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I have created brief stubs for cyclotomic field and anti-cyclotomic field. No real content there for the moment, just so as to make cross-links work and have a place to record references.
I suppose what I am really looking for regarding discussion here in another thread is a concept of anti-cyclotomic spectrum.
have added to Topos in the section on limits of toposes the description of the pullback of toposes by pushout of their sites of definition.
added to group character brief remarks on
brief category:people-entry for hyperlinking references at permutation representation
I found myself writing permutation (since I had linked to it) and realised that I could even redirect symmetric group there (which has been linked to for some time).
Pretty stubby now.
stub for isogeny
started stable vector bundle, but still vague
minimum page for disambiguation (“slope” used to redirect to slope filtration)
added to localic geometric morphism the reference to Johnstone’s factorization-article and the statement that the internal locale corresponding to a localic geometric morphsm f:F→E is f*(ΩF).
finally added some pointers at Spec(Z) – As a 3-dimensional space containing knots and cross-linked a bit.
For completeness of cross-links, I have added this to the Examples-section:
In the Zariski topology on an algebraic variety Spec(R), the closed points correspond to the maximal ideals in R (this Prop.).
In particular the prime numbers correspond to the closed points in Spec(Z) (this Example).
brief category:people-entry for hyperlinking references at fractional D-brane, boundary conformal field theory and orientifold
created Amitsur complex
stub, for the moment just as to cross-link Martindale ring of quotients with the rest of the world
Added this paragraph to the Idea-section:
While, hence, presheaves are just functors (on small categories), one says “presheaf” to indicate a specific perspective or interest, namely interest in the sheafification of the functor/presheaf, or at least interest in the functor category as a topos (the presheaf topos). Hence “presheaf” is a concept with an attitude.
brief category:people entry for hyperlinking references at Dynkin quiver and Bridgeland stability condition
am starting some minimum here, am really trying to see what is known regarding the following:
Since, by the McKay correspondence, we may identify each vertex of a Dynkin quiver with the isomorphism class of an irreducible representation of the corresponding finite subgroup of SU(2) GADE⊂SU(2), a Bridgeland stability condition on representations of a Dynkin quiver directly restricts to a stability function on GADERep.
But it feels that stability functions on the representation ring
Rℂ(GADE)=K(GADERep)⟶ℂought to have a really elementary expression in terms of basic objects of representation theory. Can one say anything here?
In particular, the immediate reaction when asked to present a complex-valued function on reps is to just use their characters, maybe evaluated at some chosen conjugacy class, and probably normalized in some way.
Is this known? Are there at least examples of stability functions on GADE-representation which have an elementary representation-theoretic expression, hopefully in terms of characters?
This seems like it should almost be the first non-trivial example of stability conditions, but I have trouble finding any source that would make this explicit.
Started to write up a homotopy-theoretic version of James construction following ideas of Brunerie’s IAS talk at filtered topological space
some minimum, and disambiguation from minimal model
I added some closure properties of the class of proper morphisms of toposes and the proposition saying that a morphism of toposes is proper iff it satisfies the stable weak Beck-Chevalley condition to proper geometric morphism.
I added a Definition-section to AKSZ sigma-model with a bit of expanded discussion
I added the comment
Equivalently, a symmetric monoidal (∞,1)-category is a commutative algebra in an (infinity,1)-category in the (infinity,1)-category of (infinity,1)-categories.
to the introduction of symmetric monoidal (infinity,1)-category. I hope that’s correct…
I also added the reference
(and also to E-infinity-ring).
Someone started additive analytic geometry.
Late last night I was reading in Science of Logic vol 1, “The objective logic”.
I see that the idea of cohesion is pretty explicit there, not in the first section of the first book (Determinateness, which has the discussion of “being and becoming” that Lawvere is alluding to in the Como preface) but in the second section of the first book, “The magnitude”.
There the discussion is all about how the continuous is made up from discrete points with “repulsion” to prevent them from collapsing to a single and with “attraction” that keeps them together nevertheless.
This “attraction” is clearly just the same idea as “cohesion”. One can play this a bit further and match Hegel’s Raunen to formal expressions involving the flat modality and the shape modality pretty well. I made some quick notes in the above entry.
On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think
Added some remarks, mostly about extensivity and exactness, to quasitopos.
Now there is Sylow p-subgroup.
Is there a compilation, somewhere, of the results “the (obvious) automorphisms of a small 𝔄 A are transitive on A’s maximal 𝔅s?” The only other example ready in my head is that the maximal tori in a compact Lie group are conjugate, but I know I’ve seen more.