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added to canonical form references (talk notes) on canonicity or not in the presence of univalence
higher coinductive types in dependent type theory, to fill out a link from the higher observational type theory article
in order to satisfy links, but maybe really in procrastination of other duties, I wrote something at quantum gravity
01-bounded semilattices, to distinguish from the bounded semilattices as discussed in semilattice#BoundedAndPseudo.
created nilradical
For all sections of
for which there is a corresponding nLab entry (most of them) I have added this pointer to the entry.
I worked on synthetic differential geometry:
I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.
Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.
seeing Eric create diffeology I became annoyed by the poor state that the entry diffeological space was in. So I spent some minutes expanding and editing it. Still far from perfect, but a step in the right direction, I think.
(One day I should add details on how the various sites in use are equivalent to using CartSp)
Just to be clear, if at wrapped cycle ϕ*[Σ] is a multiple of a cycle c, would we or wouldn’t we say it wrapped it?
stub for Blakers-Massey theorem. Need to add more references…
created Tietze extension theorem
added to quiver a very brief remark on the Gabriel classification theorem
some minimum, for the moment mostly to record this item:
created Diaconescu-Moore-Witten anomaly, so far just with a bunch of (briefly commented) references
I have added a reference to Cheng-Gurski-Riehl to two-variable adjunction, and some comments about the cyclic action.
have a quick suggestion for a definition at embedding,
This was the first idea that came to mind when reading Toby’s initial remark there, haven’t really thought much about it.
added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces
By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?
added pointer to
here and also at 3-manifold and 3-sphere
Is there a particular reference where these (or rather, their super analogs) are computed for super Riemann surfaces?
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).
Created a stub entry for norm map, for the moment just so as to make cross-links work.
I am splitting off homotopy category of a model category from model category. Have spelled out statement and proof of the localization construction there.
added hyperlinks to the text at induced representation. Made sure that it is cross-linked with Frobenius reciprocity.
I made some changes to bivector. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. M M Postnikov, Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power.
Bivector in a vector space V is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power – in general dimension just such elements in Λ2V have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to 3. Similarly, trivectors form a vector space just in the dimensions up to 4. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.
I am trying to collect citable/authorative references that amplify the analog of the mass gap problem in particle phenomenology, where it tramslates into the open problem of computing hadron masses and spins from first principles (due to the open problem of showing existence of hadrons in the first place!).
This is all well and widely known, but there is no culture as in mathematics of succinctly highlighting open problems such that one could refer to them easily.
I have now created a section References – Phenomenology to eventually collect references that come at least close to making this nicely explicit. (Also checked with the PF community here)
Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.
Wrote an article Eudoxus real number, a concept due to Schanuel.
added to the Properties-section at Hopf algebra a brief remark on their interpretation as 3-vector spaces.
I came to think that the term geometric type theory for the type theory internal toi sheaf toposes should exists. Thanks to Bas Spitter for pointing out that Steve Vickers had already had the same idea (now linked to at the above entry).
Also created geometric homotopy type theory in this vein, with some evident comments.
I have expanded slightly at coalgebra – Properties – As filtered colimits of finite dimensional pieces.
And I have added and cross-linked with corresponding remarks at dg-coalgebra, at pro-object, at L-infinity algebra and at model structure for L-infinity algebras.
https://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves
In the article here there is a confusing error in remark 2.2: in the diagram, A is contravariant, so that either the number of arrows must change or the diagram must be reversed. For comparison, the article
https://ncatlab.org/nlab/show/(infinity,1)-sheaf
in proposition 2.1 gets essentially the same correct.
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.
This is a bare list of references, to be !include-ed into the References-lists of relevant entries (such as at anyon, topological order, fusion category, unitary fusion category, modular tensor category).
There is a question which I am after here:
This seems to be CMT folklore, as all authors state it without argument or reference.
Who is really the originator of the claim that anyonic topological order is characterized by certain unitary braided fusions categories/MTCs?
Is it Kitaev 06 (which argues via a concrete model, in Section 8 and appendix E)?
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}
The book is no longer in progress, but published 8 years ago. I added the detail and a link to the AMS page.
I think it would be good to include a paragraph on the claims in the book about (∞,2)-categories the authors explicitly say they don’t prove and can’t find a proof in the literature. Just flagging this for now. Ultimately, when the papers finishing the proofs of these claims land on the arXiv, these can be cited.
Added:
\tableofcontents
A model for monoidal (∞,1)-categories.
A monoidal relative category is a monoidal category equipped with a relative category structure such that the monoidal product preserves weak equivalences.
The canonical functor from the quasicategorical localization of the relative category of monoidal relative categories, monoidal relative functors, and monoidal Dwyer–Kan equivalences to the quasicategory of monoidal quasicategories is a weak equivalence.