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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 carrying
I added to star-autonomous category a mention of “*-autonomous functors”.
Created:
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.
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 M→N are polynomial maps M→N, i.e., elements of SymM*⊗kN.
A commutative algebra A can be identified with a product-preserving functor FinPolyk→Set, where FinPolyk is the full subcategory of Polyk on finitely generated free modules. The value A(X) for X∈FinPolyk can be thought of as the space of regular functions SpecA→X, 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 X∈CartPoly)
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 kn→M, together with a way to compose a polynomial map kn→M with a regular function SpecA→kn, obtaining a regular map SpecA→M. Interpreting M as the module of sections of a quasicoherent sheaf over SpecA, a regular map SpecA→M 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)→Mthat 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.
have cross-linked with entry with
the author’s pages
further relevant entries, such as Frölicher space
Created Beck module, mentioned it (once) on the tangent category page.
Created:
The abstract notion of a derivation corresponding to that of a Beck module.
Given a category C with finite limits, a Beck module in C over an object A∈C 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 M∈Ab(C/A), a Beck derivation A→M is a a morphism idA→UA(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 A→M are in bijection with morphisms of Beck modules
ΩA→M,generalizing the universal property of Kähler differentials.
For ordinary commutative algebras, Beck derivations coincide with ordinary derivations.
For C^∞-rings, Beck derivations coincide with C^∞-derivations.
The original definition is due to Jon Beck. An exposition can be found in Section 6.1 of
added to the Properties-section at Hopf algebra a brief remark on their interpretation as 3-vector spaces.
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
Started this, following this comment.
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.
added publication details for this reference:
and am copying it over to compactly generated topological space, too
I have added to orthogonal factorization system
in the Definition-section three equivalent explicit formulations of the definition;
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.
I treid to clean up Whitehead tower a bit:
I rewrote and expanded the Idea/Definition part.
Then I moved David Roberts' material that was there to the appropriate section at the new Whitehead tower in an (infinity,1)-topos. (There I tried to add some introductory remarks to it but will try to further highlight David's results here in a moment).
At Whitehead tower I left just a new section that says that there is a notion of Whitehead towers in more general contexts with a pointer to Whitehead tower in an (infinity,1)-topos
I finally gave the Connes-Lott-Chamseddine-Barrett model its own entry. So far it contains just a minimum of an Idea-section and a minimum of references.
This was prompted by an exposition on PhysicsForums Insights that I wrote: Spectral standard model and String compactifications
tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.
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).
started an Examples-section at geometric quantization
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 (g−2)-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.
I took the liberty of incorporating material from Andre Joyal's latest message to the CatTheory mailing list into the entry dagger-category:
created sections
added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors
Initial stub to record some references. Wanted by type theoretic model category
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 (I1□I2)-cofibrations; and (here) the example of pushout products of the inclusions Sn−1↪Dn. Both without proof for the moment.