Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book 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 integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: axiom of cohesion

Bottom of Page

1 to 11 of 11

  1.  

  1. starting page on axiom C0 in cohesive homotopy type theory

    Anonymous

    v1, current

  2. renaming page to indicate that this article is about the more general notion of “axioms of cohesion” talking about axiom C0, C1, C2, and R-flat.

    Anonymous

    diff, v2, current

  3. use the singular in title

    Anonymous

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2022

    Regarding the “axiom of motivic cohesion” mentioned in the Idea section here:

    What does this refer to? Is this terminology settled already?

    I get the sense that this really means to be referring just to 𝔸 1\mathbb{A}^1-localization?

    If so and if it’s not too late, I suggest reconsidering the choice of terminology, since “unstable motivic” is really an oxymoron:

    The idea of “motives” (certainly the original idea) is primarily “that which is seen by abelian cohomology theories” and only secondarily (or tertiarily or less) about (𝔸 1\mathbb{A}^1-)homotopy invariance. In fact, with “cohomology theories” understood in sufficient generality, they need not be homotopy invariant at all. Instead, the crux of “motives” is in the stabilization.

    I am well aware that the terminology mixup regarding motives is wide-spread, but if there is a chance to help not propel it further unnecessarily, let’s do so.

    If what you are looking for is a word for an axiom that describes 𝔸 1\mathbb{A}^1-localization in a context of cohesion, then I would suggest to make it rhyme on “axiom for real cohesion”: Maybe “axiom of algebraic cohesion” or “axiom of affine cohesion” or the like?

    diff, v3, current

  5. changed “motivic” to “affine” by suggestion of Urs Schreiber

    Anonymous

    diff, v4, current

  6. added sentence in ideas section about the use of “affine cohesion” in a cohesive version of Mitchell Riley’s bunched linear homotopy type theory for motivic homotopy theory.

    Anonymous

    diff, v4, current

  7. added row in table for trivial cohesion

    Anonymous

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeNov 16th 2022

    I do wonder if there is a counterpart to real-cohesion in classical homotopy theory for the simplicial set model of homotopy theory, such that the shape modallity takes Kan complexes to types.

  9. adding reference

    Anonymous

    diff, v8, current

  10. added smooth cohesion to the list

    Anonymous

    diff, v15, current

  11. Removing all mentions of 𝔸 1\mathbb{A}_1-/motivic homotopy theory from this article. The axiom of cohesion requires a flat modality and there is no flat modality in 𝔸 1\mathbb{A}_1-homotopy theory. See the answered questions part of

    https://github.com/felixwellen/synthetic-zariski/blob/main/README.md

    Anonymouse

    diff, v20, current

1 to 11 of 11

  1.