Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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

1. • CommentRowNumber1.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 21st 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexCreated [[motivic homotopy theory]] (renamed from [[A1-homotopy theory]]). Still many blanks to fill in...

   Created motivic homotopy theory (renamed from A1-homotopy theory).

   Still many blanks to fill in…

2. • CommentRowNumber2.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 22nd 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI added descriptions of the slice filtration and the $\mathbb{A}^1$-Postnikov filtration at [[motivic homotopy theory]].

   I added descriptions of the slice filtration and the $\mathbb{A}^1$-Postnikov filtration at motivic homotopy theory.

3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 23rd 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI wrote something about [realization functors](http://ncatlab.org/nlab/show/motivic+homotopy+theory#RealizationFunctors).

   I wrote something about realization functors.

4. • CommentRowNumber4.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 24th 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI wrote something about the [six operations](http://ncatlab.org/nlab/show/motivic+homotopy+theory#SixOperations).

   I wrote something about the six operations.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeJul 26th 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI added a few sentences about the [relation to the theory of symmetric bilinear forms](http://ncatlab.org/nlab/show/motivic+homotopy+theory#relation_to_the_theory_of_symmetric_bilinear_forms_42).

   I added a few sentences about the relation to the theory of symmetric bilinear forms.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, 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...)

   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…)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexDid 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]]_.

   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.

8. • CommentRowNumber8.
   • CommentAuthorMarc Hoyois
   • CommentTimeAug 2nd 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI wrote something about [A1-coverings](http://ncatlab.org/nlab/show/motivic+homotopy+theory#A1coverings) and added the statement of Morel's connectivity theorem at [A1-Postnikov filtration](http://ncatlab.org/nlab/show/motivic+homotopy+theory#A1PostnikovFiltration). > 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 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”.

9. • CommentRowNumber9.
   • CommentAuthorMarc Hoyois
   • CommentTimeAug 12th 2013
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI wrote something about [Euler classes and splittings of vector bundles](http://ncatlab.org/nlab/show/motivic+homotopy+theory#euler_classes_and_splittings_of_algebraic_vector_bundles_58).

   I wrote something about Euler classes and splittings of vector bundles.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2013
    • (edited Aug 12th 2013)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI 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 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.

11. • CommentRowNumber11.
    • CommentAuthoradeelkh
    • CommentTimeApr 8th 2014
    • PermaLink
    Author: adeelkh
    Format: MarkdownItexI added the reference * [[Gunnar Carlsson]], Ray Joshua, _Equivariant motivic homotopy theory_, [arXiv](http://arxiv.org/abs/1404.1597v1).

    I added the reference

    • Gunnar Carlsson, Ray Joshua, Equivariant motivic homotopy theory, arXiv.
12. • CommentRowNumber12.
    • CommentAuthoradeelkh
    • CommentTimeDec 27th 2014
    • PermaLink
    Author: adeelkh
    Format: MarkdownItexI have moved the subsections on algebraic K-theory and algebraic cobordism to the pages [[algebraic K-theory spectrum]] and [[algebraic cobordism]], respectively.

    I have moved the subsections on algebraic K-theory and algebraic cobordism to the pages algebraic K-theory spectrum and algebraic cobordism, respectively.

13. • CommentRowNumber13.
    • CommentAuthortrent
    • CommentTimeJan 18th 2015
    • PermaLink
    Author: trent
    Format: MarkdownItexadded the following reference: [Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002 ](http://www.amazon.com/Motivic-Homotopy-Theory-Nordfjordeid-Universitext/dp/3540458956)

    added the following reference: Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2018
    • PermaLink
    Author: Urs
    Format: MarkdownItexI thought there was room to state the main definition here ([this Def.](https://ncatlab.org/nlab/show/motivic+homotopy+theory#MotivicHomotopyCategory)) 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]] [[localization at an object|at the]] [[affine line]] $\mathbb{A}^1$ ([1](https://ncatlab.org/nlab/show/motivic+homotopy+theory#eq:AffineLine)) of the [[(∞,1)-topos of (∞,1)-sheaves]] on the [[Nisnevich site]] $Sm/S$. and the link "[[localization at an object|at the]]" points to an actual definition of what this means. <a href="https://ncatlab.org/nlab/revision/diff/motivic+homotopy+theory/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/motivic+homotopy+theory/47">v47</a>, <a href="https://ncatlab.org/nlab/show/motivic+homotopy+theory">current</a>

    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.

    diff, v47, current

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded publication data for * [[Fabien Morel]], [[Vladimir Voevodsky]], _$\mathbb{A}^1$-homotopy theory of schemes_, Publications Mathématiques de l'IHÉS, Volume 90 (1999), p. 45-143 ([Numdam:PMIHES_1999__90__45_0](http://www.numdam.org/item/?id=PMIHES_1999__90__45_0) [K-Theory:0305](http://www.math.uiuc.edu/K-theory/0305/) ) <a href="https://ncatlab.org/nlab/revision/diff/motivic+homotopy+theory/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/motivic+homotopy+theory/50">v50</a>, <a href="https://ncatlab.org/nlab/show/motivic+homotopy+theory">current</a>

    added publication data for

    • Fabien Morel, Vladimir Voevodsky, $\mathbb{A}^1$-homotopy theory of schemes, Publications Mathématiques de l’IHÉS, Volume 90 (1999), p. 45-143 (Numdam:PMIHES_1999__90__45_0 K-Theory:0305 )

    diff, v50, current

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2019
    • (edited Mar 15th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to * [[Peter Arndt]], _Abstract motivic homotopy theory_, thesis 2017 ([web](https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full), [pdf](https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf)) also: lecture at _[Geometry in Modal HoTT](https://www.andrew.cmu.edu/user/fwellen/abstracts.html)_, 2019 ([recording I](https://www.youtube.com/watch?v=f0wpcNs8hQo)) The recording of the second talk will become available later today; will update then. <a href="https://ncatlab.org/nlab/revision/diff/motivic+homotopy+theory/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/motivic+homotopy+theory/52">v52</a>, <a href="https://ncatlab.org/nlab/show/motivic+homotopy+theory">current</a>

    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.

    diff, v52, current

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2019
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded the second recording of Peter Arndt's talk last week, [here](https://ncatlab.org/nlab/show/motivic+homotopy+theory#Arndth19) <a href="https://ncatlab.org/nlab/revision/diff/motivic+homotopy+theory/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/motivic+homotopy+theory/53">v53</a>, <a href="https://ncatlab.org/nlab/show/motivic+homotopy+theory">current</a>

    added the second recording of Peter Arndt’s talk last week, here

    diff, v53, current