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    • giving this its own little entry, for ease of hyperlinking

      v1, current

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

      v1, current

    • Moving discussion here and summarizing content in the text

      +– {: .query} Mike: Why only rings without units (that is, rngs)? Intuitively, what important properties do the above listed examples share that are not shared by rings with units?

      Zoran Skoda: I want to know the answer as well. It might be something in the self-dual axioms. For unital rings artinian implies noetherian but not other way around; though the definitions of the two notions are dual.

      Toby: The category of unital rings and unitary ring homomorphisms has no zero object.

      Mike: Ah, right. Is it protomodular? I think I will understand this definition better from some non-examples that violate each clause individually.

      walt: It is protomodular. This follows from the main theorem of Characterization of Protomodular Varieties of Universal Algebra by Bourn and Janelidze. By that theorem any variety that contains a group will be protomodular. Unital rings only fail to be semiabelian for the trivial reason that ideals aren’t subrings.

      =–

      Maybe the result on protomodularity (with citation) mentioned by walt citing Bourn and Janelidze should be moved to CRing (and also Ring, if it holds for non-commutative rings).

      diff, v34, current

    • a bare sub-section with a list of references – to be !included into relevant entries – mainly at confinement and at mass gap problem (where this list already used to live)

      v1, current

    • added English translation of this bit

      PN§260 Der Raum ist in sich selbst der Widerspruch des gleichgültigen Auseinanderseins und der unterschiedlosen Kontinuität, die reine Negativität seiner selbst und das Übergehen zunächst in die Zeit. Ebenso ist die Zeit, da deren in Eins zusammengehaltene entgegengesetzte Momente sich unmittelbar aufheben, das unmittelbare Zusammenfallen in die Indifferenz, in das ununterschiedene Außereinander oder den Raum.

      Space is in itself the contradiction of the indifferent being-apart and of the difference-less continuity, the pure negativity of itself and the transformation, first of all, to time. In the same manner time – since its opposite moments, held together in unity, immeditely sublate themselves – is the undifferentiated being-apart or: space.

      And polished a little around and following this bit.

      diff, v269, current

    • I have typed into infinitesimal interval object a detailed description of the simplicial object inuced on a microlinear space from the infinitesimal interval in immediate analogy to the construction of the finite path simplicial object induced from an interval object (as discussed there).

      I also give the inclusion of the infinitesimal simplicial object into the finite one.

      All the proofs here are straightforward checking, which I think I have done rather carefully on paper, but not typed up. What I would appreciate, though, is if somebody gave me a sanity check on the definition of the infinitesimal simplicial object (which is typed in detail).

      In the very last section, which is the one that is still just a sketch, I am hoping to describe an isomorphism from my simplicial infinitesimal object to that considered by Anders Kock, which is currently described at infinitesimal singular simplicial complex in the case that the space X satisfies Kock's assumptions (it must be a "formal manifold").

      The construction I discuss at infinitesimal interval object is supposed to generalize Kock's construction to all microlinear spaces and motivated by having that canonical obvious inclusion into the finite version at interval object.

      The isomorphism should be evident: my construction evidently yields in degree k k-tuples of pairwise first oder neighbours if the space X admits that notion. But I want to sleep over this statement one more night...

    • Added definitions of a topology being regular wrt. another topology and coupled to another topology.

      diff, v6, current

    • added this second-order-quote:


      Chen Ning Yang writes in C. N. Yang, Selected papers, 1945-1980, with commentary, W. H. Freeman and Company, San Francisco, 1983, on p. 567:

      In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of S. S. Chern in El Cerrito, near Berkeley… I said I found it amazing that gauge theory are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added: “this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.” He immediately protested: “No, no. These concepts were not dreamed up. They were natural and real.”

      diff, v6, current

    • added these two quotes:


      Yang wrote in C. N. Yang, Selected papers, 1945-1980, with commentary, W. H. Freeman and Company, San Francisco, 1983, on p. 567:

      In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of S. S. Chern in El Cerrito, near Berkeley… I said I found it amazing that gauge theory are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added: “this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.” He immediately protested: “No, no. These concepts were not dreamed up. They were natural and real.

      Yang expanded on this passage in an interview recorded as: C. N. Yang and contemporary mathematics, chapter in: Robin Wilson, Jeremy Gray (eds.), Mathematical Conversations: Selections from The Mathematical Intelligencer, Springer 2001, on p. 72 (GoogleBooks):

      But it was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the physical world is intimately tied to the deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and the beauty of form?

      diff, v3, current

    • starting something. Not done yet, but need to save

      v1, current

    • the definition, and highlighting that this coincides with Chen-Ruan cohomology of the global quotient orbifold, and with Bredon cohomology with coefficients in the representation ring functor

      v1, current

    • Started a page on the Joyal-Wraith concept.

      v1, current

    • a bare table, to be !include-ed into relevant entries.

      This is to show at a glance how various definitions used in the literature are all equivalent incarnations of rational equivariant K-theory

      v1, current

    • I split off inhabited object from inhabited set.

      (moved Mike's and Toby's old discussion query box to the new entry, too)

      I added an Examples section with a remark about this issue in the context of Models for Smooth Infinitesimal Analysis, that I happen to be looking into.

      personally, I feel I need more examples still at internal logic to follow this in its full scope. I guess I should read the Elephant one day, finally.

      In the book Moerdijk-Reyes say in a somewhat pedestrian way that existential quantifiers in the internal logic of a sheaf topos are to be evaluated on covers, hence asking internally if a sheaf F has a (internally global) element means asking if for  U \to * any cover of the point, there is a morphism  U \to F.

      That's fine with me and I follow this in as far as the purpose of their book is concerned, but I need to get a better idea of how the logical quantifiers are formulate in internal logic in full generality.

    • Adding MetaPRL, RedPRL, and proto CLF as “descendants” of Nuprl.

      diff, v9, current

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

      v1, current