Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

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

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicproper model category
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active 5 days ago

I expanded proper model category a bit.

In particular I added statement and (simple) proof that in a left proper model category pushouts along cofibrations out of cofibrants are homotopy pushouts. This is at Proper model category -- properties

On page 9 here Clark Barwick supposedly proves the stronger statement that pushouts along all cofibrations in a left proper model category are homotopy pushouts, but for the time being I am failing to follow his proof.

(??)

- Discussion Type
- discussion topicadditive category
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by nLab edit announcer
- Last Active 5 days ago

touched the formatting at

*additive category*

- Discussion Type
- discussion topicspatial locale
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 15
- Last comment by Mike Shulman
- Last Active 5 days ago

- Discussion Type
- discussion topictensor hierarchy
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicLeibniz algebra
- Category Latest Changes
- Started by zskoda
- Comments 17
- Last comment by Urs
- Last Active 6 days ago

Many additions and changes to Leibniz algebra. The purpose is to outline that the (co)homology and abelian and even nonabelian extensions of Leibniz algebras follow the same pattern as Lie algebras. One of the historical motivations was that the Lie algebra homology of matrices which lead Tsygan to the discovery of the (the parallel discovery by Connes was just a stroke of genius without an apparent calculational need) cyclic homology. Now, if one does the Leibniz homology instead then one is supposedly lead the same way toward the Leibniz homology (for me there are other motivations for Leibniz algebras, including the business of double derivations relevant for the study of integrable systems).

Matija and I have a proposal how to proceed toward candidates for Leibniz groups, that is an integration theory. But the proposal is going indirectly through an algebraic geometry of Lie algebras in Loday-Pirashvili category. Maybe Urs will come up with another path if it drags his interest.

- Discussion Type
- discussion topicSimon Covez
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*Leibniz algebra*

- Discussion Type
- discussion topicSimen Bruinsma
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicVladislav Kupriyanov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicHigher Structures in M-Theory 2018
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicRoberto Zucchini
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicIntroduction to Homotopy Theory
- Category Latest Changes
- Started by Oscar_Cunningham
- Comments 3
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topictheory of objects
- Category Latest Changes
- Started by Thomas Holder
- Comments 1
- Last comment by Thomas Holder
- Last Active 6 days ago

- Discussion Type
- discussion topicquasicoherent sheaf
- Category Latest Changes
- Started by Urs
- Comments 46
- Last comment by Dmitri Pavlov
- Last Active 7 days ago

- following up on our discussion in the thread <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=593&page=1#Item_12">oo-vector bundle (forum)</a> here on the forum, I have now spent a bit of time on expanding the entry quasicoherent sheaf

- I fixed the formulas in the section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As sheaves on Aff/X</a>. They were a bit rough and typo-ridden in the first version (which likely was my fault, not anyone else's). Since there are 50 variants in the literature to state this, I also pointed to page and verse in a lecture by Goerss where the statement is given explicitly the way it now appears there in the entry

- then I added a new section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#AsSheaves">As homs into the stack of modules</a> where I aim to describe in great detail how this definition is equivalent to the even simpler one, where we just say that the category of quasicoherent sheaves on a sheaf is the Hom-category for the stack of modules, classifying the canonical bifibration . This is the statement that my discussion at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> was secretly based on, which I promised to make more explicit.

- then I added a section <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Higher">Higher/derived quasicoherent sheaves</a>, where I indicate the now obvious oo-categorification discussed in more detail at <a href="http://ncatlab.org/schreiber/show/%E2%88%9E-vector+bundle">oo-vector bundle (schreiber)</a> and point out how this gives the derived QC sheaves used by Ben-Zvi et al as discussed at geometric infinity-function theory

- finally I wrote a fairly detailed <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf#Idea">Idea</a> section for quasicoherent sheaves, that previews the content of all these sections.

- Discussion Type
- discussion topicLie algebra object
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 7 days ago

am splitting this off from

*Lie algebra*, for ease of cross-linking.

- Discussion Type
- discussion topicLoday-Pirashvili category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

I have completed publication data (and missing words in title) for the single reference:

- Jean-Louis Loday, Teimuraz Pirashvili,
*The tensor category of linear maps and Leibniz algebras*, Georg. Math. J. vol. 5, n.3 (1998) 263–276 (doi:10.1023/B:GEOR.0000008125.26487.f3)

- Jean-Louis Loday, Teimuraz Pirashvili,

- Discussion Type
- discussion topicChristian Cuvier
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

brief

`category:people`

-entry for hyperlinking references at*Leibniz algebra*

- Discussion Type
- discussion topicJakob Palmkvist
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

brief

`category:people`

-entry for hyperlinking references at*tensor hierarchy*and at*Borcherds algebra*

- Discussion Type
- discussion topicSylvain Lavau
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicGelfand duality
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Dmitri Pavlov
- Last Active May 22nd 2020

I am splitting off Gelfand duality from Gelfand spectrum. Want to state the actual equivalence theorem here. But just a moment…

- Discussion Type
- discussion topicempty graph
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Thomas Holder
- Last Active May 22nd 2020

- Discussion Type
- discussion topicpremonoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 28
- Last comment by Mike Shulman
- Last Active May 22nd 2020

Somebody named Adam left a comment box a while ago at premonoidal category saying that naturality of the associator requires three naturality squares. I believe that this is true when phrased explicitly in terms of one-variable functors, but the slick approach using the “funny tensor product” allows us to rephrase it as a single natural transformation between functors $C\otimes C\otimes C\to C$. I’ve edited the page accordingly. I also added the motivating example (the Kleisli category of a strong monad) and a link to sesquicategory.

There is a comment on the page that “It may be possible to weaken the above make $(Cat,\otimes)$ a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category”. However, the Power-Robinson paper says that “We remark that $(C \otimes -) : Cat \to Cat$ is not a 2-functor,” which seems to throw some cold water on the obvious approach to that idea. Was the thought to define a different 2-categorical structure on $Cat$ than the usual one, e.g. using unnatural transformations? It seems that at least one would still have to explicitly require centrality of the coherence isomorphisms.

- Discussion Type
- discussion topicMartin-Löf dependent type theory
- Category Latest Changes
- Started by Zhen Lin
- Comments 18
- Last comment by Mike Shulman
- Last Active May 22nd 2020

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?

- Discussion Type
- discussion topiccommutative diagram
- Category Latest Changes
- Started by dppes
- Comments 3
- Last comment by dppes
- Last Active May 22nd 2020

- Discussion Type
- discussion topicDay convolution
- Category Latest Changes
- Started by Harry Gindi
- Comments 47
- Last comment by Théo de Oliveira S.
- Last Active May 21st 2020

See Day convolution

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.

- Discussion Type
- discussion topicnatural numbers object
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Todd_Trimble
- Last Active May 21st 2020

started an Examples-section at natural numbers object

btw, the natural numbers objects of a topos is unique, up to isomorphism, right? if so, we should say that

- Discussion Type
- discussion topic(∞,1)-limit
- Category Latest Changes
- Started by Dean
- Comments 2
- Last comment by Dean
- Last Active May 21st 2020

- Discussion Type
- discussion topiccubical type theory
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by nLab edit announcer
- Last Active May 21st 2020

started

*cubical type theory*using a comment by Jonathan Sterling

- Discussion Type
- discussion topicuniverse
- Category Latest Changes
- Started by Urs
- Comments 65
- Last comment by Todd_Trimble
- Last Active May 21st 2020

I noticed that there was a neglected stub entry universe that failed to link to the fairly detailed (though left in an unpolished state full of forgotten discussions) Grothendieck universe.

I renamed the former to universe > history and made “universe” redirect to “Grothendieck universe”

- Discussion Type
- discussion topicbiactegory
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by nLab edit announcer
- Last Active May 21st 2020

- I created biactegory following my 2006 work and being prompted by overlapping work of a student of Nikshych which appeared on the arXiv today.

- Discussion Type
- discussion topicAdS3-CFT2 and CS-WZW correspondence
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active May 21st 2020

I have split off from

*holographic principle*and then expanded a good bit a few paragraphs on*AdS3-CFT2 and CS-WZW correspondence*together with a few commented references, trying to amplify how this case is given by an actual well-known theorem and at the same time seems to be generic for the more general cases of AdS-CFT which are currently much more vague.