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Homotopy coherence datum for composition of Becker-Gottlieb transfers

I have a question about certain detail in following answer by Denis Nardin adressing the concept of presheaves with transfer (mostly known in constructions in motivic homotopy theory) from viewpoint ...
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4 votes
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Every stable homotopical functor factors through $\mathbf{SH}$

In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...
Alexey Do's user avatar
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12 votes
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Who proved the motivic 6-functor formalism?

In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that when $...
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Compatibility of different exchange structures $\operatorname{Ex}^*_{\#},\operatorname{Ex}^*_*,\operatorname{Ex}_{\# *}$

Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...
Alexey Do's user avatar
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Geometric conditions on motivic fibrations

What are the geometric conditions on a map of varieties/schemes being a motivic fibration (i.e. a fibration in the motivic model structure on simplicial presheaves on affine schemes)? For example, are ...
Grisha Taroyan's user avatar
2 votes
1 answer
200 views

Removing quasi-projective assumption in the formalism of four operations

In Ayoub's thesis, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Ayoub proved that given a stable homotopical $2$-functor (Definition 1.4.1) $\...
Alexey Do's user avatar
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3 votes
0 answers
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Direct images commute with homotopy colimits

In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...
Alexey Do's user avatar
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9 votes
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The relation between the motivic Galois group and the motivic Steenrod algebra

There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back ...
Jonathan Beardsley's user avatar
4 votes
0 answers
410 views

In which "sense" unramified Milnor-Witt K-groups are unramified

Let $X$ be an integral locally noetherian smooth scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation ring. ...
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3 votes
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Pushout homotopy squares in motivic homotopy theory

I am using Vladimir Voevodsky's notes on motivic homotopy theory and I am having trouble understanding the proofs of Corollary 2.20 and Lemma 2.21. Both of these deal with homotopy pushout squares and ...
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What are some concrete applications of Grothendieck's six operations?

In Gallauer's An introduction to six-functor formalisms I read: Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...
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Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$

My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$. It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...
Tintin's user avatar
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4 votes
1 answer
224 views

Functoriality conjectures on the slice filtration

Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...
Tintin's user avatar
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1 vote
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Localization with or without transfers

Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\...
user197402's user avatar
3 votes
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359 views

Stable $\infty$-category of motives

In nLab motive, it defines the derived category of motives as the full sub-$\infty$-category of the $\infty$-category of functors $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
Aoi Koshigaya's user avatar
6 votes
1 answer
351 views

Nisnevich topology inspired by Adeles

I'm quite a newbe in the field of motives & A1 homotopy theory, so please forgive me if the question is too elementary: In the intro from wikipedia on Nisnevish topology is remarked that it's ...
user267839's user avatar
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2 votes
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128 views

A basic computation with spectra

Let $\mathbb{E}=\big(E_n, \sigma_n\colon T\wedge E_n\to E_{n+1}\big)_{n\in\mathbb{N}}$ be a $T$-spectrum, either in the topological setting (with $T=S^1$) or in the algebraic setting (with $T=\mathbb{...
Tintin's user avatar
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2 votes
0 answers
208 views

$G$-torsor over $\mathbb{A}^1_S$ where characteristic of $S$ does not divide $|G|$

I am reading the paper $\mathbb{A}^1$-homotopy theory of schemes by Morel and Voevodsky from 1999. There is a proposition saying that Let $G$ be a finite étale group scheme over $S$ of order prime to ...
XT Chen's user avatar
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Inverting objects in a symmetric monoidal category

In Voevodsky’s ICM address: https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf In theorem 4.3 it is claimed that given a symmetric monoidal ...
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5 votes
1 answer
483 views

$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$

The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first ...
Xing Gu's user avatar
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1 vote
0 answers
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Non-examples of mixed Tate motives

I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written ...
Arpith's user avatar
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A fibration is a map which has the right lifting property with respect to injections that are weak equivalences

As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram $$\require{AMScd} \begin{CD} A @>>> X;\\ @VV{i}V @VVV \\ B @>&...
XT Chen's user avatar
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2 votes
1 answer
150 views

The multiplicativity of the (complex) geometric realization of motivic cohomology

Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the ...
Xing Gu's user avatar
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4 votes
1 answer
178 views

(Algebraic) cobordism and the rank function

I write the question for algebraic cobordism but I have the analogue question for classic cobordism. The spectrum representing algebraic cobordism $$ \mathbf{MGL}=(*, \mathrm{Th}(1) , \ldots , \mathrm{...
Tintin's user avatar
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6 votes
1 answer
418 views

$\pi_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu_n)\simeq\mathbb{G}_m/\mathbb{G}_m^n$

I have been reading Asok and Doran - $\mathbb A^1$-homotopy groups, excision, and solvable quotients. In the Example 2.17, page 1155, there is a claim that for $m>1$ and $n\geq 1$, $\pi_0^{\mathbb{...
Evans Gambit's user avatar
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0 answers
169 views

$\mathbb{A}^1$ connectedness of open subsets of $\mathbb{A}^n$

Let $k$ be an infinite field. There is a claim in the article Remark 2.16, page 1155, that if $U\subset \mathbb{A}^n_k$ is an open subset such that the complement of $U$ in $\mathbb{A}^n_k$ is of ...
Evans Gambit's user avatar
5 votes
0 answers
465 views

Is this etale motivic or motivic cohomology?

I am trying to reconcile my understanding of motivic cohomology (based on the Lecture Notes by Mazza-Voevodsky-Weibel) with the homotopic point of view. I am currently struggling to answer this ...
J.P. Gimori's user avatar
4 votes
0 answers
316 views

Universal six-functor formalism on an $\infty$-category

In the article The Universal Six-Functor Formalism by Brad Drew and Martin Gallauer it is proved that for an ordinary category $S$ with a wide subcategory $P$ of 'smooth morphisms' containing all ...
Bastiaan Cnossen's user avatar
6 votes
1 answer
536 views

Representable cohomology theories in motivic homotopy theory

I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question: Which cohomology theories on $Sm/k$ are representable, i.e. ...
Nikolai Opdan's user avatar
6 votes
0 answers
235 views

Bigraded endomorphisms of the motivic sphere over a field

In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge ...
Maxime Ramzi's user avatar
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26 votes
1 answer
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What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?

I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction. I ...
display llvll's user avatar
6 votes
0 answers
170 views

Topological realization of fiber sequences of motivic spaces

Consider the realization functor $t^{\mathbb{C}}:H^{mot}\to H^{top}$ from the unstable motivic homotopy category with base field $\mathbb{C}$ to the homotopy category of topological spaces. In this ...
Xing Gu's user avatar
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3 votes
1 answer
292 views

A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$

In the following paper N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces, on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...
Xing Gu's user avatar
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3 votes
0 answers
216 views

Loop spaces of motivic Eilenberg-Mac Lane spaces

Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$. For an abelian group A and the ...
Xing Gu's user avatar
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3 votes
0 answers
257 views

Descent and Chow groups

One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have $$[X,K(\mathbb{Z}(n),2n)]\...
curious math guy's user avatar
6 votes
0 answers
298 views

Is there an exponential map in $\mathbb A^1$ homotopy theory?

Let $k$ be a field, and let $Z \subset X$ be a smooth subscheme of a smooth scheme $X$. When $k = \mathbb C$, there is a distinguished isotopy class of (topological) open embeddings $N_Z \to X$. In ...
Phil Tosteson's user avatar
4 votes
1 answer
234 views

Suspension Theorem in $\mathbb{A}^1$-homotopy

In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have $$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$ So I'm wondering if this has an analogue in ...
curious math guy's user avatar
5 votes
0 answers
262 views

Stalk of motivic homotopy sheaves

In contrast to "classical" homotopy theory, in the motivic homotopy theory, we don't have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the ...
curious math guy's user avatar
3 votes
1 answer
401 views

Whitehead Theorem in $\mathbb{A}^1$-homotopy theory

I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (https://arxiv.org/pdf/1902.08857.pdf). There they have a version of Whitehead's Theorem, namely Prop ...
curious math guy's user avatar
4 votes
0 answers
155 views

Relation between $\mathbb{A}^1$-homotopy theory and derived algebraic geometry [duplicate]

I've often heard that one of the benefits of derived algebraic geometry, next to a cleaner intersection theory, is that "provides natural settings " for the $\mathbb{A}^1$-homotopy theory (...
curious math guy's user avatar
3 votes
0 answers
124 views

Preserves naively $\mathbb{A}^{1}$-homotopic maps

I've been studying $\mathbb{A}^{1}$-homotopy recently and would like some guidance with the question below. Thank you so much. Setup Fix $k$ a field of characteristic zero. Let $Sm_{k}$ denote the ...
Faye3's user avatar
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11 votes
0 answers
616 views

Does Merkurjev's argument help Voevodsky's program?

In the talk Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract) Voevodsky mentioned that he was ...
David Roberts's user avatar
  • 34.2k
5 votes
0 answers
176 views

What is the fibrant replacement of an Eilenberg-MacLane space in unstable motivic homotopy theory?

One can take an Eilenberg-MacLane space $K(A, n)$ for some abelian group $A$, and view it as (locally) constant simplicial sheaf on $Sch/k$, the category of schemes, or smooth schemes, over a field $k$...
user155861's user avatar
4 votes
0 answers
110 views

Filtrations of motivic spectral sequences

I had a general question about motivic spectral sequences. In order to derive them we first begin with a filtration of the algebraic $K$-theory spectra. Something like this $\cdots \rightarrow W^2(X)\...
user127776's user avatar
  • 5,851
4 votes
0 answers
192 views

Direct image and infinite suspension

I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
Tintin's user avatar
  • 2,791
2 votes
1 answer
236 views

Grayson filtration and weight filtration

I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can ...
user127776's user avatar
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1 vote
1 answer
262 views

Dimension of $\ell$-adic Eilenberg-Maclane space

I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$ In some sense, this is a representable functor, i.e. there exists an $\...
curious math guy's user avatar
9 votes
1 answer
278 views

Is the $\mathbb A^1$ homotopy type of a punctured curve independent of the choice of puncture?

Let $C$ be a smooth connected algebraic curve over $\mathbb C$. Assume that $C$ admits no automorphisms. Let $p_1, p_2 \in C$ be two distinct points. Question: Are the $\mathbb A^1$ homotopy ...
user1092847's user avatar
  • 1,327
6 votes
1 answer
390 views

Could a motivic spectrum have a "zeta function"?

I'm currently learning about zeta functions, so I apologize in advance if this is riddled with nonsense. Suppose you have a sequence $E=(E_0,E_1,...)$ of motivic spaces along with structure maps $s_i:\...
Stephen McKean's user avatar
3 votes
1 answer
289 views

Infinite loop space of ring spectra: the cup product

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
Tintin's user avatar
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