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    • brief category:people-entry for hyperlinking references

      v1, current

    • starting something – remains a stub for the moment, to be continued

      v1, current

    • Decidable language =/= decidable type checking.

      Arvid Marx

      diff, v23, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • a bare list of references, to be !include-ed into the references lists of relevant entries

      v1, current

    • absorption monoids, monoids that have an absorbing element that behaves like zero in the multiplicative monoid of the natural numbers

      Anonymous

      v1, current

    • creating article on tensor product of commutative monoids

      Anonymous

      v1, current

    • Just noticed that we have a duplicate page Jon Sterling.

      I have now moved the (little but relevant) content (including redirects) from there to here.

      Unfortunately, the page rename mechanism seems to be broken until further notice, therefore I am hesitant to clear the page Jon Sterling completely, for the time being.

      diff, v3, current

    • I have created a stub for dependent type theory.

      This used to redirect to just type theory, but in that entry it is being escaped to Martin-Löf type theory, so clearly either it should redirect there or have a separate entry. I guess a separate entry is better, since there is dependent type theory that is not of Martin-Löf “type”.

    • 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

    • a bare list of references, to be !include-ed into the References-lists of relevant entries (such as at anyon and quantum Hall effect) for ease of updating and synchronizing

      v1, current

    • starting disambiguation page for “evil” in mathematics

      Abe

      v1, current

    • created computational trinitarianism, combining a pointer to an exposition by Bob Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.

    • I added to walking structure a 2-categorical theorem that implies that usually “the underlying X of the walking X is the initial X”. This fact seems like it should be well-known, but I don’t offhand know a reference for it, can anyone give a pointer?

    • polished a bit and expanded a bit at interval category (nothing deep, just so that it looks better)

    • Created a stub for referencing purposes.

      v1, current

    • starting page on the On-Line Encyclopedia of Integer Sequences

      Abe

      v1, 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

    • starting page on evil numbers in number theory

      Abe

      v1, current

    • starting page on Elwyn Berlekamp for the sake of a reference at evil number

      Abe

      v1, current

    • starting page on odious numbers in number theory

      Abe

      v1, current

    • starting page on the Thue–Morse sequence from number theory

      Abe

      v1, current

    • I’ve added Peter May’s Galois theory example to M-category in a section “Applications”.

    • what’s the point of this article?

      It seems to just be one giant discussion page that was the redirection target of the article functor (discussion). Seems like it would be better to delete this article and post the contents of this page at the nForum discussion thread for the functor article.

      Abe

      diff, v28, current

    • Started presentation of a category by generators and relations. This is probably an evil definition (there was an old discussion on this in the context of quotient category), and there is perhaps a more modern way to do this, so feel free to change the entry. I used “quotient category” as in CWM and mentioned that this is not the definition in the nLab.

    • Mention the adjoints to the discrete category construction.

      diff, v24, current

    • removing query boxes

      +– {: .query} Madeleine Birchfield: Wouldn’t a cardinal number be an object of the decategorification of the category Set, just as a natural number is an object of the decategorification of the category FinSet? =–

      +– {: .query} Roger Witte First of all sorry if I am posting in the wrong place

      While thinking about graphs, I wanted to define them as subobjects of naive cardinal 2 and this got me thinking about the behaviour of the full subcategories of Set defined by isomorphism classes. These categories turned out to be more interesting than I had expected.

      If the background set theory is ZFC or similar, these are all large but locally small categories with all hom sets being isomorphic. They all contain the same number of objects (except 0, which contains one object and no non-identity morphisms) and are equinumerous with Set. Each hom Set contains N NN^N arrows. In the finite case N!N! of the morphisms in a particular hom set are isomorphisms. In particular, only 0 and 1 are groupoids. I haven’t worked out how this extends to infinite cardinalities, yet.

      If the background theory is NF, then they are set and 1 is smaller than Set. I haven’t yet worked out how 2 compares to 1. I need to brush up on my NF to see how NF and category theory interact.

      I am acutely aware that NF/NFU is regarded as career suicide by proffesional mathematicians, but, fortunately, I am a proffesional transport planner, not a mathematician.

      Toby: Each of these categories is equivalent (but not isomorphic, except for 0) to a category with exactly one object, which may be thought of as a monoid. Given a cardinal NN, if you pick a set XX with NN elements, then this is (up to equivalence, again) the monoid of functions from XX to itself. The invertible elements of this monoid form the symmetric group, with order N!N! as you noticed. Even for infinite cardinalities, we can say N NN^N and N!N!, where we define these numbers to be the cardinalities of the sets of functions (or invertible functions) from a set of cardinality NN to itself.

      From a structural perspective, there's no essential difference between equivalent categories, so the fact that these categories (except for 0) are equinumerous with all of Set is irrelevant; what matters is not the number of objects but the number of isomorphism classes of objects (and similarly for morhpisms). That doesn't mean that your result that they are equinumerous with Set is meaningless, of course; it just means that it says more about how sets are represented in ZFC than about sets themselves. So it should be no surprise if it comes out differently in NF or NFU, but I'm afraid that I don't know enough about NF to say whether they do or not.

      By the way, every time you edit this page, you wreck the links to external web pages (down towards the bottom in the last query box). It seems as if something in your editor is removing URLs. =–

      Anonymous

      diff, v46, current

    • brief category:people-entry for hyperlinking references

      v1, current

    • It should be clarified that type checking is decidable, not e.g Type inhabitation or some other property.

      Anonymous

      diff, v10, current

    • starting page on dependent type theory with type variables

      Anonymouse

      v1, current

    • the concept of a weak type theory as a dependent type theory where all type formers are weak in the sense that they use identity types instead of judgmental equality in the computation and uniqueness rules

      Anonymouse

      v1, current

    • I may have written something at Kervaire invariant, but it is at best a stub for the moment

    • I gave regular cardinal its own page.

      Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.

      Looks better to me, but let me know what you think.

    • Creating a new page on this topic

      Ivan

      v1, current

    • http://ncatlab.org/nlab/show/Isbell+duality

      Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality

      However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf

      seems to use another definition. Could someone please clarify?

    • Add some discussion of the equivalence between the two definitions, and how in practice we usually use the family-of-collections-of-morphisms one.

      diff, v70, current

    • Discussion of the formulas for the standard characteristic forms has been missing in various entries (e.g. at Chern class at characteristic form, etc.). Since there is little point in discussing the Chern forms independently from the Pontrjagin forms etc. I am now making it a stand-alone section to be !include-ed into relevant entries, to have it all in one place.

      Not done yet, though, but it’s a start.

      v1, current

    • I was about to create a new entry “characteristic differential form” when I discrovered this old entry.

      Have added more redirects to it and more cross-links with Chern-Weil homomorphism.

      diff, v8, current

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