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Created motivic homotopy theory (renamed from A1-homotopy theory).
Still many blanks to fill in…
I added descriptions of the slice filtration and the $\mathbb{A}^1$-Postnikov filtration at motivic homotopy theory.
I wrote something about realization functors.
I wrote something about the six operations.
I added a few sentences about the relation to the theory of symmetric bilinear forms.
Thanks, Marc, for all the work! Glad that you are looking into these $n$Lab entries on motivic stuff.
(I’ll read your aditions later, can’t right now…)
Did look at it right now after all. One tiny comment: the link labeled “symmetric space” but pointing to bilinear form should go with a bit more of a comment, since there is also symmetric space.
I wrote something about A1-coverings and added the statement of Morel’s connectivity theorem at A1-Postnikov filtration.
Did look at it right now after all. One tiny comment: the link labeled “symmetric space” but pointing to bilinear form should go with a bit more of a comment, since there is also symmetric space.
Sorry, I hadn’t noticed. I replaced “symmetric space” with “symmetric bilinear form”.
I wrote something about Euler classes and splittings of vector bundles.
I have added to the entry anchors to the definition of $H(S)$ and $SH(S)$ and cross-links back to these definitions, to make it easier for the reader to jump into the middle of this entry and still know what the notation means.
I added the reference
I have moved the subsections on algebraic K-theory and algebraic cobordism to the pages algebraic K-theory spectrum and algebraic cobordism, respectively.
added the following reference: Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002
I thought there was room to state the main definition here (this Def.) more clearly, by saying more explicitly what it means to localize “over a site equipped with an interval”.
It used to say:
The motivic homotopy category $\mathrm{H}(S)$ over $S$ is the homotopy localization of the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site $Sm/S$ equipped with the interval object $\mathbb{A}^1$.
Now I made it say:
The motivic homotopy category $\mathrm{H}(S)$ over $S$ is the homotopy localization at the affine line $\mathbb{A}^1$ (1) of the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site $Sm/S$.
and the link “at the” points to an actual definition of what this means.
added publication data for
added pointer to
Peter Arndt, Abstract motivic homotopy theory, thesis 2017 (web, pdf)
also: lecture at Geometry in Modal HoTT, 2019 (recording I)
The recording of the second talk will become available later today; will update then.
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