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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
created stub for Wick's lemma, for the moment just so as to record a pointer to a reference
this is a bare list of references, to be !include-ed into relevant entries (such as D-brane, Dirac charge quantization and D-brane charge quantization in K-theory).
In fact, the list is that which has been in each of these entries all along, and it has been a pain to synchronize the parallel lists. So this here now to ease the process.
have split off semi-simplicial object from semi-simplicial set.
added to differential operator the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the co-Kleisli category of the Jet bundle comonad.
Created:
Proceedings of the Workshop on Higher Segal Spaces and their Applications to Algebraic K-Theory, Hall Algebras, and Combinatorics. Contemporary Mathematics.
A proceedings volume for the conference Higher Segal Spaces and their Applications to Algebraic K-Theory, Hall Algebras, and Combinatorics.
Walker Stern: The 2-Segal space perspective: associativity and triangulations. PDF.
Philip Hackney: The decomposition space perspective. arXiv.
Viktoriya Ozornova : 2-Segal spaces and the S-construction
Martina Rovelli: The Waldhausen construction as an equivalence between stable augmented double Segal spaces and 2-Segal spaces. arXiv.
Benjamin Cooper, Matthew Young: Hall algebras via 2-Segal spaces. arXiv.
Tobias Dyckerhoff: Higher Segal spaces
Maru Sarazola, Brandon Shapiro, and Inna Zakharevich: A course on CGW-categories through the lens of homology.
Hiro Lee Tanaka: 2-Segal spaces and Fukaya categories
Julie Bergner and Walker Stern: Cyclic 2-Segal spaces. arXiv.
Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks: Decomposition spaces in combinatorics. arXiv.
Created:
A connective differential graded -ring is a (homologically graded with nonnegative degrees) real commutative differential graded algebra equipped with a structure of a C^∞-ring on .
A coconnective differential graded -ring is a (cohomologically graded with nonnegative degrees) real commutative differential graded algebra equipped with a structure of a C^∞-ring on such that the differential in degree 0 is a C^∞-derivation.
In the unbounded case, Carchedi–Roytenberg proposed the following definition:
An unbounded differential graded -ring is a (homologically graded with arbitrary degrees) real commutative differential graded algebra equipped with a structure of a C^∞-ring on .
\begin{remark} Every coconnective differential graded C^∞-ring in the sense defined above is also an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees, since the kernel of a C^∞-derivation is a C^∞-ring. The converse is false: an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees has a C^∞-ring structure on its 0-cocycles only, which is not enough to reconstruct a C^∞-structure on the whole degree 0 part or ensure that the degree 0 differential is a C^∞-derivation. The stronger condition is essential for some theorems about coconnective differential graded C^∞-rings, such as the one that states that smooth differential forms form the free C^∞-DGA on smooth functions. \end{remark}
Differential graded C^∞-rings can be equipped with the model structure transferred from the projective model structure on chain complexes via the forgetful functor.
Restricting to connective differential graded C^∞-rings, the resulting model structure is Quillen equivalent to the model category of simplicial C^∞-rings equipped with the model structure transferred along the forgetful functor to simplicial sets equipped with the Kan–Quillen model structure. The right adjoint functor is the normalized chains functor, which sends a simplicial C^∞-rings to its normalized chains equipped with the induced structure of a differential graded C^∞-ring. This is analogous to the monoidal Dold–Kan correspondence.
In this form, the statement was first proved in Taroyan 2023. Similar, but not entirely equivalent results can be found in Nuiten 2018, Remark 2.2.11, which uses a homotopy coherent variant of C^∞-rings and does not explicitly identify the adjoint functors.
The essential ingredients (C^∞-Kähler differentials and C^∞-derivations) appear in
The earliest known occurrence of differential graded C^∞-rings is in the paper
where at the bottom of page 28 in arXiv version 1 one reads:
The underlying algebra in degree 0 can be generalized to an algebra over some Lawvere theory. In particular in a proper setup of higher differential geometry, we would demand to be equipped with the structure of a C^∞-ring.
Additional references:
On unbounded differential graded rings for arbitrary Fermat theories (including C^∞-rings):
On the equivalence of connective differential graded C^∞-rings and simplicial C^∞-rings via the normalized chains functor:
Edit to: standard model of particle physics by Urs Schreiber at 2018-04-01 01:15:37 UTC.
Author comments:
added textbook reference
I have given the definition of (infinity,1)-pretopos an entry, from appendix A of Jacob Lurie’s “Spectral Algebraic Geometry”.
Since this is defined via a variant of the Giraud-Rezk-Lurie axioms with some of the infinitary structure made finitary, and since there is a large supply of non-Grothendieck examples (the sub--categories of coherent objects in any Grothendieck -topos are -pretoposes), it would be interesting if -pretoposes were examples of elementary (infinity,1)-toposes, for some definition of the latter.
Are they?
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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}
type variable used to redirect to dependent type theory with type variables, but some simple type theories like System F also have type variables, so started disambiguation page for type variable
I’ve made a few changes at flat module since I wanted to know what one was and the nLab page simply confused me further. It seemed to be saying that a module over a ring/algebra is flat if tensoring with is a flat functor. That seemed absurd so I changed it. The observation “everything happens for a reason” was a little curt so once I’d worked out what it meant I expanded it a bit and put in the analogy to bases. It wasn’t clear from the way that it was phrased whether this condition was due to Wraith and Blass or the fact that it can be put in a more general context. Lastly, the last sentence was originally in the same paragraph as the penultimate sentence where it didn’t seem to fit (or was at best ambiguous) and it also claimed that the module could be non-unital which seemed a little odd.
If an expert could kindly check that I’ve done no lasting damage to the page, I’d be grateful.
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 -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.
I tried to polish and impove the idea-section at lax natural transformation after pointing to it from MO
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
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 is a multiple of a cycle , 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
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.