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    • Added reference to Bruce Bartlett’s thesis.

      diff, v5, current

    • started Brauer group, collecting some references on the statement that/when Br(X)≃H et 2(X,𝔾 m) torBr(X) \simeq H^2_{et}(X, \mathbb{G}_m)_{tor} and moved notes from a talk by David Gepner on ∞\infty-Brauer groups to there.

    • Added material on the free join-semilattice on a poset.

      diff, v12, current

    • Created:

      Idea

      Maharam’s theorem states a complete classification of isomorphism classes of the appropriate category of measurable spaces.

      In the σ-finite case, the theorem classifies measure spaces up to an isomorphism. Here an isomorphism is an equivalence class of measurable bijections ff with measurable inverse such that ff and f −1f^{-1} preserve measure 0 sets.

      As explained in the article categories of measure theory, for a truly general, unrestricted statement for non-σ-finite spaces there are additional subtleties to consider: equality almost everywhere must be refined to weak equality almost everywhere, and σ-finiteness should be relaxed to a combination of Marczewski-compactness and strict localizibility.

      In this unrestricted form, by the Gelfand-type duality for commutative von Neumann algebras, Maharam’s theorem also classifies isomorphism classes of localizable Boolean algebras, abelian von Neumann algebras, and hyperstonean spaces (or hyperstonean locales).

      Statement

      Every object in one of the above equivalent categories canonically decomposes as a coproduct (disjoint union) of ergodic objects. Here an object XX is ergodic if the only subobjects of XX invariant under all automorphisms of XX are ∅\emptyset and XX itself.

      Furthermore, an ergodic object XX is (noncanically, using the axiom of choice) isomorphic to 𝔠×2 κ\mathfrak{c}\times 2^\kappa, where κ\kappa is 0 or infinite, and 𝔠\mathfrak{c} is infinite if κ\kappa is infinite. Here the cardinal 𝔠\mathfrak{c} is known as the cellularity of XX and κ\kappa is its Maharam type.

      In particular, if κ=0\kappa=0, we get a classification of isomorphism classes of atomic measure spaces: they are classified by the cardinality 𝔠\mathfrak{c} of their set of atoms.

      Otherwise, κ\kappa is infinite, and we get a classification of isomorphism classes of ergodic atomless (or diffuse) measure spaces: such spaces are isomorphic to 𝔠×2 κ\mathfrak{c}\times 2^\kappa, where 𝔠\mathfrak{c} and κ\kappa are infinite cardinals.

      Thus, a completely general object XX has the form

      ∐ κ𝔠 κ×2 κ,\coprod_\kappa \mathfrak{c}_\kappa\times 2^\kappa,

      where κ\kappa runs over 0 and all infinite cardinals, 𝔠 κ\mathfrak{c}_\kappa is a cardinal that is infinite or 0 if κ≠0\kappa\ne0, and 𝔠 κ≠0\mathfrak{c}_\kappa\ne0 only for a set of κ\kappa.

      References

      The original reference is

      • Dorothy Maharam, On homogeneous measure a lgebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942) 108-111. doi.

      A modern exposition can be found in Chapter 33 (Volume 3, Part I) of

      v1, current

    • I created Bishop’s constructive mathematics by moving some material from Errett Bishop and adding some more discussion of what it is and isn’t. Comments and suggestions are very welcome; I’m still trying to figure out the best way to describe the relationship of this theory to other things like topos logic.

    • brief category:people-entry for hyperlinking references

      v1, current

    • I am starting something at six operations.

      (Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)

    • Fixed pdf link to “Towards an understanding of Girard’s transcendental syntax”

      ALH

      diff, v6, current

    • Have added more of the original (“historical”) References with brief comments and further pointers.

    • starting an entry, for the moment mainly in order to record the fact that “crossed homomorphisms” are equivalently homomorphic sections of the corresponding semidirect product group projection. This is obvious, but is there a reference that makes it explicit?

      v1, current

    • Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.

    • moved the discussion of the Lie algebra 𝔰𝔲(2)\mathfrak{su}(2) out of the middle of “Properties – General” into its own subsection “Properties – Lie algebra”.

      Will copy this also over to su(2)

      diff, v18, current

    • I corrected an apparent typo:

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η∘a\theta_a \colon 1 \Rightarrow \eta \circ a

      to

      A 2-monad TT as above is lax-idempotent if and only if for any TT-algebra a:TA→Aa \colon T A \to A there is a 2-cell θ a:1⇒η A∘a\theta_a \colon 1 \Rightarrow \eta_A \circ a

      It might be nice to say η A\eta_A is the unit of the algebra….

      diff, v22, current

    • For the purposes of negative thinking, it may be useful to recognise that every ∞\infty-category has a (−1)(-1)-morphism, which is the source and target of every object. (In the geometric picture, this comes as the (−1)(-1)-simplex of an augmented simplicial set.)

      Jonathan Arnoult has pointed out on CT Zulip that this is misleading: it sounds like it implies that every ∞\infty-category is monoidal! And John Baez pointed out that the analogy to augmented simplicial sets fails because in an augmented simplicial set each 0-simplex has only one face, rather than a separate “source” and “target” that are both the same (−1)(-1)-simplex.

      I suggest we just remove this paragraph and the query box following it, since I can’t think of a way to rephrase it that would be more helpful than unhelpful. But I’m open to other suggestions.

      diff, v16, current

    • Added section on irreflexive comparisons, which generalises linear orders in constructive mathematics

      Anonymous

      diff, v15, current

    • Stub. For the moment just for providing a place to record this reference:

      • Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)

      v1, current

    • This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).

      Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.

      There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).

      A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).

      Bruno Gavranović

      v1, current

    • For now creating page, it needs to be (much) further expanded.

      v1, current

    • Added a reference to Tall–Wraith. Changed ∘:P⊗ kP→P\circ: P \otimes_k P \to P to ∘:P⊙ kP→P\circ: P \odot_k P \to P. Added redirects.

      diff, v3, current

    • Several references and a link to “writings” page

      diff, v2, current

    • Added definition of Tambara-Yamagami (TY) fusion category, the simplest example of non-invertible symmetries in the 2d defect language.

      v1, current

    • created a “category: reference”-page The Stacks Project

      I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.

      So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on CRing opCRing^{op} and either give a general account or call this The Algebraic Stacks Project ? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.

    • A class of structures similar to quantum groups.

      v1, current

    • I needed an entry to be able to point to which collects pointers to the various entries on “dualities” in string theory. So I created one: duality in string theory.

    • Starting page on Tr(ϕ 3)\mathrm{Tr}(\phi^3)

      Anonymouse

      v1, current

    • Canonical bases for the (representations of) quantum groups and Lie groups.

      v1, current

    • I reverted back to

      This polynomial is an invariant of the knot, the Alexander polynomial of the knot.

      I see no reason for a strange comma before “is”. The second comma is before the appositional phrase.

      diff, v23, current

    • I have edited at Tychonoff theorem:

      1. tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)

      2. moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here

      Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.

      At then end of the entry there is a line:

      More details to appear at Tychonoff theorem for locales

      which however has not “appeared” yet.

      But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.

    • Started lift.

      weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.

      Would it be better to have these redirect to lift?

    • created quick stub for framed bicategory

      but my machine's battery will die any second now...

    • I added a reference on Gabriel filters on quantales.

      Very strange: version one on show had a reference on Etendues (entered by T. Holder), but when I clicked edit there was none in the edit window, just in show window. I edited and one can not see it in any history, nowhere. So there was something in show cache from 2014 which is not recorded in any history edit.

      diff, v2, current

    • Add a strengthening of the coherence result, also due to Power.

      diff, v7, current

    • starting page on surfaceology

      Anonymouse

      v1, current

    • Explained how to prove the commutativity of addition in EFA

      diff, v11, current

    • I gave Fourier-Mukai transform a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also added a few more pointers to literature.

    • Anonymous “helpfully” changed the statement

      A matrix is a list of lists.

      to

      A matrix is a function M:[n]×[m]→XM:[n]\times[m]\rightarrow X from the Cartesian product [n]×[m][n]\times[m] to a set XX.

      which I have reverted back.

      diff, v12, current

    • a minimum of an Idea-section, but mainly to record some references

      v1, current

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