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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nlab noncommutative noncommutative-geometry number number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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    • brief category: people-entry for hyperlinking references

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      Anonymous

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      Anonymous

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      Anonymous

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    • added to identity type a mentioning of the alternative definition in terms of inductive types (paths).

    • the entries dependent type and indexed set did not know of each other.

      I have now cross-linked them minimally in their “Related entries”-sections. But this would deserve to be expanded on for exposition…

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      Anonymous

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      Anonymous

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    • Created left cancellative category. This is a useful technical term.

      One natural (and non-posetal) example is the category of fields with ring homomorphisms as the morphisms—provided that the zero ring is removed from it.

      Incidentally: is there a usual technical term for, when considering any category C\mathsf{C}, the full subcategory obtained by removing all terminal objects of C\mathsf{C}?

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      Anonymous

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      Anonymous

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    • starting something – not done yet

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    • I noticed that “isomorphism class” was just re-directing to decategorification. Just for completeness, I am giving it its own little entry hereby.

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    • adding paragraph about equivalence classes in homotopy type theory

      Anonymous

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    • brief category:people-entry for hyperlinking references

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    • brief category:people-entry for hyperlinking references

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

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    • brief category: people-entry for hyperlinking references

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    • brief category: people-entry for hyperlinking references

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    • brief category: people-entry for hyperlinking references

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    • starting something, just the bare some minimum for the moment

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    • brief category: people-entry for hyperlinking references

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    • brief category: people-entry for hyperlinking references

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    • starting some minimum, also redirecting for “barcode”

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    • brief category: people-entry for hyperlinking references

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    • brief category: people-entry for hyperlinking references

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    • brief category: people-entry for hyperlinking references

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    • starting something. not done yet but need to save

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      Anonymous

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      Anonymous

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    • I gave the entry logical relation an Idea-section, blindly stolen from a pdf by Ghani that I found on the web. Please improve, I still don’t know what a “logical relation” in this sense actually is.

      Also, I cross-linked with polymorphism. I hope its right that “parametricity” may redirect there?

    • brief category: people-entry for hyperlinking references

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    • felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.

      This has much room for being further expanded, of course.

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    • brief category: people-entry for hyperlinking references

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    • I just discovered that, all along, the term “quiver representation” was just redirecting to representation. Have started this dedicated page now, with the bare minimum

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    • Added a reference to get the ball rolling. I’ll see if Jon would like to write something.

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    • I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.

      Here is what i did:

      • Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.

        • The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from \infty-topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all σ\sigma-model quantum field theories of “\infty-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.

          The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.

        • The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.

      • Apart from this I added more references and some cross-links.

      I know that the entry is still very imperfect. If you feel like pointing out all the stuff that is still missing, consider adding at least some keywords directly into the entry.

    • starting something, but not done yet

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    • I worked a little on the entry separated presheaf. Apart from some general editing I

      • added construction and proof of the separafication functor;

      • began a section on the full notion of bi-separated presheaf.

      More deserves to be done here, but I have to stop for the moment.

    • Created:

      Elements of Mathematics (French: Éléments de mathématique) is a series of books by Nicolas Bourbaki.

      Website: https://www.bourbaki.fr/Ouvrages.html

      Most of these books were quite influential, some of them especially so.

      In particular, Bourbaki’s presentation of multilinear algebra in Chapter III of the second book (Algèbre) was the first detailed expository presentation of the Grassmann algebra.

      Another influential book was Groupes et algèbres de Lie, its presentation of root systems and related topics in Chapter 4–6 was the first of its kind.

      Some criticisms raised against this series:

      • Almost complete absence of categories (a deliberate choice);

      • Théorie des ensembles receives a lot of criticism for its somewhat nonstandard treatment of logic, as well as the rather awkward notion of a structure, which occupies a similar niche to categories.

      • The use of Daniell’s approach in Intégration, mostly ignoring abstract measure spaces.

      • Chapter 10 of Algèbre (homological algebra) only covers the very classical results, (intentionally) ignoring derived categories.

      Related entries

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    • Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.

    • I gave continuous map a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.

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      Anonymous

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    • Created page, to be renamed to “stably compact space”, more to come.

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