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    • Linked to from https://ncatlab.org/nlab/show/orthogonal+factorization+system and https://ncatlab.org/nlab/show/final+functor

      Todo: add more proofs of this result.

      For some reason the xymatrixes were causing errors so I had to comment them out to submit. Here is an example error:

      An error occurred when running pdflatex on the following diagram. \xymatrix@=5em{e \ar[r]^\gamma \ar[dr]_{\gamma’} & GFc \ar[d]^{Gf} \ & GFc’} The error was: Timed out

      How can I fix this?

      v1, current

    • 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

    • added to polynomial functor the evident but previously missing remark why it is called a “polynomial”, here.

    • the entry braid group said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

    • Create a new page to keep record of PhD theses in category theory (with links to the documents where possible), particularly older ones that are harder to discover independently. At the moment, this is just a stub, but I plan to fill it out more when I have the chance.

      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

    • the entry group algebra had been full of notation mismatch and also of typos. I have reworked it now.

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