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nLab > Latest Changes: axiom of cohesion

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  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

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