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• am splitting off this entry from exotic smooth strcuture, in order to facilitate linking to specifically the case of exotic 7-sphere.

Accordingly, so far the bulk of the entry is just copied over from the corresponding section at exotic smooth structure,

But I also added a new paragraph,

and that is what motivated me to split this off. Namely it occured to me that from the point of view of M-theory on 8-manifolds, Milnor’s classical construction of exotic smooth 7-spheres as boundaries of 8-manifolds is very particularly the construction of near horizon limits of black M2-brane spacetimes in the context of M-theory on 8-manifolds.

This must be known in the literature, and I’d like to collect what is known about it. So far I found a brief comment in this direction, in section 3.2 of

Will be adding more as I find more.

• brief category:people-entry for hyperlinking references

• added to KK-theory brief remark and reference to relation to stable $\infty$-categories / triangulated categories

• brief category:people-entry for hyperlinking references

• Sum types are distributive coproducts, not just coproducts. I’ll make a page for distributive coproduct next.

• Unfortunately, there are two entries on the same topic, both created by Urs: quantum Hall effect (redirecting also fractional quantum Hall effect what should eventually split off) with some substance, and the microstub quantum hall effect. I would like to create quantum spin Hall effect and I think I should rename/reclaim the stub quantum hall effect for this. Do others agree ? Urs ?

As the action is now delayed I record here the reference which I wanted to put there

• B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, Quantum spin Hall rffect and topological phase transition in HgTe quantum wells, Science 15 December 2006: 314, n. 5806, pp. 1757-1761 doi

Somewhat surprisingly, the authors and roughly this work of them are mentioned (though not in the list of references) in a paper in algebraic geometry

which considers the mirror symmetry and topological states of matters (topological insulators in particular) as main applications.

• a bare list of references, to be !include-ed into the References-section of relevant entries (such as at braid group representation and at semi-metal).

Had originally compiled this list already last April (for this MO reply) but back then the nLab couldnt be edited

• (This is my first foray into nLab, so sorry if I'm making elementary errors.)
In the definition of left adjoint of a functor U:C→ D, the claim is that it's a functor F:D → C s.t. ∃ natural transformations
ι:id_C → F;U
ϵ:U;F → id_D
But F;U is a morphism in D and U;F is a morphism in C.
Is something wrong here, have I misunderstood the notation F;U, is there a more general version of a natural transformation being used here, or what?
Thank you.
• brief category:people-entry for hyperlinking references

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

Anonymous

• the term “pre-spectrum” used to re-direct to spectrum and was explained a little more at sequential spectrum. Am giving the term its own page now, for better clarification, prompted by the discussion in another thread (here)

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Anonymous

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Anonymous

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Anonymous

• starting discussion thread for this post

Anonymous

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Anonymous

• I have touched H-space, slightly expanded here and there and slightly reorganized it.

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Anonymous

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Anonymous

• starting something – remains a stub for the moment, to be continued

• adding explicit definition of a 2-poset

Anonymous

• This is a bare list of references, to be !include-ed into the References-lists of relevant entries (such as at anyon, topological order, fusion category, unitary fusion category, modular tensor category).

There is a question which I am after here:

• This seems to be CMT folklore, as all authors state it without argument or reference.

Who is really the originator of the claim that anyonic topological order is characterized by certain unitary braided fusions categories/MTCs?

Is it Kitaev 06 (which argues via a concrete model, in Section 8 and appendix E)?

• brief category:people-entry for hyperlinking references

• $A\times B$ is not the coproduct of $A$ and $B$ in Rel.

Anonymous

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Anonymous

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Anonymous

• On the page hom-functor, it says

There is also a contravariant hom-functor

$hom(-,c) : C^{op} \to Set,$

where $C^{op}$ is the opposite category to $C$, which sends any object $x \in C^{op}$ to the hom-set $hom(x,c)$.

If you write it like this, should you really call it “contravariant”? When you write $C^{op}$, I thought you should call it just “functor” or “covariant”. By saying it is contravariant AND writing $C^{op}$, it seems like double counting.

I hope to add some illustrations to these pages. It is a shame there are not more illustrations on the nLab since nStuff is so amenable to nice pictures.

• copied from HoTT wiki

Anonymous

• brief category:people-entry for hyperlinking references

• Todd,

you added to Yoneda lemma the sentence

In brief, the principle is that the identity morphism $id_x: x \to x$ is the universal generalized element of $x$. This simple principle is surprisingly pervasive throughout category theory.

Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element $id_x$ is a tautological statement that does not need or imply the Yoneda lemma, it seems.

• added to complete Segal space a discussion of what an ordinary category looks like when regarded as a complete Segal space.

(This is meant to be pedagogical, therefore the recollection of all the basics at the beginning.)

• copying text from HoTT wiki

Anonymous

• added text from the HoTT wiki

Anonymous

• added text from HoTT wiki

Anonymous

• a stub, need the link to work, but nothing to be seen here just yet

• brief category:people-entry for fixing the long requested links at structured cospan

• This is my first (substantial) contribution to the nLab, so forgive my likely ineptitude. This wants to be a initial stub, everything is basically scraped from the reference and this post: https://golem.ph.utexas.edu/category/2019/07/structured_cospans.html Clearly a lot of material can be added, included a better definition and clearer examples. It’s also quite necessary to make a page for decorated cospans. I might start it myself later this month.

mattecapu

Anonymous

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• copied from HoTT wiki

Anonymous

• fixed broken links in references.

• copied from HoTT wiki

Anonymous

• copied from HoTT wiki

Anonymous

• copied from HoTT wiki

Anonymous

• copied from the HoTT wiki

Anonymous

• copied from HoTT

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• copied from HoTT wiki, added context sidebar

Anonymous

• copied from HoTT wiki

Anonymous