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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) , a Bridgeland stability condition on representations of a Dynkin quiver directly restricts to a stability function on .
But it feels that stability functions on the representation ring
ought 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 -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.
Added a recent reference on Peirce’s Gamma graphs for modal logic. This describes his first approach via broken cuts rather than the later tinctured sheet approach. I keep meaning to see if there’s anything in the latter close to LSR 2-category of modes approach.
According to the broken-cut method, possibility is broken cut surrounding solid cut, while necessity is solid cut surrounding broken cut. Since solid cut is negation, broken cut signifies not-necessarily. Easy to see as the same pattern of three cuts, etc.
In the Alpha case, we’re to think of negated propositions as though written elsewhere on another sheet (or the back of the sheet). There seems to be a three-dimensionality to the graphs, e.g., the conditional as like a tube from one sheet to another, Wikipedia. I gather his later ideas on tinctured graphs had this idea of being inscribed on different sheets.
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 are transitive on ’s maximal s?” 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.
I am giving this generalized homology theory its own little entry, so that it becomes possible to refer to it more specifically, beyond broadly pointing to just “stable homotopy groups”.
(Curious that things are set up such that the most fundamental of homology theories is almost un-nameable, since its canonical name clashes with the name of the whole subject. Curious circularity there.
The other day I was visiting the Grand Mosque. It’s qibla wall has a huge mosaique displaying the 99 names of God in 99 flowers, plus one flower with no name it in, to represent the un-nameable (one can see it well here, only that the sheer size of it is not brought across by photographs). )
brief category:people entry for hyperlinking of references at McKay correspondence and fractional D-brane
brief category:people entry for hyperlinking references at motivic integration
gave the statement an entry with a pointer to a proof: Hausdorff implies sober, then added pointer to this at sober space where it was claimed without proof or citation, and at Hausdorff space where it had previously not been mentioned yet.
brief category:people entry for hyperlinking references at geometric fixed point spectrum
slightly expanded the Idea-section,
added pointer to the lecture notes by Andrew Blumberg,
cross-linked with Mackey-functor and enriched (∞,1)-functor
stub entry, for the moment just for the purpose of facilitating cross-links from/to Mackey functor
discovered this ancient entry (while searching for occurences of “permutation matrix” on the nLab). This was in very bad shape, with a ill-rendering floating toc and big query box right at the beginning, then a little bit of content, and then some speculation by a contributor who we had to persuade to leave, long ago.
I did a minimum of cleaning up, in particular removed the query box, since it had been dealt with. This is what it had said:
+– {.query}
Zoran: there are several things called “Birkhoff’s theorem” in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff’s. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff’s factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal nad Pressley). I would like that the lab does not mislead by distinguishing one of the several famous Bikhoff labels without mentioning and directing to 2-3 others.
Ian Durham: Good point. I think this probably ought to be renamed the “Birkhoff-von Neumann theorem.” Is that a good enough label or should we get more specific with it?
Toby: I have moved it. See also the new page Birkhoff’s theorem, which is basically just Zoran's comment above. =–
brief category:people-entry for hyperlinking references at de Sitter spacetime
brief category:people-entry for hyperlinking references at Bayesian interpretation of quantum mechanics
brief category:people-entry for the purpose of hyperlinking references at Feit-Thompson theorem and at permutation representation