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    • brief category:people-entry for hyperlinking references

      v1, current

    • Some references on particular constructions for gravity and Maxwell theory on n-plectic manifolds.

      diff, v31, current

    • If GG is an arbitrary monoid with multiplication m:G×G→Gm:G\times G\to G then mm induces a map m *:Fun(G,k)→Fun(G×G,k)m^*:Fun(G,k)\to Fun(G\times G,k), m *(f):f↦f∘mm^*(f):f\mapsto f\circ m. We say that f∈Fun(G,k)f\in Fun(G,k) is representative if m *(f)m^*(f) is in the image of the canonical map Fun(G,k)⊗Fun(G,k)↪Fun(G×G,k)Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k). Equivalently, ff is representative if the span of all functions g⋅f:h↦f(h⋅g)g\cdot f : h\mapsto f(h\cdot g) is finite dimensional. It follows then that m *(f)m^*(f) is in fact in (the image of) R(G)⊗R(G)R(G)\otimes R(G) where R(G)R(G) is the space of all representative functions on GG.

      Peter-Weyl theorem says that the continuous representative functions form a dense subspace of the space of all continuous functions on a compact Lie group GG.

      diff, v4, current

    • Added a stub of definition, together with a reference to an ’easy’ characeterization of arrows

      diff, v4, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • This entry largely overlaps with the new entry distribution on an affine algebraic group, but for the moment, due different tradition and minor differences in scope and in definitions , the entries are (at least temporarily) separate.

      diff, v4, current

      Some minor additions and changes at the page.

    • I found the section-outline of the entry distribution was a bit of a mess. So I have now edited it (just the secion structure, nothing else yet):

      a) There are now two subsections for “Operations on distributions”,

      b) in “Related concepts” I re-titled “Variants” into “Currents” (for that’s what the text is about) and gave “Hyperfunctions and Coulombeau distributions” its own subsection title.

      c) split up the References into “General” and “On Coulombeau functions”.

      (I hope that this message is regarded as boring and non-controversial.)

    • Created:

      This article is meant to give an exhaustive list of explicitly constructed nontopological functorial field theories in dimension 2 and higher. All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.

      Free field theories

      Posthuma

      Kandel

      Tener

      Field theories with interaction

      Pickrell

      S(Φ)=∫ Σ2 −1(‖dΦ‖ 2+m 2Φ 2)+P(Φ)S(\Phi)=\int_\Sigma 2^{-1}(\|d\Phi\|^2+m^2\Phi^2)+P(\Phi)

      Liouville field theory

      • Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas, Segal’s axioms and bootstrap for Liouville Theory, arXiv.

      v1, current

    • I am starting a page about the pentagon relation for multiplicative unitaries and related mathematics. The page for pentagon relation should be a separate page, as one does not really need the real forms and unitarity condition for the pentagon to work; this pentagon relations is sometimes called pentagon equation. nnLan uses pentagon equation as a redirect to pentagon identity from the axioms of (coherent) monoidal category, which is usually called pentagon identity indeed, and the terms relations and equation are more used in the context of dilogarithms, quantum groups, operator algebras and alike subjects, all related. The pentagon coherence is in fact related to all of these in a large subset of cases which can be directly expressed categorically, but the literature is quite different in flavour and eventually I will build 3 different pages with redirects and other superstructure, and references to the related terms like Drinfeld associator.

      v1, current

    • Minimal content fitting into the context.

      v1, current

    • I added some more information under Frobenius algebra. I would like to add the axioms in picture form, but I haven't figure out how to upload pictures yet. I'm sure I could figure it out if I wanted...

    • Added definition of Tambara-Yamagami (TY) fusion category, the simplest example of non-invertible symmetries in the 2d defect language.

      v1, current

    • Created a stub to record a new article and to satisfy a link.

      v1, current

    • Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.

    • Put proof of “second-countable space are Lindelöf” added/will add links to this page accordingly.

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • This comment is invalid XHTML+MathML+SVG; displaying source. <div> <p>created <a href="https://ncatlab.org/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a></p> <p>the secret title of this entry is "Schreier theory done right". (where "right" is right from the <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a>)</p> <p>this is the first part of the answer to</p> <blockquote> What is going on at <a href="https://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>? </blockquote> <p>The second part of the answer is the statement:</p> <blockquote> The same. </blockquote> <p>;-)</p> <p>I'll expand on that eventually.</p> </div>

    • the graphics at the old entry horizontal composition comes out wrong on my system. What’s going on? This is included as SVG.

    • Cat in here says

      you can define Cat\Cat to be the 2-category of all U′U'-small categories, where U′U' is some Grothendieck universe containing UU. That way, you have Set∈Cat\Set \in \Cat without contradiction.

      Do you agree with changing this to

      ” you can define Cat\Cat to be the 2-category of all U′U'-small categories, where U′U' is some Grothendieck universe containing UU. That way, for every small category JJ, you have the category Set J\Set^J an object of Cat\Cat without contradiction. This way, e.g. the diagram in Cat used in this definition of comma categories is defined. “

      ?

      Reason: motivation is to have the pullback-definition of a comma category in (For others, it’s about the diagram here) defined, or rather, having Cat provide a way to make it precise. Currently, the diagrammatic definition can either be read formally, as a device to encode the usual definition of comma categories, or a reader can try to consult Cat in order to make it precise. Then they will first find only the usual definition of Cat having small objects only, which does not take care of the large category

      Set ISet^I

      used in the pullback-definition. Then perhaps they will read all the way up to Grothendieck universes, but find that option not quite sufficient either since it only mentions Set, but not Set IntervalSet^{Interval} . It seems to me that large small-presheaf-categories such as Set IntervalSet^{Interval} can be accomodated, too, though.

      (Incidentally, tried to find a “canonical” thread for the article “Cat”, by using the search, but to no avail. Therefore started this one.)

    • promted by demand from my Basic-Course-On-Category-Theory-Students I expanded the entry 2-category:

      • mentioned more relations to other concepts in the Idea-section;

      • added an Examples-section with a bunch of (classes of) examples;

      • added a list of references. Please add more if you can think of more!

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • fix wrong definition of free group action

      Alexey Muranov

      diff, v32, current

    • 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?

    • I have added the adjoint modality of Even⊣OddEven \dashv Odd on (ℤ,≤)(\mathbb{Z}, \leq).

      This example is from adjoint modality (here). But it was actually a little wrong there. I have fixed it and expanded there and then copied over to here.

      diff, v3, current

    • I just see that in this entry it said

      Classically, 1 was also counted as a prime number, …

      If this is really true, it would be good to see a historic reference. But I’d rather the entry wouldn’t push this, since it seems misguided and, judging from web discussion one sees, is a tar pit for laymen to fall into.

      The sentence continued with

      …[[ the number 1 is ]] too prime to be prime.

      and that does seem like a nice point to make. So I have edited the entry to now read as follows, but please everyone feel invited to have a go at it:

      A prime number is a natural number which cannot be written as a product of two smaller numbers, hence a natural number greater than 1, which is divisible only by 1 and by itself.

      This means that every natural number n∈ℕn \in \mathbb{N} is, up to re-ordering of factors, uniquely expressed as a product of a tuple of prime numbers:

      n=2 n 13 n 25 n 37 n 411 n 5⋯ n \;=\; 2^{n_1} 3^{n_2} 5^{n_3} 7^{n_4} 11^{ n_5 } \cdots

      This is called the prime factorization of nn.

      Notice that while the number 1∈ℕ1 \in \mathbb{N} is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in mathematics of objects being too simple to be simple one might say that 1 is “too prime to be prime”.

      diff, v13, current

    • a stub entry, nothing here yet for the moment

      v1, current

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