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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.
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.
At overt space there was a remark that since the definition quantifies over “spaces”, the overtness of a single space might depend on the general meaning chosen for “space”, but that no example was known to the author. I added an example involving synthetic topology, which may not be quite what the author of that remark was thinking of, but which I think is interesting.
I incorporated some of my spiel from the blog into the page type theory.
There has GOT to be a better photograph than that! Is there anyone here in Oxford? Can they go and get a picture for us?
I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.
Added some content to display map from Taylor’s book. Not very deep, mostly as a reference to the respective section for me.
Created basic outline with some important connections. Yang-Mills measure, after all the main concept which makes this special case interesting, and references will be added later.
Edit: Crosslinked D=2 Yang-Mills theory on related pages: D=2 QCD, D=4 Yang-Mills theory, D=5 Yang-Mills theory.