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

• starting something, not done yet

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

• am giving this an entry of its own, split off from motivic homotopy theory.

Nothing much here yet, just a bare minimum so far

• I felt we were lacking an entry titled simplicial homotopy theory that usefully collects the relevant entries that we do have. So I am starting one.

• derived algebraic geometry

Also I'm wondering whether the first sense, i.e. the study of derived categories of coherent sheaves, is really a common use of the term "derived algebraic geometry"; I've always seen it in the second sense.

And perhaps we should mention homotopical algebraic geometry as well?

• added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces

By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?

• I am moving the following old query box exchange from orbifold to here.

old query box discussion:

I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely does not view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.

Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

end of old query box discussion

• Andrea Fontanella, Tomas Ortin, On the supersymmetric solutions of the Heterotic Superstring effective action (arxiv:1910.08496)
• Corrected a statement about Lie groups.

• Is Met really Complete? Wikipedia argues that not, since indeed |R^|R isn’t even First-Countable so let alone Metrizable, when it would be the induced Topological Space underlying an Uncountable Product in Met…

GrothenDitQue:

• Began this entry.

• 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.
• Started this since it was mentioned elsewhere. So it’s just the opposite one object 2-category, isn’t it?

• This is intended to continue the issues discussed in the Lafforgue thread!

I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.

I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.

Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.

• I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category $V^{A^{op}}$ and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).

This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.

• brief category: people-entry for hyperlinking references at pillowcase orbifold

• starting something

• am starting something

• Edit to: GUT by Urs Schreiber at 2018-04-01 01:21:13 UTC.

• Starting the background for explaining the connection to partial logic.

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

Anonymous

• at Atiyah Lie groupoid was this old query box discussion, which hereby I am moving from there to here:

+– {: .query} What is all of this $diag$ stuff? I don't understand either $(P \times P)/_{diag} G$ or $(P_x \times P_x)_{diag} G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

Toby: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)

Todd: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $g p = p g^{-1}$.

Toby: H'm, maybe we should write diagonal action if there's something interesting to say about it. =–

• I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.