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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
The entry used to start out with the line “not to be confused with neutral element”. This was rather suboptimal. I have removed that sentence and instead expanded the Idea-section to read now as follows:
Considering a ring , then by the unit element one usually means the neutral element with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.
But more generally a unit element in a unital (!) ring is any element that has an inverse element under multiplication.
This concept generalizes beyond rings, and this is what is discussed in the following.
expanded concrete sheaf: added the precise definition and some important properties.
stub for Hilbert’s sixth problem
added pointer to:
added to the entry on David Hilbert a pointer to this remarkable recording:
Added this pointer also, cross-link wise, at Galileo Galilei and at The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Adding reference
Anonymouse
Unfortunately, there are two entries on the same topic, both created by Urs: quantum Hall effect (redirecting also fractional quantum Hall effect what should eventually split off) with some substance, and the microstub quantum hall effect. I would like to create quantum spin Hall effect and I think I should rename/reclaim the stub quantum hall effect for this. Do others agree ? Urs ?
As the action is now delayed I record here the reference which I wanted to put there
Somewhat surprisingly, the authors and roughly this work of them are mentioned (though not in the list of references) in a paper in algebraic geometry
which considers the mirror symmetry and topological states of matters (topological insulators in particular) as main applications.
Todd had created subdivision.
I interlinked that with the entry Kan fibrant replacement, where the subdivision appears.
created a minimum at function monad (aka “reader monad”, “environment monad”)
mathematical physics with a slight distinction from physical mathematics which points to the same entry. The relation to theoretical physics has been discussed, but I am not sure yet if we should have theoretical physics as a separate entry so I do not put is as another redirect.
added to gerbe
definition of -gerbes;
classification theorem by -cohomology;
the notion of banded -gerbes.
I gave the category:people entry Daniel Freed a bit of actual text. Please feel invited to edit further. Currently it reads as follows:
Daniel Freed is a mathematician at University of Texas, Austin.
Freed’s work revolves around the mathematical ingredients and foundations of modern quantum field theory and of string theory, notably in its more subtle aspects related to quantum anomaly cancellation (which he was maybe the first to write a clean mathematical account of). In the article Higher Algebraic Structures and Quantization (1992) he envisioned much of the use of higher category theory and higher algebra in quantum field theory and specifically in the problem of quantization, which has – and still is – becoming more widely recognized only much later. He recognized and emphasized the role of differential cohomology in physics for the description of higher gauge fields and their anomaly cancellation. Much of his work focuses on the nature of the Freed-Witten anomaly in the quantization of the superstring and the development of the relevant tools in supergeometry, and notably in K-theory and differential K-theory. More recently Freed aims to mathematically capture the 6d (2,0)-superconformal QFT.
am starting curvature characteristic form and Chern-Simons form.
But still working…
I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset .
Here is some of the past discussion I’m now exporting to the nForum:
The cycle category may be defined as the subcategory of Cat whose objects are the categories which are freely generated by the graph , and whose morphisms are precisely the functors of degree (seen either at the level of nerves or via the embedding given by on the level of objects, the rest being obvious).
The simplex category can be identified with a subcategory of , having the same objects but with fewer morphisms. This identification does not respect the inclusions into , however, since and are different categories.
started cubical type theory using a comment by Jonathan Sterling
Inspired by a discussion with Martin Escardo, I created taboo.
Created polymorphism.
I added this to the entry for Nima Arkani-Hamed.
Urs (or anyone else) do you know anything about Nima’s recent interest in category theory?
“six months ago, if you said the word category theory to me, I would have laughed in your face and said useless formal nonsense, and yet it’s somehow turned into something very important in my intellectual life in the last six months or so” (@ 44:05 in The End of Space-Time July 2022)
A combinatorial notion in the study of total positivity.
for completeness, to go with the other entries in coset space structure on n-spheres – table
I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:
A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted . The underlying set is the completion of the ordered field of rational numbers: the result of adjoining to suprema for every bounded subset with respect to the natural ordering of rational numbers.
The set of real numbers also carries naturally the structure of a topological space and as such is called the real line also known as the continuum. Equipped with both the topology and the field structure, is a topological field and as such is the uniform completion of equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces for natural numbers – the real line is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.
added publication data for these two items:
Rui Loja Fernandes, Marius Crainic, Integrability of Lie brackets, Ann. of Math. 157 2 (2003) 575-620 [arXiv:math.DG/0105033, doi:10.4007/annals.2003.157.575]
Rui Loja Fernandes, Marius Crainic, Lectures on Integrability of Lie Brackets, Geometry & Topology Monographs 17 (2011) 1–107 [arxiv:math.DG/0611259, doi:10.2140/gtm.2011.17.1]
have created enriched bicategory in order to help Alex find the appropriate page for his notes.
Created:
The correct notion of a Kähler differential for C^∞-rings
See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for motivation and definition and the article smooth differential forms form the free C^∞-DGA on smooth functions for further developments and applications like the Poincaré lemma.
Created:
The correct notion of a derivation for C^∞-rings
See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for motivation and definition and the article smooth differential forms form the free C^∞-DGA on smooth functions for further developments and applications like the Poincaré lemma.
Created:
See the article Kähler C^∞-differentials of smooth functions are differential 1-forms for the necessary background for this article, including the notions of C^∞-ring, C^∞-derivation, and Kähler C^∞-differential.
In algebraic geometry, (algebraic) differential forms on the Zariski spectrum of a [commutative ring (or a commutative -algebra ) can be defined as the free commutative differential graded algebra on .
This definition does not quite work for smooth manifolds: as already explained in the article Kähler C^∞-differentials of smooth functions are differential 1-forms, the notion of a Kähler differential must be refined in order to extract smooth differential 1-forms from the C^∞-ring of smooth functions on a smooth manifold .
Thus, in order to get the algebra of smooth differential forms, the notion of a commutative differential graded algebra must likewise be adjusted.
\begin{definition} A commutative differential graded C^∞-ring is a real commutative differential graded algebra whose degree 0 component is equipped with a structure of a C^∞-ring in such a way that the degree 0 differential is a C^∞-derivation. \end{definition}
With this definition, we can recover smooth differential forms in a manner similar to algebraic geometry, deducing the following consequence of the Dubuc–Kock theorem for Kähler C^∞-differentials.
\begin{theorem} The free commutative differential graded C^∞-ring on the C^∞-ring of smooth functions on a smooth manifold is canonically isomorphic to the differential graded algebra of smooth differential forms on . \end{theorem}
The Poincaré lemma becomes a trivial consequence of the above theorem.
\begin{proposition} For every , the canonical map
is a quasi-isomorphism of differential graded algebras. \end{proposition}
\begin{proof} (Copied from the MathOverflow answer.) The de Rham complex of a finite-dimensional smooth manifold is the free C^∞-dg-ring on the C^∞-ring . If is the underlying smooth manifold of a finite-dimensional real vector space , then is the free C^∞-ring on the vector space (the real dual of ). Thus, the de Rham complex of a finite-dimensional real vector space is the free C^∞-dg-ring on the vector space . This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain complex on the vector space . The latter cochain complex is simply with the identity differential. It is cochain homotopy equivalent to the zero cochain complex, and the free functor from cochain complexes to C^∞-dg-rings preserves cochain homotopy equivalences. Thus, the de Rham complex of the smooth manifold is cochain homotopy equivalent to the free C^∞-dg-ring on the zero cochain complex, i.e., in degree 0. \end{proof}
gave this reference item some more hyperlinks:
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 strongly disagree with the statement in Grothendieck category that the Grothendieck category is small. The main examples like are not! What did the writer of that line have in mind ?
I added to the “abstract nonsense” section in free monoid a helpful general observation on how to construct free monoids. “Adjoint functor theorem” is overkill for free monoids over .
added at Grothendieck universe at References a pointer to the proof that these are sets of -small sets for inaccessible . (also at inaccessible cardinal)
The entry lax morphism classifier was started two yeats ago, is actually empty!
I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.