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    • starting article about the ascending chain condition on principal ideals

      Anonymouse

      v1, current

    • starting article on atomic domains

      Anonymouse

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    • starting page on commutative operations of arbitrary finite arity

      Anonymouse

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    • Beginning an article on δ\delta-rings in the sense of Joyal, partly spurred by the recent additions by Anton Hilado.

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    • Harry Gindi points out that “infinity-field” redirects here, clashing with the unrelated entry of the same name.

      I can’t fix it right now. Maybe later.

      diff, v8, current

    • brief category:people-entry for hyperlinking references

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    • starting something – not done yet

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    • brief category:people-entry for hyperlinking references

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    • brief category:people-entry for hyperlinking references

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    • brief category:people-entry for hyperlinking references

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    • an absolute minimum, just to make the term linkable

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    • I rewrote a good bit of the entry sheaf, trying to polish and strengthen the exposition.

      The rewritten material is what now constituttes the section “Definition”. This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries.

      I left the discussion of sheaves and the general notion of localization untouched (it is now in the section “Sheaves” and localization”). This would now need to be harmonized notationally a bit better. Maybe later.

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

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    • For some text I need to explain the relation between sequents in the syntax of dependent type theory and morphisms in their categorical semantics.

      I wanted to explain this table:

      \, types terms
      (∞,1)-topos theory XEType\;\;\;\;X \stackrel{\vdash \;\;\;\;E}{\to} \;\;\Type Xt XE\;\;\;\;X \stackrel{\vdash \;\;\;t}{\to} {}_X \;\;E
      homotopy type theory x:XE(x):Typex : X \vdash E(x) : Type x:Xt(x):E(x)x : X \vdash t(x) : E(x)

      So I was looking for a place where to put it. This way I noticed that sequent used to redirect to sequent calculus. I think this doesn’t do justice to the notion and so I have

      • split off a new entry sequent

      • added a brief Idea-blurb

      • added my table and some explanation leading up to it

      leaving the whole entry in genuinely stubby state. But no harm done, I think, if we compare to the previous state of affairs.

    • Created a stub for this concept.

      v1, current

    • 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.

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    • 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 AA is equivalent to the category of ordinary modules over the underlying real algebra of AA.

      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 kk is a commutative ring. Denote by Poly kPoly_k the following category. Objects are kk-modules. Morphisms MNM\to N are polynomial maps MNM\to N, i.e., elements of SymM * kNSym M^*\otimes_k N.

      A commutative algebra AA can be identified with a product-preserving functor FinPoly kSetFinPoly_k\to Set, where FinPoly kFinPoly_k is the full subcategory of Poly kPoly_k on finitely generated free modules. The value A(X)A(X) for XFinPoly kX\in FinPoly_k can be thought of as the space of regular functions SpecAXSpec A\to X, where SpecASpec A is the Zariski spectrum of AA.

      The starting observation is that a module MM over a commutative kk-algebra AA can be identified with a dinatural transformation (dinatural in XCartPolyX\in CartPoly)

      η:Poly k(X,M)×A(X)M.\eta\colon Poly_k(X,M)\times A(X)\to M.

      We require η\eta to be linear in the first argument.

      That is to say, to specify an AA-module MM, we have to single out polynomial maps k nMk^n\to M, together with a way to compose a polynomial map k nMk^n\to M with a regular function SpecAk nSpec A\to k^n, obtaining a regular map SpecAMSpec A\to M. Interpreting MM as the module of sections of a quasicoherent sheaf over SpecASpec A, a regular map SpecAMSpec A\to M can be restricted to the diagonal SpecASpec A, obtaining an element of MM as required.

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

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

      η:C (X,M)×A(X)M\eta\colon C^\infty(X,M)\times A(X)\to M

      that is linear in the first argument. If η\eta is also continuous in the first argument, we say that MM 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

    • Created:

      Idea

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

      Definition

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

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

      U A:Ab(C/A)C/A.U_A\colon Ab(C/A)\to C/A.

      Given MAb(C/A)M\in Ab(C/A), a Beck derivation AMA\to M is a a morphism id AU A(M)id_A \to U_A(M) in C/AC/A.

      If U AU_A has a left adjoint Ω A\Omega_A, then Ω A\Omega_A is known as the Beck module of differentials over AA. Thus, Beck derivations AMA\to M are in bijection with morphisms of Beck modules

      Ω AM,\Omega_A\to M,

      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(x jx i)\prod_{1\leq i\lt j\leq n} (x_j - x_i) 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

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    • brief category:people-entry for hyperlinking references

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    • 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

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    • Added a few additional descriptions of 1\Box_{\leq 1}, which is the same as Δ 1\Delta_{\leq 1}.

      diff, v18, current

    • 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.

    • 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).

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

      v1, current

    • brief category:people-entry for hyperlinking references

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

      Anonymous

      v1, current

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

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

      diff, v5, current