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finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).
I’ll be preparing here notes for my lectures Categories and Toposes (schreiber), later this month.
Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.
Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.
I would try to say what the idea is, but have to dash.
a bare subsection with a list of references, to be !include-ed at super Riemann surface and at moduli space of super Riemann surfaces, for ease of synchronization
created a stub for super Riemann surface, just to record Witten’s latest
I have added a little bit to supermanifold, mainly the definition as manifolds over superpoints, the statement of the equivalence to the locally-ringed-space definition and references.
At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of ℝ∞=lim⟵nℝn. And I touched the description of this example in the main text, now here.
Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category Set. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group G on a set X, and looks what happens in the vector space of functions into a field K. As we know, for a group element g the definition is, (gf)(x)=f(g−1x), for f:X→K is the way to induce a representation on the function space KX. The latter representation is called the permutation representation in the standard representation theory books like in
I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.
Edit: new (related) entries for Claudio Procesi and Arun Ram.
Added:
There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.
Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.
The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.
The nLab uses “F-space” to refer to the non-locally convex notion and “Fréchet space” to refer to the locally convex notion.
I added a Definition section to Burnside ring (and made Burnside rig redirect to it).
stub for quantum computation
added at adjoint functor
more details in the section In terms of universal arrows;
a bit in the section Examples
Added to Dedekind cut a short remark on the ¬¬-stability of membership in the lower resp. the upper set of a Dedekind cut.
I have added some accompanying text to the list of links at monad (disambiguation).
One question: in the entry Gottfried Leibniz it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?
I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?
added reference to derived category
added pointer to:
Julian Schwinger: Quantum Kinematics and Dynamics, CRC Press (1969, 1991) [ISBN:9780738203034, pdf]
Julian Schwinger (ed.: Berthold-Georg Englert): Quantum Mechanics – Symbolism of Atomic Measurements, Springer (2001) [doi:10.1007/978-3-662-04589-3]
with a link to arguments that Schwinger secretly (re-)invented groupoid algebra, in these books.
a stub entry, for the moment just to make a link work that has long been requested at Handbook of Quantum Gravity
for the equivariant+twisted version I added further pointer to
El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)
Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)
Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)
just the other day I was searching for good references on “asymptotic symmetries”, not finding much. But today appears the useful
and so I am starting an entry hereby
a bare list of references, to be !include-ed into the References-sections of relevant entries (such as knot homology, topological string theory) for ease of synchronization
added at Grothendieck universe at References a pointer to the proof that these are sets of κ-small sets for inaccessible κ. (also at inaccessible cardinal)
Together with my PhD students, I have been thinking a lot recently about the appropriate notion of a module over a C^∞-ring, i.e., something with better properties than Beck modules, which boil down to modules over the underlying real algebra in this case.
We stumbled upon the article C-infinity module (schreiber).
It says: “a C-infty algebra A is a copresheaf A∈Quantities=CoPrSh(CartesianSpaces) which becomes a copresheaf with values in algebras when restricted along FinSet↪CartesianSpaces,”
Why are we restricting to FinSet here? The underlying commutative real algebra is extracted by restricting to the Lawvere theory of commutative real algebras, i.e., CartesianSpaces_Poly, the subcategory of cartesian spaces and polynomial maps. Restricting to FinSet^op (as opposed to FinSet) extracts the underlying set only. It is unclear what is being meant by restricting along FinSet→CartesianSpaces, since the latter functor does not preserve finite products, so restricting along it does not produce a functor between categories of algebras over Lawvere theories.
In this entry, generating functional redirects to generating function. This use does not seem to match the context.
added some formatting and some cross-links (nilpotent groups!) and added pointer to:
stub at locally compact locale
Added:
Introducing hyperstonean spaces:
Added to T-duality a section with the discussion of the usual path-integral heuristics for why the two sigma-models on T-dual backgrounds yield equivalent quantum field theories.