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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 $\mathbb{A}^1$-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 ($\mathbb{A}^1$-)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 $\mathbb{A}^1$-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?
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
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