Manifolds, Annals of Mathematics Second Series, *84** 3 (1966) 555-571 (jstor:1970460)

(prompted by discussion in another thread, here)

• made explicit the conclusion that forming Lie groupoid convolution algebras is a (2,1)-functor

$C^\ast(-) \;\colon\; DiffStack^{prop} \overset{\phantom{AAA}}{\longrightarrow} C^\ast Alg^{op}_{bim}$

here

• added to groupoid a section on the description in terms of 2-coskeletal Kan complexes.

• had added to finite group two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres.

What I’d really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.

• I am splitting off an entry classification of finite rotation groups from ADE classification in order to collect statements and references specific to the classification of finite subgroups of $SO(3)$ and $SU(2)$.

Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.

• Hi,

I was going to add some details to Godement product, but I can’t reproduce what is there and suspect a typo.

For categories $A,B,C$, if $\alpha: F_1\to G_1 : A\to B$ and $\beta: F_2\to G_2 : B\to C$ are natural transformations of functors, the components $(\alpha * \beta)_M$ of the Godement product $\alpha * \beta: F_2\circ F_1\to G_2\circ G_1$ are defined by any of the two equivalent formulas:

$(\beta * \alpha)_M = \beta_{F_2 M}\circ G_1(\alpha_M)$ $(\beta * \alpha)_M = G_2(\alpha_M)\circ\beta_{F_1 M}$

Following MacLane (page 42), the natural transformation $\alpha:F_1\Rightarrow G_1$ implies the existence of a morphism $\alpha_M:F_1(M)\to G_1(M)$. This, together with the natural transformation $\beta:F_2\Rightarrow G_2$, implies

$\array{ F_2\circ F_1(M) & \stackrel{F_2(\alpha_M)}{\to} & F_2\circ G_1(M) \\ \beta_{F_1(M)}\downarrow && \downarrow \beta_{G_1(M)} \\ G_2\circ F_1(M) & \stackrel{G_2(\alpha_M)}{\to} & G_2\circ G_1(M) } \,.$

I thought the component of the Godement product, i.e. horizontal composition in Cat, should be the diagonal of this diagram so that

$(\beta\circ\alpha)_M = \beta_{G_1(M)}\circ F_2(\alpha_M) = G_2(\alpha_M)\circ \beta_{F_1(M)}.$

Is there a typo on the page or am I completely missing the mark?

Note: Interchanging $F_2\leftrightarrow G_1$ in the two formulas on the page would give my two formulas.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

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

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Switch commuting reasoning to use an implication for clarity.

• Created page to link with several references.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Disambiguated the article, moved some content to polytope.

• I have touched formatting and wording of this entry, in an attempt to polish it up a little.

• B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, corollary 15.3.4 of Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985
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• This term used to redirect to Hopf algebra. But we need it also in another sense at real projective space and elsewhere, so I am making this here a brief category:diambiguation-entry.

Tim

• am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

• brief category:people-entry for hyperlinking references

• added references for higher curvature corrections in 11d supergravity

• Fixed broken link to David Spivak’s thesis, (was: http://math.berkeley.edu/~dspivak/thesis2.pdf)

Anonymous

• starting an entry (finally) on “singular cohesion”: the square of cohesions of $\infty$-presheaves over the global orbit category with values in an $\infty$-topos which itself is cohesive over $Grpd_\infty$.

Not done yet, but need to save

• 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)

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• I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.

• Thanks to an alert by Dmitri here, I realized that this entry had two spurious relative entries, one titled

• “Chern-Simons 2-gerbe”

which did nothing but point here

and one titled

• “Chern-Simons gerbe”

which did not even do that.

I have cleared these entries and instead made their titles be redirects to here.

• Apparently an orphaned article. Do we still need it?

Anonymous

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

Anonymous