Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have tried to brush-up existential quantifier a little more. But not really happy with it yet.
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 R, then by the unit element one usually means the neutral element 1∈R 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
Todd had created subdivision.
I interlinked that with the entry Kan fibrant replacement, where the subdivision nerve∘Face 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 G-gerbes;
classification theorem by AUT(G)-cohomology;
the notion of banded G-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.
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 ℤ.
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 [n]Λ which are freely generated by the graph 0→1→2→…→n→0, and whose morphisms Λ([m],[n])⊂Cat([m],[n]) are precisely the functors of degree 1 (seen either at the level of nerves or via the embedding Ob[n]Λ→R/Z≅S1 given by k↦k/(n+1)modZ on the level of objects, the rest being obvious).
The simplex category Δ can be identified with a subcategory of Λ, having the same objects but with fewer morphisms. This identification does not respect the inclusions into Cat, however, since [n] and [n]Λ 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
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 R (or a commutative k-algebra R) can be defined as the free commutative differential graded algebra on R.
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 M.
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 A whose degree 0 component A0 is equipped with a structure of a C^∞-ring in such a way that the degree 0 differential A0→A1 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 M is canonically isomorphic to the differential graded algebra of smooth differential forms on M. \end{theorem}
The Poincaré lemma becomes a trivial consequence of the above theorem.
\begin{proposition} For every n≥0, the canonical map
R[0]→Ω(Rn)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 M is the free C^∞-dg-ring on the C^∞-ring C∞(M). If M is the underlying smooth manifold of a finite-dimensional real vector space V, then C∞(M) is the free C^∞-ring on the vector space V* (the real dual of V). Thus, the de Rham complex of a finite-dimensional real vector space V is the free C^∞-dg-ring on the vector space V*. This free C^∞-dg-ring is the free C^∞-dg-ring on the free cochain complex on the vector space V*. The latter cochain complex is simply V*→V* 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 V is cochain homotopy equivalent to the free C^∞-dg-ring on the zero cochain complex, i.e., R in degree 0. \end{proof}
gave this reference item some more hyperlinks:
I strongly disagree with the statement in Grothendieck category that the Grothendieck category is small. The main examples like RMod 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 Set.
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.
Added related concepts section with links to coherent category, coherent hyperdoctrine, Pos, and Frm
Anonymouse
Added table of contents and links to geometric category and geometric hyperdoctrine
Anonymouse
I have added some things to frame. Mostly duplicating things said elsewhere (at locale and at (0,1)-topos), but I need these statements to be at frame itself.