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    • I added to initial object the theorem characterizing initial objects in terms of cones over the identity functor.

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • starting some minimum, for the moment mainly to record some references

      v1, current

    • starting page on ℤ\mathbb{Z}-functors

      Anonymouse

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      v1, current

    • am too tired to do it now, but on occasion of an MO discussion:

      remind me to insert at smooth manifold the statement and proof that smooth manifolds are equivalently the locally representable sheaves on CartSp (more precisely: the 𝒢=CartSp\mathcal{G} = CartSp-schemes).

    • Created a stub for the conference.

      v1, current

    • I worked on synthetic differential geometry:

      I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

      Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.

    • changed “an English mathematician of Egyptian origin” to “a British-Lebanese mathematician”.

      In checking his “origin” on Wikipedia…

      …I see that Wikipedia says that Sir Michael Atiyah has died. Today.

      (!?)

      diff, v6, current

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

      v1, current

    • The name was the incorrect “semi-locally simply topological connected space”

      diff, v8, current

    • a minimum entry, for the moment just so as have a place to record the fact that two smooth functions are smoothly homotopic as soon as they are (continuously) homotopic

      (It sure feels like we must have recorded this somewhere already, but I couldn’t find it…)

      v1, current

    • In the definition, the article states "every object in C is a small object (which follows from 2 and 3)". The bracketed remark doesn't seem quite right to me, since neither 2 nor 3 talk about smallness of objects. Presumably this should better be phrased as in A.1.1 of HTT, "assuming 3, this is equivalent to the assertion that every object in S is small".

      Am I right? I don't (yet) feel confident enough with my category theory to change this single-handedly.

    • 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

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