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In the category:people-entry “William Lawvere” I have created a subsection “Motivation from foundations of physics” where I want to collect pointers to where and how Lawvere was/is motivated from finding foundations for (classical continuum) physics.
Explicit evidence for this that I am aware of includes notably the texts Toposes of laws of motion and the introduction to the book Categories in Continuum Physics.
The Wikipedia entry has this about motivation from physics:
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell. He learned of category theory found it a promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll. meeting on “Categories in Continuum Physics” in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry which had evolved from the spatial part of Lawvere’s categorical dynamics program). Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.
Question: Can anyone point me to more on this early phase of the story (graduate student is supposed to start to look into continuum mechanics, starts to wonder “What is a vector field, really?, what a differential equation?” and ends up revolutionizing the foundations of differential calculus)?
added to equivariant K-theory comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.
(Also finally added references to Green and Julg at Green-Julg theorem).
This all deserves to be prettified further, but I have to quit now.
Began Freyd cover. What’s it for?
stub for quantum computation
Added to Hopf monad the Bruguières-Lack-Virelizier definition and some properties.
edited reflective subcategory and expanded a bit the beginning
stub for braid group statistics (again, for the moment mainly in order to record a reference)
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
starting an entry on the integer Heisenberg group.
For the moment it remains telegraphic as far as the text is concerned (no Idea-section)
but it contains a slick (I find) computation of the modular transformation of Chern-Simons/WZW states from the manifest modular automorphy of certain integer Heisenberg groups.
Hope to beautify this entry a little more tomorrow (but won’t have much time, being on an intercontinental flight) or else the days after (where I am however at a conference, but we’ll see).
Created:
In algebraic geometry, the module of Kähler differentials of a commutative ring corresponds under the Serre–Swan duality to the cotangent bundle of the Zariski spectrum of .
In contrast, the module of Kähler differentials of the commutative real algebra of smooth functions on a smooth manifold receives a canonical map from the module of smooth sections of the cotangent bundle of that is quite far from being an isomorphism.
An example illustrating this point is , since in the module of (traditionally defined) Kähler differentials of we have , where is the exponential function. That is to say, the traditional algebraic notion of a Kähler differential is unable to deduce that 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 (where is an -module) is a derivation if and only if for any real polynomial the chain rule holds:
Indeed, taking and recovers the additivity and Leibniz property of derivations, respectively.
Observe also that is an element of the free commutative real algebra on elements, i.e., .
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 is canonically isomorphic to the module of sections of the cotangent bundle of . \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 Café 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 ) and indicated the possibility of another definition, along the lines that we are discussing on the Café
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.
following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be !include-ed into the References sections of relevant entries (mainly at homotopy theory and at algebraic topology) for ease of updating and syncing these lists.
The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.
I’ll just check now that I have all items copied, and then I will !include this entry here into homotopy theory and algebraic topology. It may best be viewed withing these entries, because there – but not here – will there be a table of contents showing the subsections here.
(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