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

• I noticed that an entry bifunctor was still missing, though requested by some existing entries. So I briefly added something.

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Anonymous

• To provide an author link.

• Describing the arrangements which have been made for funding of the nLab in collaboration with the Topos Institute. The page, linked to from the home page, is intended to be fairly general; specific requests for donations can be made elsewhere.

• have created enriched bicategory in order to help Alex find the appropriate page for his notes.

• Some tidying up and additions at simplex category, in particular a section on its 2-categorical structure, and more on universal properties.

I’ve edited the definition to focus more on the augmented simplex category $\Delta_a$ instead of the ’topologists’ $\Delta$’, but I haven’t changed their names, because it seemed to me that that was the best way to keep everyone involved in the discussion at that page happy. (I also changed the ordinal sum functor from $+$ to $\oplus$, after Tim’s suggestion.)

• New sections at coring:

• base extension of corings
• morphisms in the 1-category of corings over variable base rings.
• This article has a weird claim on top, highlighted in yellow (see the second line):

Redirected from “local Langlands correspondence”.

Note: local Langlands conjecture and local Langlands conjecture both redirect for “local Langlands correspondence”.

• Redirect: Barwick-Kan equivalence.

Renamed.

Also tried to fix the ordering of the rest of the reference, but I give up for the moment. This needs attention by some expert

• There was a big chunk of technical discussion on this page. I am moving this to algebraic cobordism, where it belongs.

• This is a bare list of references, to be !include-ed into relevant entries, such as at swampland and 24 branes transverse to K3, for ease of cross-linking and updating.

I am taking the liberty of including a pointer to our upcoming M/F-Theory as Mf-Theory which has some details on a precise version of the conjecture and a proof (from Hypothesis H).

• I have added at HomePage in the section Discussion a new sentence with a new link:

If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page Welcome to the nForum (nlabmeta). I re-doscivered it only after my recent related comment here.

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Anonymous

• I started an article about Martin-Löf dependent type theory. I hope there aren't any major mistakes!

One minor point: I overloaded $\mathrm{cases}$ by using it for both finite sum types and dependent sum types. Can anyone think of a better name for the operation for finite sum types?

• Tim Porter added references to microbundle and I edited the formatting of the entry a bit

• Removed the following discussion to the nForum:

Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map $p:E\to B$ and a numerable covering of $B$ such that the restrictions $p^{-1}(U)\to U$ for every $U$ in the covering is a Hurewicz fibration then $p$ is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s Elements of homotopy theory is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).

Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.

Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that $\Top$; for example to the classical model structure on $Cat$?

• Collected some basic facts.

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• Added some basic examples from HTT. There doesn’t seem to be a page for the corresopnding 1-categorical notion. This notion is used pretty heavily in $\infty$-category theory, but it’s not so familiar from 1-category theory. But I’d have to think a 1-categorical treatment exists somewhere, right?

• (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.
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Anonymous

• Mention relevance to (bo, ff) factorization system.

• The entry unit of an adjunction had a big chunk of mixed itex+svg code at the beginning to display an adjunction. On my machine though the output of that code was ill typeset. So I have removed the code and replaced it by plain iTex encoding of an adjunction.

(Just in case anyone deeply cares about the svg that was there. It’s still in the history. If it is preferred by anyone, it needs to be fixed first.)

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Anonymous

• starting something, not done yet

• Idea-section and one further reference at Thomason model structure.

I remember Mike once said on the blog somewhere that there might be some problem with Thomason's original claim that cofibrant objects in this structure are posets. I made a brief remark on this, but I can't find Mike's original comment.

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

• Created an entry for this.

I’ve adopted the existing convention at nLab in the definition of $Tw(C)$ (which is also the definition I prefer).

Since the opposite convention is used a lot (e.g. by Lurie), I’ve decided it was worth giving it notation, the relation between the versions, and citing results in both forms. Since I didn’t have any better ideas, I’ve settled on $\overline{Tw}(C)$.

• I added the description of lax (co)limits of Cat-valued functors via (co)ends and ordinary (co)limits. I should probably flesh this out more.

I’ve adopted the convention on twisted arrows at twisted arrow category, which is opposite of that in GNN.

In the case of ordinary 2-category, when the diagram category is a 1-category, is the expression of lax (co)limits via ordinary weighted (co)limits really as simple as taking the weights $C_{\bullet/}$ or $C_{/\bullet}$? I can’t find a reference that spells that out clearly; if there really is such a simple description it should be put on the lax (co)limit page.

• 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)

• some bare minimum, for the moment just a glorified list of references

• considerably expanded the Idea-section

• starting something – not done yet

• Added material to injective object, including a proof of Baer’s criterion for injective modules, and the result that for modules over Noetherian rings, direct sums of injective modules are injective.

• Added link to Geoffrey Lewis’s 1974 thesis (under Kelly) “Coherence for a Closed Functor”

• this is a bare list of references, to be !include-ed as a subsection in the References-sections of relevant entries

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• Hello, I thought that a new entry would be a good thing. Just a sketch for now.