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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 𝔸1-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 (𝔸1-)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 𝔸1-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?
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.
adding reference
Anonymous
Removing all mentions of 𝔸1-/motivic homotopy theory from this article. The axiom of cohesion requires a flat modality and there is no flat modality in 𝔸1-homotopy theory. See the answered questions part of
https://github.com/felixwellen/synthetic-zariski/blob/main/README.md
Anonymouse
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