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    • Todd,

      when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?

      Thanks!

    • Created a stub for the concept.

      v1, current

    • I added to star-autonomous category a mention of “*-autonomous functors”.

    • externalizing the table, so we can update it across several pages

      v1, current

    • Created:

      Beck modules

      The category of Beck modules over a C^∞-ring A is equivalent to the category of ordinary modules over the underlying real algebra of A.

      This is established using the proof given at Beck module for ordinary rings, using the fact that ideals of C^∞-rings coincide with ideals in the ordinary sense and the square zero extension construction used there can be promoted to a C^∞-ring using Taylor expansions.

      Furthermore, the resulting notion of a Beck derivation coincides with that of a C^∞-derivation.

      Kainz–Kriegl–Michor modules

      A different, nonequivalent definition was proposed by Kainz–Kriegl–Michor in 1987.

      Suppose k is a commutative ring. Denote by Polyk the following category. Objects are k-modules. Morphisms MN are polynomial maps MN, i.e., elements of SymM*kN.

      A commutative algebra A can be identified with a product-preserving functor FinPolykSet, where FinPolyk is the full subcategory of Polyk on finitely generated free modules. The value A(X) for XFinPolyk can be thought of as the space of regular functions SpecAX, where SpecA is the Zariski spectrum of A.

      The starting observation is that a module M over a commutative k-algebra A can be identified with a dinatural transformation (dinatural in XCartPoly)

      η:Polyk(X,M)×A(X)M.

      We require η to be linear in the first argument.

      That is to say, to specify an A-module M, we have to single out polynomial maps knM, together with a way to compose a polynomial map knM with a regular function SpecAkn, obtaining a regular map SpecAM. Interpreting M as the module of sections of a quasicoherent sheaf over SpecA, a regular map SpecAM can be restricted to the diagonal SpecA, obtaining an element of M as required.

      The proposal of Kainz–Kriegl–Michor is then to replace polynomial maps with smooth maps:

      A C^∞-module over a C^∞-ring A is a Hausdorff locally convex topological vector space M together with a dinatural transformation

      η:C(X,M)×A(X)M

      that is linear in the first argument. If η is also continuous in the first argument, we say that M is a continuous C^∞-module.

      Topological vector spaces in the above definition can be replaced by any notion of a vector space that allows for smooth maps, e.g., convenient vector space etc.

      Related concepts

      References

      • G. Kainz, A. Kriegl, P. Michor, C∞-algebras from the functional analytic view point, Journal of Pure and Applied Algebra 46:1 (1987), 89-107. doi

      v1, current

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      Idea

      The abstract notion of a derivation corresponding to that of a Beck module.

      Definition

      Given a category C with finite limits, a Beck module in C over an object AC is an abelian group object in the slice category C/A.

      The forgetful functor from modules to rings is modeled by the forgetful functor

      UA:Ab(C/A)C/A.

      Given MAb(C/A), a Beck derivation AM is a a morphism idAUA(M) in C/A.

      If UA has a left adjoint ΩA, then ΩA is known as the Beck module of differentials over A. Thus, Beck derivations AM are in bijection with morphisms of Beck modules

      ΩAM,

      generalizing the universal property of Kähler differentials.

      Examples

      For ordinary commutative algebras, Beck derivations coincide with ordinary derivations.

      For C^∞-rings, Beck derivations coincide with C^∞-derivations.

      References

      The original definition is due to Jon Beck. An exposition can be found in Section 6.1 of

      v1, current

    • added a remark (here) that the expression 1i<jn(xjxi) changes sign under exchange of any pair of variables.

      Also tried to beautify the formatting throught the entry.

      diff, v4, current

    • Started this page. No doubt it could be more elegant.

      v1, current

    • brief category:people-entry for hyperlinking references

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    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • I started an idea section at transgression, but it could probably use some going over by an expert. I hope I didn’t mess things up too badly. I was reading Urs’ note on “integration without integration” on the train ride home and fooled myself into thinking I understood something.

      By the way, this reminded me of a discussion we had a while back

      Integrals: Loops space vs target space

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

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    • brief category:people-entry for hyperlinking references

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    • I wrote a little piece at general covariance on how to formalize the notion in homotopy type theory. Just for completeness, I also ended up writing a little blurb at the beginning about the genera idea of general covariance.

    • creating this minimal entry, just to make the term linkable

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

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

      Anonymous

      v1, current

    • Mentioned the alternative terminology “Zappa–Szép product” and added redirects.

      diff, v5, current

    • The cotangent complex theorem

      Natalie Stewart

      diff, v3, current

    • Adding the actual definition.

      Natalie Stewart

      diff, v5, current

    • I have added to orthogonal factorization system

      1. in the Definition-section three equivalent explicit formulations of the definition;

      2. in the Properties-section the statement of the cancellability property.

      Wanted to add more (and to add the proofs). But have to quit now. Maybe later.

    • added section labels and a table of contents

      Anonymous

      diff, v6, current

    • starting page on antithesis partial orders

      Anonymouse

      v1, current

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

      v1, current

    • starting page on zero-dimensional rings

      Anonymouse

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • starting page on rings with tight apartness

      Anonymouse

      v1, current

    • tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

    • brief category:people-entry for hyperlinking references

      v1, current

    • as mentioned in another thread, I have expanded the Idea-section at polarization in order to highlight the relation to canonical momenta (which I also edited accordingly).

    • 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)-entry here and also at quantum gravity here.

    • I have been expanding and polishing the entry Heisenberg group.

      This had existed in bad shape for quite a while, but now it’s maybe getting into better shape.

      I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that

      • there are different Lie groups for a given Heisenberg Lie algebra,

      • and the appearance of an “i” in [q,p]=i may be all understood as not picking the simply conncted ones of these;

      I also added remarks on the relation to Poisson brackets, and symplectomorphisms.

      In this context: either I am dreaming, or there is a mistake in the Wikipedia entry Poisson bracket - Lie algebra.

      There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.

    • starting page on inequality rings or rings with inequality

      Anonymouse

      v1, current

    • starting page on residually discrete local rings

      Anonymouse

      v1, current

    • working on writing out how the “inversion” morphism of a groupoid object naturally arises from this structure.

      Jonathan Beardsley

      diff, v54, current

    • The entry test category which I wrote some time ago, came into the attention of Georges Maltsiniotis who kindly wrote me an email with a kind praise on nlab and noting that his Astérisque treatise on the topic of Grothendieck’s homotopy theory is available online on his web page and that the Cisinski’s volume is sort of a continuation of his Astérisque 301. Georges also suggested that we should emphasise that a big part of the Pursuing Stacks is devoted to the usage of test categories, so I included it into the bibliography and introductory sentence. I hinted to Georges that when unhappy with a state of an nlab entry he could just feel free to edit directly.

    • Have added to pushout-product the statement (here) that pushout product of I1-cofibrations with I2-cofibrations lands in (I1I2)-cofibrations; and (here) the example of pushout products of the inclusions Sn1Dn. Both without proof for the moment.