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    • added second initial to page name, for better disambiguation

      diff, v2, current

    • Removed completely unrelated reference to a fictional movie


      diff, v5, current

    • Add a reference for string diagrams in closed monoidal categories


      diff, v42, current

    • added to KK-theory brief remark and reference to relation to stable \infty-categories / triangulated categories

    • a stub entry, for the moment just to make the link work

      v1, current

    • brief category:peopleentry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category: people-entry for hyperlinking references

      v1, current

    • stub entry

      for the moment just to make links work

      v1, current

    • created microlinear space

      One thing I might be mixed up above:

      in the literature I have seen it seems to say that

      $ X^D x_X X^D \simeq X^{D(2)}$


      $ D(2) = { (x_1,x_2) \in R \times R | x_i x_j = 0} $.

      But shouldn't it be

      $ D(2)' = { (x_1,x_2) \in R \times R | x_i^2 = 0} $.


    • See Day convolution

      I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category V A opV^{A^{op}} and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).

      This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.

    • brief category:people-entry for hyperlinking references

      v1, current

    • added pointer to:

      removed the following ancient query box discussion:

      +–{.query} Left I could understand, but right? —Toby

      The way I rewrote it explains it. It is unfortunate that the Eilenberg-Watts theorem treated in Bass was using only right adjoint functors so later they dropped word right. – Zoran

      Thanks. —Toby =–

      diff, v12, current

    • added various references, notably on computation of graviton scattering amplitudes.

      diff, v15, current

    • Todd,

      you added to Yoneda lemma the sentence

      In brief, the principle is that the identity morphism id x:xxid_x: x \to x is the universal generalized element of xx. This simple principle is surprisingly pervasive throughout category theory.

      Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element id xid_x is a tautological statement that does not need or imply the Yoneda lemma, it seems.

    • added these pointers:

      • Sergio Ferrara, Supersymmetry: Some Reflections on the Future of a Symmetry from the Future,

        talk at 2020 Breakthrough Prize Symposium (November 2019) [video:yt]

        talk at Colloqui della Classe di Scienze (December 2019) [video:yt]

      diff, v8, current

    • added pointer to yesterday’s

      • Jim Gates, Yangrui Hu, S.-N. Hazel Mak, Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, 𝒩=1\mathcal{N} = 1 Superspace (arXiv:2002.08502)

      diff, v13, current

    • made page name singular, added references on Wilson line observables

      diff, v2, current

    • following up on our discussion in the thread <a href="">oo-vector bundle (forum)</a> here on the forum, I have now spent a bit of time on expanding the entry quasicoherent sheaf

      - I fixed the formulas in the section <a href="">As sheaves on Aff/X</a>. They were a bit rough and typo-ridden in the first version (which likely was my fault, not anyone else's). Since there are 50 variants in the literature to state this, I also pointed to page and verse in a lecture by Goerss where the statement is given explicitly the way it now appears there in the entry

      - then I added a new section <a href="">As homs into the stack of modules</a> where I aim to describe in great detail how this definition is equivalent to the even simpler one, where we just say that the category of quasicoherent sheaves on a sheaf  X is the Hom-category  QC(X) = Hom(X,QC) for QC : Ring \to Cat the stack of modules, classifying the canonical bifibration  Mod \to Rings. This is the statement that my discussion at <a href="">oo-vector bundle (schreiber)</a> was secretly based on, which I promised to make more explicit.

      - then I added a section <a href="">Higher/derived quasicoherent sheaves</a>, where I indicate the now obvious oo-categorification discussed in more detail at <a href="">oo-vector bundle (schreiber)</a> and point out how this gives the derived QC sheaves used by Ben-Zvi et al as discussed at geometric infinity-function theory

      - finally I wrote a fairly detailed <a href="">Idea</a> section for quasicoherent sheaves, that previews the content of all these sections.
    • I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.

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

    • recorded some recent surveys of the status of MOND at MOND

    • brief category:people-entry foe hyperlinking references

      v1, current

    • stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.

    • just a stub for the moment, in order to make links work

      v1, current

    • the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.

      I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).

      (That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)

    • brief category:people-entry for hyperlinking references

      v1, current

    • changed entry title to full name,

      added “category:people” tag,

      updated webpage url,

      added section “Related nLab entries”, so far with a pointer to Grothendieck construction

      diff, v5, 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

    • Since I gathered them for my recent talk, I may as well provide a list here of work in this area. I need to add names, etc.

      v1, current

    • I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

      Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

    • Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.

    • Thought I’d start something in the hope that experts will say more.

      v1, current

    • starting a category:reference-entry.

      Just a single item so far, but this entry should incrementally grow as more preprints appear (similar to what we have been doing at Handbook of Quantum Gravity and similar entries).

      I know that a soft deadline for submissions of at least one of the sections is this December, so I am guessing this is planned to appear in 2024.

      v1, current

    • Add some explanation why Kan condition explains composition and inverse from the groupoid point of view.

      Chenchang Zhu

      diff, v4, current

    • I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.

    • I have started on a revision of algebraic K-theory. The old version launched straight into a particular nPOV, which really just summarised the Blumberg et al paper, and did not mention any of the other ideas in the area. At present I have just put in some historical stuff, but given the importance of the subject e.g. in modern C*-algebra the page needs a lot more work.