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    • I keep trying to see to which degree one may nail down supergeometry via the yoga of adjoint modalities.

      My the starting point (maybe there is a better one, but for the time being that’s the best I have come up with) is that the inclusion of commutative algebras into supercommutative algebras is reflective and coreflective, the reflector quotients out the ideal generated by the odd part, the coreflector picks the even subalgebra.

      Passing to sheaves over the sites of formal duals to algebras, this gives an adjoint modality which (and you may not like that now, but keep in mind that it’s just notation which doesn’t reall matter) I decided to denote

      ⇉⊣⇝ \rightrightarrows \; \dashv \; \rightsquigarrow

      (The mnemonic is this: in Feynman diagrams ⇝\rightsquigarrow is the symbol for the bosonic particles, so that denotes taking the bosonic subspace. Similarly a single fermion in a Feynman diagram appears as →\to, so fermion bilinears look like ⇉\rightrightarrows and that has to suffice to remind you of general even numbers of fermions.)

      The details (there are not many, it’s straightforward) are at super formal smooth infinity-groupoid.

      Now, while nice, this falls short of characterizing supercommutative algebras.

      For instance the inclusion of commutative algebras into commutative algebras equipped with ℤ/nℤ\mathbb{Z}/n\mathbb{Z}-grading (for any 2≤n<∞2 \leq n \lt \infty, with no extra signs introduced when swapping factors) works just as well. (For the inclusion into ℤ\mathbb{Z}-graded algebras there are also left and right adjoints but they coincide, and so I declare that this case is excluded by demanding faithfulness/non-degeneracy of the model.)

      So the question is, which natural-looking further conditions could we impose on the above adjoints such as to narrow in a bit more on supercommutative algebras?

      Here is one observation:

      with the setup as described at super formal smooth infinity-groupoid we also have the reduction modality ℜ\Re which, on function rings, takes away the nilpotent ideal in an algebra, and its right adjoint, the de Rham stack functor ℑ\Im:

      ℜ⊣ℑ. \Re \; \dashv \; \Im \,.

      For both supercommutative algebras and for commutative algebras with grading we have inclusions of images of these functors

      ⇉ ⊣ ⇝ ⊥ ⊥ ⇝ ⊣ R ∨ ∨ ℜ ⊣ ℑ \array{ \rightrightarrows &\dashv& \rightsquigarrow \\ \bot && \bot \\ \rightsquigarrow &\dashv& \R \\ \vee && \vee \\ \Re &\dashv& \Im }

      On the other hand, for supercommutative algebras but not for commutative algebras with grading, there is also an inclusion of images diagonally

      ⇝ℑ≃ℑ. \rightsquigarrow \Im \; \simeq \; \Im \,.

      Because evaluating this via Yoneda on representables, then by adjunction this means that on these

      ℜ⇉≃ℜ \Re \rightrightarrows \; \simeq \; \Re

      hence that the reduced part of the even part is the reduced part of the full algebra.

      But this says that the odd part of the algebra is nilpotent! This is something that is true for supercommutative algebras, but which need not be true for any old commutative algebras with grading.

      It is tempting to say that this is a kind of Aufhebung , though it differs from what is currently discussed in that entry in that the diagonal inclusion is along nw-se instead of ne-sw. But I think that just means the entry should be generalized.

      In summary, while the Aufhebung-condition here gets closer on narrowing in the axioms on supercommutative algebras, it still does not quite characterize them. So far these axioms still allow in particular also (sheaves on formal duals to) cyclically-graded algebras whose non-0 graded parts are nilpotent.

      The first question seems to be: do we have further natural-looking axioms on the modalities that would narrow in further on genuine supergeometry?

      But possibly a second question to consider is: might there be room to declare that the further generality allowed by the axioms is something to consider instead of to discard. If there is a nice axiomatics that characterizes something a tad more general than supergeometry, maybe that’s indication that this generalization is of interest?

      Not sure yet.

    • I have added the pair of references

      • John Francis, Derived algebraic geometry over ℰ n\mathcal{E}_n-Rings (pdf)

      • John Francis, The tangent complex and Hochschild cohomology of ℰ n\mathcal{E}_n-rings (pdf)

      to the References-section at various related entries, such as at derived noncommutative geometry.

      (Thanks to Adeel in the MO-comments here. He watched me ask the question there on three different forums, before then giving a reply on the fourth ;-)

    • Someone created a page Gabriel-Zisman. I have changed the name to Gabriel and Zisman and made a start on a (stub) entry.

    • Just added a references to the entry Morita equivalence. Noticed that the entry is in a woeful state.

      Can’t edit right now, but hereby I move some ancient query boxes that were sitting there from there to here:

      — forwarded query boxes:

      +–{: .query} David Roberts: More precisely, a Morita morphism is a span of Lie groupoids such that the 'source leg' has an anafunctor pseudoinverse. Anafunctors are only examples of Morita morphisms, in the sense that open covers ∐U i→M\coprod U_i \to M are examples of surjective submersions.

      I’m also not sure that this should be called the folk model structure, as I don’t think it exists for groupoids internal to DiffDiff. Details of the model structure are in a paper by Everaert, Kieboom and van der Linden, but seem to be tailored towards groupoids internal to categories of algebraic things (e.g semiabelian categories). I think the best one can do is a category of fibrant objects, but that is not something I’ve looked at much.

      Toby: For me, an anafunctor involves a surjective submersion rather than an open cover, which is how that got in there. The important thing is to have equivalent hom-categories.

      David Roberts: I’m not what I was thinking at the time, but you are pretty much right: the definition of anafunctors depends on a choice of a subcanonical singleton Grothendieck pretopology, so it was remiss of me to demand the use of open covers :) As for the definition of Morita morphism, I now can’t remember if that referred to the span which is an arrow in the localised 2-category or the arrow in the unlocalised 2-category that is sent to an equivalence under localisation. At least for me, the terminology Morita morphism evokes the generalisation from a Morita equivalence (a span of weak equivalences) to a more general morphism in that setting.

      Toby: I think that a Morita morphism should be a span, although now I'm not sure that this is what the text says, is it? I should check a reference and then change it.

      David Roberts: I’m fairly sure I’ve heard Lie groupoid people (Ping Xu springs immediately to mind) speak about a Morita morphism as being a fully faithful, essentially surjective (in the appropriate sense) internal functor, but I disagree with their usage. If this is indeed the case, we could note the terminological discrepancies =–

      +– {: .query}

      So is it true that there is a model category structure on algebras such that Morita equivalences of algebras are spans of acyclic fibrations with respect to that structure?

      Zoran Škoda: Associative (nonunital) algebras make a semi-abelian category, ins’t it ? So one could then apply the general results of van den Linden published in TAC to get such a result, using regular epimorphism pretopology, it seems to me. It is probably known to the experts in this or another form.

      =–

    • I fixed two grey links by creating pages for Whitehead’s two papers on Combinatorial Homotopy. If I get around to it I may add in a table of section headings with some comments. The entries are stubs at the moment.

    • I gave “adjoint cylinder” its own entry.

      This is the term that Lawvere in Cohesive Toposes and Cantor’s “lauter Einsen” (see the entry for links) proposes for adjoint triples that induce idempotet (co-)monads, and which he proposes to be a formalization of Hegel’s “unity of opposites”.

      In the entry I expand slightly on this. I hope the terminology does not come across as overblown. If it does, please give it a thought. I believe it is fun to see how this indeed formalizes quite well several of the examples from the informal literature.

      I am not sure if “adjoint cylinder” is such a great term. I like “adjoint modality” better. Made that a redirect.

    • I added two links to interesting looking ArXiv papers at Snigdhayan Mahanta. I also updated the link to his webpage which was no longer valid.

    • I have been adding stuff related to Green-Schwarz sigma models on super-AdS target spaces – added a list of references here – , and in the course of this touched a bunch of related entries and created a few more minimal entries.

      These references show (more or less) that the relevant super Lie algebra cocycles all lift through the Inönü-Wigner contraction from the super anti-de-Sitter/superconformal group to the super Poincaré-group. What I am really after is seeing which of the cocycles lift to super Möbius space, i.e. not to the coset of the super adS/superconrormal group by a super-Lorentz subgroup, but by a parabolic conformal subgroup. This I don’t understand yet at all, but it might as well be trivial, and I wouldn’t see it at this stage.

      Anyway, I have touched/created

      local and global geometry - table, anti de Sitter group, superconformal group, super anti de Sitter spacetime

      and maybe others, too. At conformal group I have added alightning statement of the key isomorphism with a pointer to the literature.

    • I am still after getting a more fine-grained idea of how to best and systematically attach words (notions) to the pluratily of structures encoded by adjoint pairs of modal operators.

      Over in the GoogleGroup “nLab talk” Mike and I were discussing this in the thread called “modal and co-modal”, I am now moving this to here.

      This issue is to a large extent just a linguistic one, and so should maybe better be ignored by readers with no tolerance for that.

      Myself, I may be very slow in discussing this, as I come back to the question every now and then when something strikes me. For the moment I am just looking more closely at our examples to explore further what in their known situation is natural terminology.

      Here is one random thought:

      given a sharp modality, then we look for the “concrete” objects XX for which the unit X↪♯XX \hookrightarrow \sharp X is a monomorphism. In a sense the anti-modal objects for which ♯X≃*\sharp X\simeq \ast are at the “opposite extreme” of these where a concrete object is all “supported on points”, a sharp-anti-modal object is maximally not supported on points.

      By the way, regarding just the terminology for this special case: I have come to think that this is what should be referred to by “intensive quantity” and “extensive quantity”. In modern language an extensive quantity in thermodynamics is a differential form in positive degree (e.g. the mass density 3-form on Euclidean 3-space) while an intensive quantity is a 0-form, hence a function (for instance a temperature function on Euclidean 3-space).

      Now the sheaf of functions ℝ\mathbb{R} is indeed such that ℝ↪♯ℝ\mathbb{R}\hookrightarrow \sharp \mathbb{R}, while the sheaves Ω p\mathbf{\Omega}^p of differential forms in positive degree are indeed such that ♯Ω p≃*\sharp \mathbf{\Omega}^p \simeq \ast. The objects XX with ♯X≃*\sharp X\simeq \ast are “extensive” in the literal sense that they “extend further than a single point”. This is much the “extension” by which also Grassmann’s Ausdehnungslehre refers to his forms.

      A related random thought: so given any modal operator ◯\bigcirc we should probably be looking at objects XX such that X→◯XX \to \bigcirc X is a mono, or is nn-truncated for general nn.

      For instance the “separated objects” for the constant *\ast-modality are the mere propositions. Generally, the nn-truncated objects are maybe more naturally thought of as being part of what the *\ast-modality encodes. From this perspective the nn-truncation monads on the one hand and the monads appearing in cohesion on the other play somewhat different roles in the whole system, which maybe explains some of the terminological mismatch (if that’s what it was) that Mike and I were struggling with in the earlier thread.

    • I tried to slightly polish the article Dold--Kan correspondence (section: Details) to reduce my confusion concerning the concept of "normalized Moore complex" for a simplicial abelian group. Before I did this work, two different definitions of this concept were given, and it wasn't made vividly clear that these two definitions are naturally isomorphic. In fact that's made clear on the page Moore complex, but I didn't think to look there at first. So, I've tried to make this page a bit more clear and self-contained.

      This page could still use lots of work. There's a half-completed proof, and I suspect the notation and terminology regarding "normalized Moore complex", "alternating face complex", etc. is not completely consistent throughout the $n$Lab.

      I'm also quite sure that in the abelian case, about 100 times as many people have heard of the alternating face complex --- except they call it the "normalized chain complex" of a simplicial abelian group. It may be okay to force people to learn about the normalized Moore complex, which has the advantage of applying to nonabelian simplicial groups. But, it's good to explain what's going on here.

    • inspection of the original sources shows that since we have an entry cohesion we should also have an entry elasticity. Created it with some minimum of pointers.

    • I recently obtained a copy of Richard Statman’s thesis, “Structural Complexity of Proofs”, so I created a little entry (Structural Complexity of Proofs) to describe some of its contents. For now there is just his definition of the “genus of a proof” and some references.

    • Wonderful to have the nForum back. Testing with the following message.

      I can’t remember everything I did at the nLab over the weekend (I guess I could look up Recently Revised), but I did add to Hopfian group, and linked to my current favored proof that epis in GrpGrp are surjections.

    • added the following to the Baruch Spinoza and Spinoza’s System pages:

      Spinoza and Motifs

      Spinoza seeks, in the vein of deep unification programs in mathematics and natural science, to find structural uniformities behind the segregative diversities analytic philosophy so prizes. Spinoza is “musical” in this search for unity and unification of all being(s), prizing a motif (as [a tragic soul mate of Spinoza’s], Grothendieck, so wisely dubbed it three hundred years later) - a single structural archetype idea, recurring throughout the theory of cognition, the theory human (and not “intentional’) action, the study of the mathematics of space-time, the nature of God and the interlocked natures of power and right - in short a motif unifying all beings and all actions in Nature.

      Everything in Its Right Place: Spinoza and Life by the Light of Nature, Joseph Almog, pg xi

    • created as stub for Zeno’s paradox of motion, for the moment mainly in order to record pointers to texts that expand on its relation to the modern concept of convegence and differentiation, for which I found

      • Carl Benjamin Boyer, The history of the Calculus and its conceptual development, Dover 1949

      Added cross-pointer to this from the References-section of differential calculus, convergence, differentiation and analysis

    • Someone started a page, Taub-NUT spacetime, on this with no content. I have added a link to a wikipedia page.

    • I have added a stubby category:reference entry for Spivak’s Calculus on Manifolds. Then I have allowed myself to put the quote from the preface there

      There are good reasons why the theorems should all be easy and the definitions hard.

      into the entries theorem and definition.

    • Thought I should put some content into coercion. I’m sure it could be much better phrased. What should be said about the kind of c:(A→B)c:(A \to B) that can be used to coerce?

    • I have fine-tuned the definition of manifolds in differential cohesion a bit more (here).

      I think now a good axiomatization is like this:

      Let VV be a differentially cohesive homotopy type equipped with a framing. Then a VV-manifold is an object XX such that there exists a VV-cover, namely a correspondence

      U ↙ ↘ V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

      such that both morphisms are formally étale morphisms and such that U→XU \to X is in addition an effective epimorphism.

      This style of definition very naturally leads to a good concept of integrable G-structures (in differential cohesion).

      What I find particularly charming is that if we take such a correspondence and “prequantize” it in the sense of prequantized Lagrangian correspondences, i.e. if we pick a differential coefficient object B𝔾 conn\mathbf{B}\mathbb{G}_{conn} and complete to a correspondence in the slice

      U ↙ ↘ V ⇙ ≃ X L WZW↘ ↙ L WZW X B𝔾 conn \array{ && U \\ & \swarrow && \searrow \\ V && \swArrow_{\simeq} && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}^X_{WZW}}} \\ && \mathbf{B}\mathbb{G}_{conn} }

      then this captures precisely the globalization problem of WZW terms that we have been discussing elsewhere: on the left we pick a WZW term on the model space, and completing the diagram to the right means finding a globalization of this term to XX that locally restricts to the canonical term, up to equivalence.

      I think I have now full proof of one direction of the corresponding obstruction (details in this pdf):

      Theorem Given VV a differentially cohesive ∞\infty-group, XX a VV-manifold, and L WZW\mathbf{L}_{WZW} an equivariant WZW-term on VV, then an obstruction to L WZW X\mathbf{L}_{WZW}^X to exist as above is the existence of an integrable QuantMorph(L WZW 𝔻 V)QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V})-structure on XX,

      (i.e. a lift of the structure group of the frame bundle to the quantomorphism n-group of the restriction L WZW 𝔻 e V\mathbf{L}_{WZW}^{\mathbb{D}^V_e} of the WZW term to the infinitesimal neighbourhood of the neutral element in VV, such that this lift restricts over a VV-cover UU to the canonical QuantMorph(L WZW 𝔻 e V)QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V_e})-structure on VV).

      I still need to prove that this is not just a necessary but also a sufficient condition. This is harder…

    • When somebody asked me about the sSet enrichment of cosimplicial objects, I noticed that the nLab didn’t have the pointers. So I have now split off a stub cosimplicial object from simplicial object and added a bare minimum of pointers. No time for more at the moment.

    • I gave Lawvere’s Cohesive Toposes and Cantor’s lauter Einsen a category:reference-entry.

      The article makes an interesting claim: that Cantor’s original use of terminology was distorted by its editors to become what we now take to be the standard meaning in set theory. But that instead Cantor really meant “cohesive types” when saying “Menge” and used “Kardinale” really for what we call the underlying set of a cohesive type.

      The article also contains the proposal that adjoint modalities capture Hegel’s “unity of opposites”.

      Notice that where the English translations of Science of Logic say “the One”, the original has “Das Eins”, which might just as well be translated with “The Unit”. In view of this and looking through Hegel’s piece on discreteness and repulsion, I think it is clear that Hegel’s “Einsen” is precisely Cantor’s “Einsen” as recalled by Lawvere. Namely: copies of the unit type.

    • Just for completeness I should say that in the course of a recent discussion I had created a bare minimum at term model.

    • Since the Idea section was so big, I put most of it in a Summary section instead. I also a few remarks about it along the way.

    • Steiner gave a beautiful description of the so-called Crans-Gray tensor product in the paper that I linked which gives a simple description of the tensor product of strict ω-categories that admit a basis with certain loop-freeness assumptions.

      Among these objects are the objects of Θ, the orientals, the qbicals (defined by Crans), and a ton of other families of primitive shapes.

      I’m surprised that this hasn’t gotten more coverage, especially since it’s about a trillion times easier to make sense of than Crans’s original paper, which is made obsolete by this paper, which is better in every way. It should be the standard reference.

      I have added this link and a short description to Crans-Gray tensor product.

      Also, not to disrespect Crans, but other mathematicians gave constructions of this tensor product before him (I believe there is an earlier construction using orientals by Street), and the nLab is the only place that calls it the Crans tensor product, so therefore, I recommend a name change to something like that “Lax Gray Tensor Product”.

      It might also be instructive to show that the 2-categorical version is obtained by taking the 2-category whose 2-cells are connected 2-components and similarly that the same holds for the cartesian product of 1-categories.

    • (…)

    • I couldn’t find an open discussion on the page descent spectral sequence so I figured I’d start one and just everyone know that I added a little bit there to the introduction to try to put it into context. I’m not sure how well it fits with the general nlab POV, but thought it might be good just to have a little more there than what was there before.

    • I noticed that Todd had just edited atom. I had been meaning to add something explaining the difference between “atomic” and “atomistic” so I added the following at atom#remarks_on_terminology.

      “Atomic” and “Atomistic” differ for the simple example of the divisor lattice for some number nn. The atoms in this lattice are prime numbers while it may also contain semi-atoms which are powers of primes. This lattice is atomic because any object not the bottom (11) is divisible by a prime. However it is not atomistic but instead uniquely semi-atomistic (every non-bottom object is the product of a unique set of semi-atoms), which is one way of stating the fundamental theorem of arithmetic, also known as the unique factorization theorem.

      EDIT: I improved my nLab edit and reflected it above.

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