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I added to excluded middle a discussion of the constructive proof of double-negated LEM and how it is a sort of “continuation-passing” transform.
I’ve been inactive here for some months now; I hope this will significantly change soon.
I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:
Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?
Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?
In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products C, and form a new 2-category with 2-products Mon(C) whose 0-cells are pseudomonoids in C, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate Mon(−), starting with C=Cat. Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace Mon(−) with Monnorm(−) (where all the units are forced to coincide up to isomorphism)?
Apologies if these are naive questions; I am not very familiar with the literature.
a bare minimum, for the moment just so as to satisfy links from graded modality
I think the line between the two types of Kan extension (weak versus pointwise) is drawn at the wrong place. Am I missing something?
copied over the homotopy-theoretic references from modal type theory to here.
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?
added to G2 the definition of G2 as the subgroup of GL(7) that preserves the associative 3-form.
collected some references on the interpretation of the !-modality as the Fock space construction at !-modality.
Cross-linked briefly with he stub entries_Fock space_ and second quantization.
Added to noetherian ring a homological chacaterization: a ring is Noetherian iff arbitrary direct sums of injective modules are injective.
I have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here
Added appropriate axioms for the various definitions of affine space, along with another definition in terms of a single quaternary operation.
created dg-nerve
starting page on right triangles since the paper
talks about right triangles too
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I have expanded vertex operator algebra (more references, more items in the Properties-section) in partial support to a TP.SE answer that I posted here
We should have an entry on large N limit gradually. But sometimes it can be treated as a semiclassical limit. I quoted a reference by Yaffe where I originally read of that approach to the entry semiclassical expansion.
Move to clopen subset (since it's a relative notion, agreeing with open subset and closed subsetl
More examples added at principal ideal domain.
See Day convolution
I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category VAop and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).
This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.
I wanted to be able to use the link without it appearing in grey, so I created a stub for general relativity.
moving material about the limited principle of omniscience from principle of omniscience to its own page at limited principle of omniscience
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moving material about the lesser limited principle of omniscience from principle of omniscience to its own page at lesser limited principle of omniscience
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