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    • starting something – not done yet, but need to save

      v1, current

    • How would people feel about renaming distributor to profunctor? I seem to recall that when this came up on the Cafe, I was the main proponent of the former over the latter, and I've since changed my mind.

    • Add a stub page to host the thesis.

      v1, current

    • fixed the statement of Example 5.2 (this example) by restricting it to 𝒞=sSet\mathcal{C} = sSet

      diff, v36, current

    • Add a students section (very sparse at the moment).

      diff, v7, current

    • I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.

    • a stub, for the moment just to satisfy links

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I added the description of lax (co)limits of Cat-valued functors via (co)ends and ordinary (co)limits. I should probably flesh this out more.

      I’ve adopted the convention on twisted arrows at twisted arrow category, which is opposite of that in GNN.

      In the case of ordinary 2-category, when the diagram category is a 1-category, is the expression of lax (co)limits via ordinary weighted (co)limits really as simple as taking the weights C /C_{\bullet/} or C /C_{/\bullet}? I can’t find a reference that spells that out clearly; if there really is such a simple description it should be put on the lax (co)limit page.

      diff, v2, current

    • Changes made only to the Universal property of the 2-category of spans section. The citations by Urs lead to another citation which, in turn, leads to another citation. With a little effort, I tracked down the a full copy of said universal property, I’ve replicated it here, added the citation used, although I left the previous citations there for convenience; a more experienced editor can remove those if they would like.

      I would like to note that the author whose work I have referenced, Hermida, also notes: “[this universal property] is folklore although we know no references for it.”

      Please make any corrections needed and clean up the language here; this is a fairly direct copy of what is written, but I imagine somebody with more knowledge of all the language used here can rewrite this universal property stuff in a cleaner way.



      diff, v57, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • there is an old article (Berends-Gastman 75) that computes the 1-loop corrections due to perturbative quantum gravity to the anomalous magnetic moment of the electron and the muon. The result turns out to be independent of the choice of (“re”-)normalization (hence what they call “finite”).

      I have added a remark on this in the (g2)(g-2)-entry here and also at quantum gravity here.

    • Moving the following old query out of the entry to here. Maybe it inspires somebody to add to the entry a remark towards the answer:

      [ begin forwarded query ]

      +– {: .query} Anonymous: Under what conditions are all injections in a category monomorphisms? Obviously injections are monomorphisms in a well-pointed topos or pretopos (those are models of particular types of set theories), but does that remain true in a (pre)topos without well-pointedness, a coherent category or an exact category?

      Anonymous: There is this stackexchange post, but the answers only refer to concrete categories with a forgetful functor to Set and a free functor from Set, rather than arbitrary abstract categories. =–

      [ end forwarded query ]

      diff, v19, current

    • Added:

      Free rigid monoidal categories

      The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor LL.

      Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category CC admits a fully faithful strong monoidal functor CL(C)C\to L(C), where L(C)L(C) is a rigid monoidal category.

      See Theorems 1 and 2 in Delpeuch \cite{Delpeuch}.

      diff, v17, current

    • A page with diagrams representing separation axioms T0-T4 as lifting properties, to be included into the separation axioms.


      v1, current

    • a bare list of entry names, to be !include-ed into the “Related concepts”-sections of the relevant entries – for ease of cross-linking

      v1, current

    • I added some discussion to Hausdorff space of how the localic and spatial versions compare in classical and constructive mathematics, including in particular the fact that I just learned (in discussion with Martin Escardo and Andrej Bauer) that a discrete locale is Hausdorff iff it has decidable equality.

    • Created:


      A measurable field of Hilbert spaces is the exact analogue of a vector bundle over a topological spaces in the setting of bundles of infinite-dimensional Hilbert spaces over measurable spaces.


      The original definition is due to John von Neumann (Definition 1 in \cite{Neumann}).

      We present here a slightly modernized version, which can be found in many modern sources, e.g., Takesaki \cite{Takesaki}.

      \begin{definition} Suppose (X,Σ)(X,\Sigma) is a measurable space equipped with a σ-finite measure μ\mu, or, less specifically, with a σ-ideal NN of negligible subsets so that (X,Σ,N)(X,\Sigma,N) is an enhanced measurable space. A measurable field of Hilbert spaces over (X,Σ,N)(X,\Sigma,N) is a family H xH_x of Hilbert spaces indexed by points xXx\in X together with a vector subspace MM of the product PP of vector spaces xXH x\prod_{x\in X} H_x. The elements of MM are known as measurable sections. The pair ({H x} xX,M)(\{H_x\}_{x\in X},M) must satisfy the following conditions. * For any mMm\in M the function XRX\to\mathbf{R} (xm(x)x\mapsto \|m(x)\|) is a measurable function on (X,Σ)(X,\Sigma). * If for some pPp\in P, the function XCX\to\mathbf{C} (xp(x),m(x)x\mapsto\langle p(x),m(x)\rangle) is a measurable function on (X,Σ)(X,\Sigma) for any mMm\in M, then pMp\in M. * There is a countable subset MMM'\subset M such that for any xXx\in X, the closure of the span of vectors m(x)m(x) (mMm\in M') coincides with H xH_x. \end{definition}

      The last condition restrict us to bundles of separable Hilbert spaces. One can also define bundles of nonseparable Hilbert spaces, but this cannot be done simply by dropping the last condition.

      Serre–Swan-type duality

      The category of measurable fields of Hilbert spaces on (X,Σ,N)(X,\Sigma,N) is equivalent to the category of W*-modules over the commutative von Neumann algebra L (X,Σ,N)\mathrm{L}^\infty(X,\Sigma,N).

      (If we work with bundles of separable Hilbert spaces, then W*-modules must be countably generated.)

      Related entries


      \bibitem{Neumann} John Von Neumann, On Rings of Operators. Reduction Theory, The Annals of Mathematics 50:2 (1949), 401. doi.

      \bibitem{Takesaki} Masamichi Takesaki, Theory of Operator Algebras. I, Springer, 1979.

      v1, current

    • Doi link for Scott 1976, and more details for Hyland 2017 (arxiv, doi, journal ref)

      diff, v18, current

    • A while ago I created reflexive object without too much content, and now I’ve revised and expanded it a bit (after hearing an inspiring talk by Dana Scott!).

    • Some stuff that Zoran wrote on recollement reminded me that I had been long meaning to write Artin gluing, which I’ve done, starting in a kind of pedestrian way (just with topological spaces). Somewhere in the section on the topos case I mention a result to be found in the Elephant which I couldn’t quite find; if you know where it is, please let me know.

    • At affine scheme, the fundamental theorem on morphisms of schemes was stated the other way round. I fixed that.

      As a handy mnemonic, here is a quick and down-to-earth way to see that the claim “Sch(SpecR,Y)CRing(𝒪 Y(Y),R)Sch(Spec R, Y) \cong CRing(\mathcal{O}_Y(Y), R)” is wrong. Take Y= nY = \mathbb{P}^n and R=R = \mathbb{Z}. Then the left hand side consists of all the \mathbb{Z}-valued points of n\mathbb{P}^n. On the other hand, the right hand side only contains the unique ring homomorphism \mathbb{Z} \to \mathbb{Z}, since 𝒪 n( n)\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}.

    • added to group cohomology

      • in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to

      • in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.

      it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...

    • Created:


      An entire functional calculus algebra is a product-preserving functor

      CartHoloSet,CartHolo \to Set,

      where CartHoloCartHolo is the category of finite-dimensional complex vector spaces and holomorphic maps.

      This is in complete analogy to C^∞-rings, and EFC-algebras are applicable in similar contexts.


      The category of globally finitely presented Stein spaces is contravariantly equivalent to the category of finitely presented EFC-algebras. The equivalence functor sends a Stein space to its EFC-algebra of global sections.

      The category of Stein spaces of finite embedding dimension is contravariantly equivalent to the category of finitely generated EFC-algebras. The equivalence functor sends a Stein space to its EFC-algebra of global sections.

      These statements can thus be rightfully known as Stein duality.

      Related concepts


      • Alexei~Yu.~Pirkovskii, Holomorphically finitely generated algebras. Journal of Noncommutative Geometry 9 (2015), 215–264. arXiv:1304.1991, doi:10.4171/JNCG/192.

      • J.~P.~Pridham, A differential graded model for derived analytic geometry. Advances in Mathematics 360 (2020), 106922. arXiv:1805.08538v1, doi:10.1016/j.aim.2019.106922.

      v1, current

    • Stonean spaces and Stonean duality

      v1, current

    • On the page Grothendieck topos, Toby Bartels raised the question whether local smallness should be part of the Giraud axioms, referring to a math overflow discussion on a related issue. A while ago I added some comments to the math overflow discussion saying that local smallness has to be part of the Giraud axioms, giving an argument with universes, but nobody seems to have noticed. I raise the issue here again since it’s beside the point of the original math overflow discussion, but clearly related to the nlab.

      Concretely, if U, V are universes with U in V, then V is not a Grothendieck topos relative to U, but satisfies all the conditions of the Giraud axioms.

      The confusion on math overflow seems to come from C2.2.8(vii) in the Elephant, where Grothendieck toposes are characterized as infty-pretoposes with a generating set of objects.

      The crucial point is that the assumption of local smallness is implicit in the definition of infty-pretopos in the Elephant, but not in the nlab.

      To actually see that one has to look fairly closely at the Elephant:

      • Before Lemma A.1.4.19, Johnstone writes ” … C is an infty-pretopos if it is an infty-positive geometric category which is effective as a regular category.

      • a geometric category is defined as one “satisfying the conditions of Lemma 1.4.18”, which can be reformulated as “regular well-powered with pullback-stable small joins of subobjects”.

      So geometric categories are well-powered, but any regular well powered category is also locally small, since Hom(A,B) can be embedded into sub(AxB) via graphs.

      On the other hand, well poweredness is not a requirement for geometric categories on the nlab.

      I also prefer the convention on the nlab, since there are certain interesting settings where we don’t have well-poweredness, and then it is possible to have small joins of subobjects w/o having small meets.

      But this means that the Giraud theorem on the nlab has to contain the assumption of local smallness explicitely.

      I will now go ahead and remove the comments of Toby Bartels and add the condition. Feel free and rollback or modify if you don’t agree.

      (By the way, my favourite definition of infty-pretopos is “exact, infty-extensive category”. I find this easier to digest and memorize)

    • I brushed up the entry power a bit: wrote an Idea-section, created an Examples-section etc.

    • I keep wanting to point to properties of the terminal geometric morphism. While we had this scattered around in various entries (such as at global sections, at (infinity,1)-topos and elsewhere – but not for instance at (infinity,1)-geometric morphism) I am finally giving it its own entry, for ease of hyperlinking.

      So far this contains the (elementary) proofs that the geometric morphism to the base SetSet/Grpd Grpd_\infty is indeed essentially unique, and that the right adjoint is equivalently given by homs out of the terminal object.

      v1, current

    • Stub. This is a technique for describing complexity classes as the total maps of some Turing category.

      v1, current

    • Stub. I don’t like this approach but I’m not sure how to make it better.

      I tried to give a type-theoretic approach to avoid the classic problem where we overspecialize on Turing machines, but I’m not great with type theory.

      v1, current

    • added pointer to

      • Heisuke Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I, Annals of Mathematics Second Series, Vol. 79, No. 1 (Jan., 1964), pp. 109-203 (95 pages) (jstor:1970486)

      • Michael Atiyah, Resolution of singularities and Division of Distributions, Communications in Pure and Applied Mathematics, vol. XXIII, 145-150 (1970)

      with some comment

      diff, v6, current

    • created equivariant de Rham cohomology with a brief note on the Cartan model.

      (I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)

    • I have removed the bulk of the Idea section that I had written, starting way back when the entry was created in 2013.

      I kept the other material that Mike (Shulman) had written.

      I see now that mine was really besides the point.

      Now that I finally understand this topic more deeply, maybe I find time to write a better explanation of what’s really going on.

      diff, v58, current