Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck 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 homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 stack 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.

Welcome to nForum
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).
    • There seem to be some misleading remarks at Čech model structure on simplicial presheaves.

      Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.

      Marc Hoyois seems to says the opposite: there is no deep relation between “Čech” in “Čech cohomology” and in “Čech model structure”.

      […] the corresponding Čech cover morphism .

      Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.

      The Čech nerve is projective-cofibrant if we assume the site has pullbacks. I don’t know how to prove it otherwise. Of course, injective-cofibrancy is trivial.

      this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be

      Based on the discussion here, it seems that the Čech model structure is not site-independent, even though it can be defined on the category of simplicial sheaves. A very strange state of affairs…

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinks references

      v1, current

    • possibly empty groups

      Anonymous

      v1, current

    • Added these pointers to today’s replacements:

      • Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas I. Compact Lie groups [arXiv:2112.07942]

      • Changha Choi, Leon A. Takhtajan: Supersymmetry and trace formulas II. Selberg trace formula [arXiv:2306.13636]

      diff, v8, current

    • starting something, but nothing much here yet

      v1, current

    • briefly recording the way to isomorph linear representations of compact groups on Hilbert spaces into unitary reps, by group-averaging

      v1, current

    • a bare list of references, to be !included into the References-section of relevant entries, for ease of synchronizing (such as at anyon but also at vortex etc.)

      v1, current

    • I added some more to Lebesgue space about the cases where 1<p< fails.

    • Added a bunch of material to inverse semigroup under subsections of “Properties”.

    • I have added some discussion to variable.

      Not sure if I ever announced this here, the original version is a few months old already. But right now I have added also some sentences on bound variables.

      Some more professional logician might please look over this. There will be lots of room to improve on those few sentences that I jotted down, mainly such as to have something there at all.

    • have made explicit the statement that for a pair of adjoint (co)monads their EM-categories are isomorphic (now this Prop).

      Also tried to to give this entry a more systematic structure, but still some way to go.

      diff, v11, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a stub, to make links work

      (This used to be a stub “quantum circuit” which I just quasi-duplicated at a more extensive entry quantum circuit diagram. But since quantum gate was already redirecting here – which is how I discovered/remembered that this entry exists – no harm is done by making that it’s new title.)

      diff, v4, current

    • I am trying to collect citable/authorative references that amplify the analog of the mass gap problem in particle phenomenology, where it tramslates into the open problem of computing hadron masses and spins from first principles (due to the open problem of showing existence of hadrons in the first place!).

      This is all well and widely known, but there is no culture as in mathematics of succinctly highlighting open problems such that one could refer to them easily.

      I have now created a section References – Phenomenology to eventually collect references that come at least close to making this nicely explicit. (Also checked with the PF community here)

      diff, v4, current

    • I have changed the name to Haar Integral – if that’s ok – since the perspective I have added to the article leaves Haar measure as a consequence of Haar integral, and not the other way around.

      edeany@umich.edu

      diff, v10, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a small entry on the quantum T- and S-gates, just for completeness.

      v1, current

    • Added link to mentioned book and possible link to mentioned paper

      Julius

      diff, v20, current

    • make lists of conferences part of the category: reference

      Valeria de Paiva

      diff, v5, current

    • The link for ’equivalent’ at the top redirected to natural isomorphism which (as I understand it) is the correct 1-categorical version of an equivalence of functors, but this initially lead me to believe that a functor was monadic iff it was naturally isomorphic to a forgetful functor from the Eilenberg-Moore category of a monad on its codomain, which would mean that the domain of the functor was literally the Eilenberg-Moore category of some adjunction since natural isomorphism is only defined for parallel functors.

      diff, v19, current

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

      v1, current

    • a bare minimum, for the moment just to record some references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • This article is weird. What is the difference between the object described by it and the article submodule? Appears to be a duplicate.

      diff, v4, current

    • Created page.

      For now Pol (the category) redirects here. Let me know if this is okay.

      v1, current

    • added the statement of the Fubini theorem for ends to a new section Properties.

      (I wish this page would eventually give a good introduction to ends. I remember the long time when I banged my head against Kelly’s book and just didn’t get it. Then suddenly it all became obvious. It’s some weird effect with this enriched category theory that some of it is obvious once you understand it, but looks deeply mystifying to the newcomer. Kelly’s book for instance is a magnificently elegant resource for everyone who already understands the material, but hardly serves as an exposition of the ideas involved. I am hoping that eventually the nLab entries on enriched category theory can fill this gap. Currently they do not really. But I don’t have time for it either.)

    • Several recent updates to literature at philosophy, the latest being

      • Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2., pdf; Structures, Learning and Ergosystems: Chapters 1-4, 6 (2011) pdf

      which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

    • Created:

      Definition

      Given vector subspaces V0 and V1 of a vector space V, we write V0V1 if V0/(V0V1) is finite-dimensional. We write V0V1 and say V0 and V1 are commensurable if V0V1 and V1V0.

      A Tate vector space is a complete Hausdorff topological vector space V that admits a basis of neighborhoods of 0 whose elements are mutually commensurable vector subspaces of V.

      Duality

      A vector subspace W of a Tate vector space V is bounded if for every open vector subspace UV we have WU.

      The dual of a Tate vector space V is Hom(V,C) equipped with a topology generated by the basis of neighborhoods of 0 whose elements are orthogonal complements to bounded subspaces of V.

      Properties

      Tate vector spaces form an pre-abelian category.

      References

      v1, current

    • disambiguation page for Ore condition

      watcher

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • renaming page to “Ore condition in a category” because there also exists the Ore condition in ring theory

      watcher

      diff, v6, current

    • I added some material to Peano arithmetic and Robinson arithmetic. At the latter, I replaced the word “fragment” (which sounds off to my ears – actually Wikipedia talks about thisterm a little) with “weakening”.

      Still some links to be inserted.

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

      Anonymous

      v1, current

    • Included the condition on sequential (co)limits that the indexing ordinal should be nonzero, which I presume to be the correct convention. (e.g. based on the description they are a special case of filtered colimits)

      diff, v8, current

    • starting disambiguation page for twisted arrow

      Anonymouse

      v1, current

    • starting page on the twisted arrow modality in simplicial type theory

      Anonymouse

      v1, current

    • 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 ¯Tw(C).

      v1, current

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

      v1, current

    • At coverage, I just made the following change: Where the sheaf condition previously read

      X(U)iIX(Ui)i,jIX(Ui×UUj),

      it now uses the variable names “j” and “k” instead of “i” and “j”:

      X(U)iIX(Ui)j,kIX(Uj×UUk).

      I’m announcing this almost trivial change because I’d like to invite objections, in which case I’d rollback that change and also would not go on to copy this change to related entries such as sheaf. There are two tiny reasons why I prefer the new variable names:

      • It’s more symmetric. The previous notation unjustly favored “i”.
      • It’s slightly easier to infer the definition of the two maps. (I had a student who was briefly confused by the original notation.)
    • Added a lemma about fully faithful functors.

      Sorry for the mess, there does not seem to be a way to preview edits.

      diff, v3, current

    • starting stub article on synthetic (,1)-category

      Anonymous

      v1, current

    • starting page on the op modality in simplicial type theory and synthetic (infinity, 1)-category theory

      Anonymouse

      v1, current

    • starting page on spatial type theory, which is modal dependent type theory with the sharp and flat modalities.

      Anonymous

      v1, current

    • The entry (infinity,1)-Kan extension is still a sad stub which you shouldn’t look at if you have better things to do. But I have now briefly added at least a few more specific pointers to HTT, in particular to the pointwise-ness issue. But just pointers, essentially no text for the moment. (If you feel energetic, be invited to turn the entry into something prettier!)

    • For completeness I have added pointer to

      • Emily Riehl, Dominic Verity, Section 6 of: Fibrations and Yoneda’s lemma in an -cosmos, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)

      though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.

      diff, v13, current