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Added
Edit to: standard model of particle physics by Urs Schreiber at 2018-04-01 01:15:37 UTC.
Author comments:
added textbook reference
cross-linked with Euler form and added these pointers:
Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:
{#MathaiQuillen86} Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)
{#Wu05} Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)
Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)
{#Nicolaescu18} Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)
I’ve added to reflexive graph a definition of the free category of a reflexive quiver.
That page needs some reorganization because everything now said there is about reflective quivers, and not say about reflective undirected simple graphs.
Maybe free category also also needs touching up and maybe a link to reflective graph. I don’t know how to justify that the paths in the free category don’t contain identity edges.
a bare list of references, to be !include-ed into the References-sections of relevant entries (such as supersymmetry and solid state physics) for ease of synchronization
a bare list of bibitems, to be !include-ed into the References-section of relevant entries (such as fractional quantum Hall effect and Laughlin wavefunctions), for ease of synchronization
Created:
In algebraic geometry, the module of Kähler differentials of a commutative ring R corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of R.
In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold M receives a canonical map from the module of smooth sections of the cotangent bundle of M that is quite far from being an isomorphism.
An example illustrating this point is M=R, since in the module of (traditionally defined) Kähler differentials of C∞(M) we have d(exp(x))≠expdx, where exp:R→R is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that exp′=exp using the Leibniz rule.
However, this is not a defect in the conceptual idea itself, but merely a failure to use the correct formalism. The appropriate notion of a ring in the context of differential geometry is not merely a commutative real algebra, but a more refined structure, namely, a C^∞-ring.
This notion comes with its own variant of commutative algebra. Some of the resulting concepts turn out to be exactly the same as in the traditional case. For example, ideals of C^∞-rings and modules over C^∞-rings happen to coincide with ideals and modules in the traditional sense. Others, like derivations, must be defined carefully, and definitions that used to be equivalent in the traditional algebraic context need not remain so in the context of C^∞-rings.
Observe that a map of sets d:A→M (where M is an A-module) is a derivation if and only if for any real polynomial f(x1,…,xn) the chain rule holds:
d(f(a1,…,an))=∑i∂f∂xi(x1,…,xn)dxi.Indeed, taking f(x1,x2)=x1+x2 and f(x1,x2)=x1x2 recovers the additivity and Leibniz property of derivations, respectively.
Observe also that f is an element of the free commutative real algebra on n elements, i.e., R[x1,…,xn].
If we now substitute C^∞-rings for commutative real algebras, we arrive at the correct notion of a derivation for C^∞-rings:
A __C^∞-derivation__ of a [[C^∞-ring]] $A$ is a map of sets $A\to M$ (where $M$ is a [[module]] over $A$) such that the following chain rule holds for every smooth function $f\in\mathrm{C}^\infty(\mathbf{R}^n)$:
$$d(f(a_1,\ldots,a_n))=\sum_i {\partial f\over\partial x_i}(x_1,\ldots,x_n) dx_i,$$
where both sides use the structure of a [[C^∞-ring]] to evaluate a smooth real function on a collection of elements in $A$.
The module of Kähler C^∞-differentials can now be defined in the same manner as ordinary Kähler differentials, using C^∞-derivations instead of ordinary derivations.
\begin{theorem} (Dubuc, Kock, 1984.) The module of Kähler C^∞-differentials of the C^∞-ring of smooth functions on a smooth manifold M is canonically isomorphic to the module of sections of the cotangent bundle of M. \end{theorem}
I gave CW-pair its own entry.
have created an entry Khovanov homology, so far containing only some references and a little paragraph on the recent advances in identifying the corresponding TQFT. I have also posted this to the nCafé here, hoping that others feel inspired to work on expanding this entry
have created geometric infinity-stack
gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in TAlgop∞Spec↪Sh∞(C) ) and indicated the possibility of another definition, along the lines that we are discussing on the nCafé
added to quantum anomaly
an uncommented link to Liouville cocycle
a paragraph with the basic idea of fermioninc anomalies
the missing reference to Witten’s old article on spin structures and fermioninc anomalies.
The entry is still way, way, stubby. But now a little bit less than a minute ago ;-
the table didn’t have the basic examples, such as Gelfand duality and Milnor’s exercise. Added now.
(Hi, I’m new)
I added some examples relating too simple to be simple to the idea of unbiased definitions. The point is that we often define things to be simple whenever they are not a non-trivial (co)product of two objects, and we can extend this definition to cover the “to simple to be simple case” by removing the word “two”. The trivial object is often the empty (co)product. If we had been using an unbiased definition we would have automatically covered this case from the beginning.
I also noticed that the page about the empty space referred to the naive definition of connectedness as being
“a space is connected if it cannot be partitioned into disjoint nonempty open subsets”
but this misses out the word “two” and so is accidentally giving the sophisticated definition! I’ve now corrected it to make it wrong (as it were).
adding references
Ming Ng, Steve Vickers, Point-free Construction of Real Exponentiation, Logical Methods in Computer Science, Volume 18, Issue 3 (August 2, 2022), (doi:10.46298/lmcs-18(3:15)2022, arXiv:2104.00162)
Steve Vickers, The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms, (arXiv:2312.05228)
Anonymouse
brief category:people-entry for hyperlinking references at equivariant principal bundle
category: people page for the reference
Anonymouse
category: people page for the reference
Anonymouse