ABSTRACT

Title of Thesis: MOTIVIC HOMOTOPY THEORY
AND SYNTHETIC SPECTRA

Charles Richard Dziedzic
Master of Arts, 2024

Thesis Directed by: Professor Niranjan Ramachandran
Department of Mathematics

Motivic homotopy studies the application of techniques from homotopy theory to algebraic

geometry, using A1 as the analogue of the unit interval. Voevodsky found early success in using

his constructions to prove Milnor’s conjecture and the Bloch-Kato conjecture. An interesting

and deep theory arises when constructing the unstable and stable motivic categories, and it has

developed into its own field of study. We begin with a survey of these constructions, detailing the

equivalences between the different model used in the construction of H(S) and SH(S). Here,

we draw connections between all the constructions one might encounter across the literature, and

provide explicit statements on their equivalence.

Stable homotopy theorists have also found utility in motivic homotopy, using the stable mo-

tivic homotopy category SH to advance computations of the stable homotopy groups of spheres,

such as in the work of Isaksen-Wang-Xu. Other work by Bachmann-Kong-Wang-Xu has made

great progress in our understanding of motivic homotopy theory.



Synthetic spectra are a construction of Pstragrowski which represent a ‘return to form’ of

some sort, as they are constructed entirely in the∞−category of spectra. However, they give rise

to a natural bigrading and a strong connection to motivic homotopy; one of the main results is

an equivalence of∞−categories with cellular motivic spaces over C, Spcell
C . We build up enough

of the general theory to establish the connection with motivic homotopy and comment on recent

applications.



MOTIVIC HOMOTOPY THEORY AND SYNTHETIC SPECTRA

by

Charles Richard Dziedzic

Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Master of Arts

2024

Advisory Committee:
Professor Niranjan Ramachandran, Chair/Advisor
Professor Jonathan M. Rosenberg
Professor William M. Goldman



© Copyright by
Charles Richard Dziedzic

2024



Acknowledgments

This thesis started out as a passion project, and every step of the process has been an

enjoyable one. That isn’t to say that it has been a stress-free experience, and it would not have

been possible without the support of a (not small) set of people.

First and foremost, I would like to thank Professor Niranjan Ramachandran for suggesting

the thesis topic, and encouraging me not to shy away from it despite the amount of background

required. It is because of this guidance that I was able to learn so much throughout the process.

I also want to thank Professor Jonathan Rosenberg for his many helpful comments on the drafts

of this thesis, and for providing a neat, self-contained, and captivating introduction to motivic

homotopy through his course in the spring of 2023. Finally, Professor Bill Goldman’s focus on

telling a story while also seeing the ‘bigger picture’ in his lectures has shaped the way I think

about mathematics – I thank him for this.

I also am indebted to Professor Paul Goerss of Northwestern University. His mentorship

and confidence in me are what inspired my strong curiosity for homotopy theory, and are the

main reason I ended up pursuing graduate studies in mathematics.

The support and encouragement of my friends Javier Reyes and Baidehi Chattopadhyay has

been invaluable throughout my time writing this thesis. It is impossible to image how I would

have managed the first two years of this program without them.

ii



Outside the world of mathematics, I owe my family special thanks for a lifetime of love,

patience, and support. I especially could not have produced this thesis without the enduring

support of my mother.

iii



Table of Contents

Acknowledgements ii

Table of Contents iv

Chapter 1: Preliminaries 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Simplicial Sets and Kan Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2: Motivic Homotopy Theory 43
2.1 Grothendieck Topologies and Simplicial Presheaves . . . . . . . . . . . . . . . . . . 43
2.2 Unstable Motivic Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3 Stable Motivic Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4 Another Construction: P1−Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5 Cellularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter 3: An∞−Categorical Construction 76
3.1 The Language of∞−Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 The Motivic Stable∞−Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Chapter 4: Synthetic Spectra 110
4.1 Spherical Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Synthetic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3 Synthetic Spectra as a Tool for Motivic Homotopy . . . . . . . . . . . . . . . . . . . 128
4.4 Applications and Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Bibliography 147

iv



Chapter 1: Preliminaries

1.1 Introduction

The main goal of this thesis is to provide a self contained treatment of the construction of

synthetic spectra and their relationship with motivic homotopy theory, culminating in the follow-

ing result of Pstragowski (which is stated in loc. cit. as Theorem 4.3.35):

Theorem 1.1.1 ([Pst23, Theorem 7.34]). The adjunction of∞−categories of Proposition 4.3.28

induces an adjoint equivalence of∞−categories

(Spcell
C )∧p (Synev

MU)∧p
(Θ∗)p
≃
Θ∗

between the p−completions for every prime p.

To accomplish this goal, we first give some relevant background on simplicial model cate-

gories in Chapter 1. Then, we construct the category of motivic spaces, and both the unstable and

stable motivic homotopy categories in Chapter 2. In Chapter 3 we introduce (∞,1)−categories

through the use of Joyal and Lurie’s theory of quasicategories, and then provide a construction

of the stable motivic ∞−category. This is essential for our comparison with synthetic spectra.

Finally, Chapter 4 gives the definitions and constructions for synthetic spectra, and proves our

desired main result.

1



1.2 Simplicial Sets and Kan Complexes

As a gentle introduction for the reader, we first give a few definitions of simplicial sets and

Kan complexes. The standard reference is [GJ99].

Notation. We denote ∆ the category of ordinal numbers n, seen as a sequence 0 → 1 → 2 →

⋅ ⋅ ⋅→ n. The morphisms of this category are order-preserving maps. Alternatively, a object n can

be thought of as a category with objects the increasing sequence of numbers, and morphisms the

maps between including composition; then, maps between ordinal numbers are precisely functors

between these categories.

Definition 1.2.1. There are two classes of maps (functors) of ∆, which are the

cofaces di ∶ n-1↪ n “skip i”

codegeneracies si ∶ n+1↠ n “hit i twice,”

where the cofaces are injective and codegeneracies surjective.

There are a list of identities that these coface and codegeneracies must satisfy called the

cosimplicial identities, and it is a standard exercise to check that these maps generate all maps in

∆. We present them here.

2



Definition 1.2.2. The cosimplicial identities are as follows:

djdi = didj−1 if i < j

sjdi = disj−1 if i < j

sjdj = 1 = sjdj+1

sjdi = di−1sj if i > j + 1

sjsi = sisj+1 if i ≤ j

Definition 1.2.3. A simplicial set is a presheaf of sets on ∆, S ∶∆op → Set. We use the notation

Sn ∶= S(n)

to denote the n−simplices of S.

Definition 1.2.4. The category of simplicial sets, denoted S or Psh(∆), has objects simplicial

sets, and morphisms maps of simplicial sets, i.e. natural transformations. More concisely, this is

a presheaf category on ∆, also called a functor category.

Example 1.2.5. The standard n-simplex ∆n is the simplicial set represented by n,

∆n ∶ hom∆(−,n)

The Yoneda lemma then tell us that to study a simplicial set S, it suffices to study the functors

(simplicial maps) of our standard n−simplex into the simplicial set S. To unpack this, the Yoneda

3



lemma gives us an isomorphism

homS(∆n, S) = homS(hom∆(−,n), S) ≅ S(n) = Sn

which explains the importance of this example (alternatively a definition or presented as an exer-

cise in [GJ99]).

With the introduction of the standard n−simplex, we are poised to introduce two partic-

ularly important subcomplexes. However, we take a brief narrative sidestep to discuss the face

and degeneracy maps. While we don’t have the topological framework one gets accustomed to

when introduced to algebraic topology (with simplicial complexes) we can borrow some of the

intuition.

A simplicial set X has n−simplices the elements of Xn(= X(n)). The n + 1 face maps

di =X(di),0 ≤ i ≤ n assign to x its n + 1 faces, where ‘faces’ means (n − 1)-simplices.

Remark 1.2.6. Explicitly, on Xn our face map di takes a simplex x ∈Xn to the face di(x) ∈Xn−1

which does not contain the ith vertex.

The case of degeneracies is a bit less intuitive at first thought but can be explained nicely.

In the case of simplicial complexes, we have ‘hidden’ simplexes which have repeat vertices. For

example, the standard 1−simplex ∣∆1∣ contains a degenerate 1−simplex as (0,0); this ‘looks’ like

a point but is technically a 1-simplex.

Remark 1.2.7. Explicitly, on Xn our degeneracy si takes a simplex x ∈ Xn and assigns it to the

n + 1 simplex

Definition 1.2.8. A simplicial set S′ is said to be a subset of the simplicial set S if there is a

monomorphism S′ ↪ S. This means that S′n ⊂ Sn and if the face and degeneracies of S′ agree

4



with S; that is S(di)∣S′ = S′(di), S(si)∣S′ = S′(si).

Definition 1.2.9. We let ιn ∈ hom∆(n,n) denote the identity map. The boundary of ∆n, ∂∆n,

is the smallest subcomplex of ∆n containing the faces dj(ιn) for 0 ≤ j ≤ n. As such, we have

∂∆n
j =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∆n
j if 0 ≤ j ≤ n − 1

degeneracies of ∆n
k for 0 ≤ k ≤ n − 1 if j ≥ n

We sometimes refer to this as the simplicial n−sphere, whose definition is sometimes re-

formulated as follows:

Definition 1.2.10. For our standard simplicial set ∆n, the ith face of ∆n, denoted ∂i∆n, is the

simplicial subset generated by di ∈ ∆n
n−1. In other words, it is the smallest simplicial subset of

∆n containing di.

We then say that the simplicial n-sphere ∂∆n is the union of the faces ∂i∆n, 0 ≤ i ≤ n,

a simplicial subset of ∆n; this is the simplicial subset generated by the maps di(ιn) ∈ ∆n
n−1 for

0 ≤ i ≤ n.

The simplicial sphere will be a key ingredient in our motivic stable homotopy category, but

we don’t want to say too much on that now – it would ruin the ‘punchline.’ However, in support

of the ultimate goal of introducing the realization-singular adjunction, it can be useful to describe

∂∆n in one other way, for which we want to provide some intuition.

Coequalizers, as a colimit of a diagram, can naively be thought of as gluing constructions.

The construction of ∂∆n can be seen as a gluing construction; consider ∂∆2. We want to glue

together the faces ∂i(ιn) for 0 ≤ i ≤ 2, and this gluing is done along the simplicial sets ∆0, seen

as the vertices 0,1,2 – yielding ∂∆2. This construction is generalized in higher dimensions; for

5



∆n, we wish to glue together n−copies of ∆n−1, and this gluing should be done along 1
2n(n + 1)

copies of ∆n−2. Concisely, we can state the definition as follows:

Definition 1.2.11. The simplicial n−sphere ∆n is the following coequalizer

∐
0≤i<j≤n

∆n−2 ∐
0≤i≤n

∆n−1 ∂∆n

Definition 1.2.12. The simplicial horn Λn
k is the union of all faces of ∆n but the kth face. Equiv-

alently, we can view it as the subset of ∆n generated by the maps di(ιn), except i = k.

In [GJ99] they present the following picture, which is the best visual reference for the kth

horn.

Example 1.2.13. In ∆2, we have that d0(ιn) = (0 → 1) ↦ (1 → 2); so the horn Λ2
0 should miss

this face,

0 0

Λ2
0 = ⊂ =∆2

1 2 1 2

Definition 1.2.14. We call the horns Λn
k for 0 < k < n the inner horns, and Λn

0 ,Λ
n
n are called the

outer horns.

We can see that (Λn
k)j = ∆n

j when j < n − 1, and (Λn
k)n−1 = ∆n

n−1 ∖ dk(ιn). Similar to our

coequalizer presentation of ∂∆n, we have one for Λn
k :

6



Lemma 1.2.15. The following is a coequalizer diagram

∆n−2 ∆n−1

∐
0≤i<j≤n

∆n−1 ∐
i≠k

∆n−1 Λn
k

∆n−2 ∆n−1

dj−1

di

di

di

Proof. A proof can be found in [GJ99]. We note that this is analogous to expressing ∂∆n as

a coequalizer, with the kth face excluded – this is immediately apparent by comparing the two

diagrams.

In fact, this could be taken as the definition of Λn
k , and it might be best to think of it that

way. The horn and the concept of a Kan complex will play a bigger role later in our discussion

of infinity categories. However, it is most appropriate to introduce the definition here, so we take

a slight detour.

Definition 1.2.16. For an arbitrary S ∈ S , A horn in S is a map of simplicial sets Λn
k → S.

There are a few equivalent ways to formulate the definition of a Kan complex. Some will

be presented later in the paper.

Definition 1.2.17. A Kan complex is a simplicial set S such that for every horn in S (a natural

transformation Λn
k → S), there is an extension

Λn
k S

∆n

The Yoneda lemma tell us that giving a map ∆n S is equivalent to giving a simplex in

7



Sn, whose faces agree with that of the horn by commutativity.

We now find it appropriate to introduce the geometric realization functor, ∣ ⋅ ∣ ∶ S → Top.

There are a few ways to introduce this construction – one which builds off our intuition with

geometric simplices, and a slicker categorical construction.

Definition 1.2.18. The simplex category ∆ ↓X of a simplicial set X consists of objects the maps

σ ∶ ∆n → X (the n-simplices of S by the Yoneda lemma) and morphisms/arrows commutative

triangles with simplicial maps (natural transformations)

∆n

X

∆m

φ

σ

τ

Where, by contravariance, any map φ ∶ hom∆(−,n) → hom∆(−,m) is induced by a unique map

m→ n (recall any map in ∆ is a composition of coface and codegeneracies, giving rise to unique

face and degeneracies between standard n−simplices). This is a specific version of a more general

concept known as a comma category; see [Rie17, p. 22]

Lemma 1.2.19. We have an isomorphism

X ≅ limÐ→
∆n →X

in ∆ ↓X

∆n

We want to take a second to parse the notation here before commenting on the proof. The

colimit is of the functor ∆ ↓ X → S which takes each σ ∶ ∆n → X to ∆n, and a morphism (φ in

the diagram) to ‘itself’. This colimit is taken over all ∆n → X for all n. This intuitively makes

8



sense; thinking of a colimit as a gluing construction, were gluing together all of our ∆n → X in

the category S , and the Yoneda lemma tells us that the collection homS(∆n,X) is precisely Xn.

Put together all the Xn and we should hope to recover X .

Proof. The proof of the lemma is a direct consequence of the result [ML78, Theorem 7.1]. There

are also other ways to prove this, using some generalizations of the Yoneda lemma – every

presheaf ends up being a colimit of representables!

Theorem 1.2.20. Any functor Cop → Set with C small can be canonically represented as a colimit

of a diagram of contravariant representable functors homC(−, c)

To construct the realization functor, we will exploit the fact that any simplicial set is the

colimit (direct limit) of standard n−simplices ∆n, and giving an explicit definition for the real-

ization of a standard n−simplex – functoriality will be a simple consequence when we extend to

all simplicial sets. The choice for such a realization functor should be obvious.

Definition 1.2.21. For a standard n−simplex, the realization is defined as

∣∆n∣ = {(t0, . . . , tn) ∈ Rn+1 ∶∑
i

ti = 1, ti ≥ 0}

Thus, we have a covariant functor ∆→ Top which sends n↦ ∣∆n∣.

While this is a bit of an abuse (or confusion) of notation, what it suggests is correct: that

we will get a functor S → Top (and factoring this functor ∆→ Top as a functor S → Top works,

but that’s not too important).

9



Definition 1.2.22. The realization functor ∣ ⋅ ∣ ∶ S → Top sends X to

∣X ∣ ≅ limÐ→
∆n →X

in ∆ ↓X

∣∆n∣

Where the colimit is of the functor ∆ ↓ X → Top (or really CGWH, compactly generated weak

Hausdorff spaces) sending (∆n →X)↦ ∣∆n∣.

Functoriality is immediate by postcomposition with a morphism ϕ ∶X → Y

∆n

X Y

∆m

φ

σ

ϕ

τ

and then passing to the colimit. There is another equivalent way to define the geometric realiza-

tion, which might be more commonly seen.

Definition 1.2.23. The geometric realization of a simplicial set X can also be defined as follows:

∣X ∣ = (∐
n

Xn × ∣∆n∣) / ∼

With Xn given the discrete topology. Our relation is given for f ∶ n → m, for (x, d) ∈ Xm × ∣∆n∣

we identify Xn × ∣∆n∣ ∋ (f∗x, d) ∼ (x, f∗d) ∈Xm × ∣∆m∣, where f∗ =X(f) and f∗ is the induced

map on ∣∆n∣→ ∣∆m∣, i.e. we take f∗(d0, . . . , dn) = (∑f(i)=j di)0≤j≤m.

10



There is a more concise way to state this as the coequalizer

∣X ∣ = colim
⎛
⎝ ∐

f ∶n→m
Xm × ∣∆n∣ ∐

n
Xn × ∣∆n∣

f∗

f∗ ⎞
⎠

Shedding some light on these definitions, we can think of our new definition as adjoin-

ing the n−simplices Xn in a way to preserve its structure, when gluing together our topological

n−simplices ∣∆n∣; taking a coequalizer of induced maps Xm →Xn and (∣∆n∣→ ∣∆m∣) tell us how

we should preserve the structure of the faces and degeneracies of X topologically. The equiva-

lence of our first definition to the second definition makes sense, then, as it is just a rephrasing of a

colimit in a simplex category as a colimit involving topological realizations of the product.

Geometric realization becomes a more powerful tool when we realize it is left adjoint to

the following functor.

Definition 1.2.24. There is a functor Sing ∶ Top→ S defined by

SingX(n) = homTop(∣∆n∣,X)

It is not complicated to check that SingX is a simplicial set, and that Sing is a functor,

which means it isn’t done in any of the standard references. We will quickly do so.

SingX is indeed a simplicial set; for ϕ ∶ n→m, we have

n homTop(∣∆n∣,X)

m homTop(∣∆n∣, Y )

ϕ

SingX

SingX

SingX(ϕ)

where our induced map SingX(ϕ) gives the map ∣∆n∣ ϕ
∗
Ð→ ∣∆m∣→X . This reverses composition

11



and respects identity, giving us SingX ∈ S

We have a commutative diagram

X SingX

Y SingY

Sing

f f∗
Sing

where f∗ ∶ SingX ⇒ SingY is the natural transformation which postcomposes n↦ homTop(∣∆n∣,X)↦

homTop(∣∆n∣, Y ), ∣∆n∣ → X
fÐ→ Y . Of course, for a composition X

fÐ→ Y
gÐ→ Z in Top, we have

that (gf)∗ = g∗f∗, as

((gf)∗)n(∣∆n∣→X) = (∆n →X
fÐ→ Y

gÐ→ Z)

= (g∗)
n(∣∆n∣→X

fÐ→Y )

= (g∗)n(f∗)n(∣∆n∣→X)

and thus a functor. We can now prove our desired result,

Theorem 1.2.25. The realization functor is left adjoint to the singular functor, ∣ ⋅ ∣ ⊣ Sing. That

is, there is an isomorphism

homTop(∣X ∣, Y ) ≅ homS(X,SingY )

which is natural in the simplicial set X and topological space Y .

Proof. We give our own explanation of the proof, following [GJ99]. Using our Lemma 1.2.19

from before, that X limÐ→
∆n→X

∆n, we have that homS(X,Sing(Y )) ≅ homS( limÐ→
∆n→X

∆n,Sing(Y )).

We now apply a result from category theory on the representable nature of (co)limits – see for

12



example, [Rie17] Theorem 3.4.7. This is the isomorphism

homS( limÐ→
∆n→X

∆n,Sing(Y )) ≅ lim←Ð
∆n→X

homS(∆n,Sing(Y ))) ≅ lim←Ð
∆n→X

homTop(∣∆n∣, Y )

where our second isomorphism is from construction of Sing. We then apply our result again

lim←Ð
∆n→X

homTop(∣∆n∣, Y ) ≅ homTop( limÐ→
∆n→X

∣∆n∣, Y ) ≅ homTop(∣X ∣, Y )

and this gives us our natural isomorphism homTop(∣X ∣, Y ) ≅ homS(X,SingY ).

We now wish to show one more important result about the geometric realization functor –

that the geometric realization of any simplicial set is, in fact, a CW complex. We first provide a

definition, then prove the result.

Definition 1.2.26. For a simplicial set X , the nth skeleton sknX is the subcomplex of X gener-

ated by simplices of degree ≤ n. sk can also be viewed as a functor S → S .

Theorem 1.2.27. The geometric realization of any simplicial set X , ∣X ∣, is a CW-complex.

Proof. We fill in all the gaps in the proof of [GJ99]. Start by noting that X is the union of all its

n−skeletons, X = ∪n≥0 skn X . We have a pushout diagram

∐
x∈Xn

∂∆n skn−1X

∐
x∈Xn

∆n sknX

where each ∐x∈Xn
is actually taken over non-degenerate simplices, in simplicial sets. Let s a

k−simplex in sknX . We can prove this as follows. First, note that, since our right vertical map

13



is inclusion, to prove that this is a pushout square is tantamount to showing that if s factors as

∆k σÐ→∆n τÐ→X where σ factors through ∂∆n, then s is in skn−1X . This is equivalent to showing

that if s is not in skn−1X , then σ does not factor through ∂∆n and τ nondegenerate.

The unique factorization holds in general. Let ∆k σÐ→ ∆m τÐ→ X be a factorization with

minimal m. Since m minimal, σ must be surjective (else choose its image for a smaller m), and

τ nondegenerate. Since s ∈ sknX tells us that m ≤ n and s ∉ skn−1X gives that m ≥ n; thus

m = n. So we have our factorization ∆k σÐ→∆n τÐ→X . Since σ surjective, it cannot factor through

any ∂∆k, and we have a pushout diagram.

We can note that the geometric realization of ∆n is ∣∆n∣ our standard topological n−simplex.

This is because our identity map ∣∆n∣ ≅ ∣∆n∣, in Top, determines an n−simplex in Sing(∣∆n∣),

which by Yoneda is equivalent to some ∆n → Sing(∣∆n∣). Thus ∣∆n∣ is its geometric realization.

Furthermore, we note that there is a coequalizer diagram

∐
0≤i<j≤n

∆n−2
n

∐
i=0

∆n−1 ∂∆n

which comes from the relation djdi = didj−1 as stated earlier in this section, inducing the diagram

∐
0≤i<j≤n

∣∆n−2∣
n

∐
i=0
∣∆n−1∣ ∣ ∂∆n∣

which formalizes the notion that the induced map ∣∂∆n∣ → ∣∆n∣ maps ∣∂∆n∣ onto the boundary

14



of ∣∆n∣, the (n − 1)-sphere. So, our resulting pushout diagram

∐
x∈Xn

∣∂∆n∣ ∣ skn−1X ∣

∐
x∈Xn

∣∆n∣ ∣ sknX ∣

tells us that we can obtain the ∣ sknX ∣ by attaching n−cells to ∣ skn−1X ∣ according to a pushout

diagram; then our space ∣X ∣ is the filtered colimit of these spaces, which is precisely the definition

of a CW complex.

Corollary 1.2.28. The geometric realization functor ∣ ⋅ ∣ ∶ S → CGHaus (compactly generated

Hausdorff) preserves finite limits.

Proof. This can be found in [GZ67].

This accounts for most of the important facts about simplicial sets and their realizations

that this paper will need. There is more to be done with them, though – we haven’t yet identified

why simplicial sets are the ‘right’ setting to do stable homotopy theory yet. This viewpoint, in

some form, will show up at the end of the following section on model categories – where we

mention that simplicial sets have a model structure on them. The following section contains a

primer on model categories.

1.3 Model Categories

Definition 1.3.1. A model category is a categoryM endowed with three special classes of maps

called cofibrations, fibrations, and weak equivalences, which are characterized by five axioms:

(M1) M is (co)complete – it has all small limits and colimits.

15



(M2) If we have a pair of morphisms M M ′ M ′′f g
inM, and any two of f, g, g○f

are weak equivalences, then so is the other.

(M3) Cofibrations, fibrations and weak equivalences are closed under retracts. That is, if g ∈

homM(W,Z) a (cofibration, fibration, weak equivalence) and f is a retract of g, i.e. the

diagram
X W X

Y Z Y

f

i

g

r

f

i′ r′

commutes, with the horizontal compositions giving identity, then f is also a (cofibration,

fibration, weak equivalence).

(M4) For a commutative diagram with i a cofibration and p a fibration,

X W

Y Z

f

i p

g

if either i or p is also a weak equivalence (called a trivial (co)fibration or an acyclic

(co)fibration) then i has the left lifting property with respect to p (and p has the right

lifting property with respect to i). This means we have a lift

X W

Y Z

f

i p

g

l

(M5) Any f ∈ homM(X,Y ) can be factored as a composition of a trivial cofibration and a

16



fibration, or a cofibration and trivial fibration,

X Z

W X

i′

g
i ∼ p′∼

p

Further, these factorizations are functorial.

Two comments are in order. We have made the choice to require all small (co)limits in

the first axiom, while it is often only required that we have finite (co)limits. We make this

choice because our motivic model category will be (co)complete. Second, we do require that the

factorization is functorial in the final axiom. This is often brushed over in the literature, so we

give an explicit definition:

Definition 1.3.2. A functorial factorization is an (ordered) pair of functors (σ, τ) ∶ homC →

homC so that, for all f ∈ homC, f = τ(f) ○ σ(f).

We also wish to make one more definition. Since we have the (strong) condition that a

model category is (co)complete, we must have an initial and final object in the category. We give

a certain class of objects relating to the initial and final objects a name.

Definition 1.3.3. We say an object in our model categoryM is fibrant if its unique map into the

terminal object F → ∗ is a fibration. Dually, we say that an object is cofibrant if the morphism

from the initial object 0→ C is a cofibration.

It should also be noted that we have a concept of (co)fibrant replacement.

Definition 1.3.4. In a model categoryM, every object X has a cofibrant replacement consisting

of a weak equivalence Z X
∼ , where Z cofibrant. This always exists by M5. The dual

notion of a fibrant replacement also exists.

17



Notation. For an object X ∈ M , we will often denote the cofibrant replacement QX , and the

fibrant replacement RX . These are functorial, and it will be clear from context which model

structure we are taking the replacement with respect to.

To get some feel for model categories, it can be useful to note some of the immediate

consequences of the axioms. We provide some facts and proofs of these facts, as proof is omitted

in the standard reference on motivic homotopy theory [DLORV].

Proposition 1.3.5. If h can be factored h = g ○ f , where g has the right lifting property with

respect to h, then h is a retract of f .

Dually, if h = g ○ f where f has the left lifting property with respect to h, then h is a retract

of g.

Proof. We have a diagram

X Z

Y Y

h

f

g

=

l

which gives rise to the retraction

X X X

Y Z Y

=

h

=

f h

l

idY

g

The dual statement has diagram

X Z X

Y Y Y

f

h

idX

l

g h

= =

18



The above proof included two statements which are dual in the context of model categories;

in general, we shall omit proofs of such statements in the future. In fact, all statements can be

(formally) dualized by the construction of model categories.

Corollary 1.3.6. A map i inM is a cofibration (resp. acyclic cofibration) if and only if it has the

left lifting property with respect to acyclic fibrations (resp. fibrations). Similarly, a map p inM

is a fibration (resp. acyclic fibration) if and only if it has the right lifting property with respect to

acyclic cofibrations (resp. cofibrations).

Proof. We will provide a proof of the first statement in the case of i a cofibration; the proof

generalizes to all the other cases. The forward implication is by (M4). For the backwards impli-

cation, let i a map with the left lifting property with respect to acyclic fibrations. By (M5) our

map factors
X Y

Z
i′
∼

i

p′

where i′ a cofibration and p′ an acyclic fibration. Then i must have a left lifting with respect to

p′, so by our Proposition 1.3.5, i is a retract of i′, and hence by (M3) i is a cofibration.

An immediate consequence of this corollary is that all (co)fibrations are completely deter-

mined by the weak equivalences and (co)fibrations, where you pick (co) for only one of the two.

It is a nice consequence of the axioms that the weak equivalences are also completely determined

by both the (co)fibrations; we can state this succinctly as:

Proposition 1.3.7. Specifying any two of the class of weak equivalences, cofibrations, or fibra-

tions completely determines the third.

19



Proof. By remarks above, it remains to show that weak equivalences are completely determined

by fibrations and cofibrations. We first claim that a map is a weak equivalence if and only if it fac-

tors as an acyclic cofibration followed by an acyclic fibration. The if direction is by composition

of weak equivalences (formally, by M2).

For the only if direction, let X
ωÐ→ Y be a weak equivalence which factors as

X Z Y

ω

i

∼
p

since i and ω are weak equivalences, p must also be (M2).

Before we produce a few more facts about model categories, it is important to note the

following fact:

Proposition 1.3.8. For any model categoryM, the opposite categoryMop is also a model cat-

egory, where the weak equivalences are preserved (under reversing arrows), and (co)fibrations

are interchanged (and taken to be the opposite). More succinctly, model categories are defined

dually.

In lieu of a proof, it is better to provide a diagram which makes it clear that our proposition

is true. Here, the i, p denote that the original maps are (co)fibrations and the shape of the arrow

denotes the flipping in the opposite category

A B

C D

i p

A consequence of this proposition is that any statement proven for a general model category gives

20



yield to the dual statement, where we swap cofibrations and fibrations, and colimits and limits.

As such, we can make our statements in terms of either, and get the dual statement for free.

We have the following lemma establishing some basic results.

Lemma 1.3.9. (acyclic) cofibrations are closed under composition, coproducts, and pushouts.

Proof. These all come from previous facts and definitions. For a composition, consider the dia-

gram
A D

B

C E

a

i

p∼

j
cj

li

c

where the dashed arrow is comes from i a cofibration and p an acyclic fibration, and this induces

a diagram

B D

C E

li

j p∼l

which gives us that ji a cofibration, by the final diagram

A D

C E

ji

a

p∼l

c

so our composition is, indeed, a cofibration.

Closure under coproducts boils down to the universal property and using the left lifting

of cofibrations. We will show for the case of one coproduct. Let i1, i2 ∶ Aj → Bj cofibrations,

21



p ∶ C1 → C2 an acyclic fibration. We have the two diagrams

A1 C1 A2 C1

B1 C2 B2 C2

i1 p∼ i2 p∼
l1 l2

then, by using the universal property of both coproducts, we end up with the following diagram,

where dashed arrows are induced by universal properties and cofibrations

A1 A1

A1∐A2 C1

B1 B2

B1∐B2 C2

p∼

so we get a cofibration between coproducts.

For pushouts, we have the following diagram:

A C

E

B D

F

a

i j

c

p∼b

lb

d

ld

22



The story behind the diagram is as follows: our left lifting property first gives the lift lb,

and then we have maps to E from both B and C giving our induced map ld from the universal

property of the pushout. We want to show that the whole diagram commutes (or really, just

the bottom right ‘square’) so that j also has the left lifting property with respect to our acyclic

fibration p, and hence is a cofibration. Our first lift lb tells us that lbi = ca and plb = db. Our second

induced map ld (we don’t know its actually a left lift a priori) tells us, by universal property of

the colimit/pushout, that c = ldj and lb = ldb. The first condition tells us we have commutativity at

E, in our second square. To prove commutativity at F , we have to use the uniqueness of the map

out of the pushout (i.e. the universal property). Combining our facts above, and using the solid

arrows in our diagram, we can see that pldb = plb = db and pldj = pc = dj; we have the diagram

C

B D E

F

j c

b

lb

db

d

ld

p

which commutes, so by the uniqueness of the mapping of the pushout we get the final relation d =

pld. This is enough to show commutativity of our entire diagram above so j has the right lifting

property and hence is a cofibration, giving closure under pushouts, completing the proof.

The above proof was worked out in detail as it gives us a good feel for how to work with

the formalism of a model category. In general, such granularity is not needed. To wrap up our

current discussion of model categories, we present Ken Brown’s lemma – which was first proved

in [Bro73].

Lemma 1.3.10 (Ken Browns Lemma). Let M a model category, and f ∶ A → B a map of

23



cofibrant objects. There is a functorial diagram

A B

Cf B

f

if pf
∼ =

jf
∼

with if a cofibration, pf an acyclic fibration, and jf a trivial cofibration. The dual statement also

holds in the expected way

Proof. Recognizing the coproduct A∐B as the pushout of the diagram A ∗ B ,

∗ B

A A∐B

Cf

B

idB

f

if

pf
∼

Where we factor the map A∐B → B given by universal property of coproduct on the maps

A
fÐ→ B,B

idBÐÐ→ B as a cofibration and acyclic fibration. By closure under pushouts, since A,B

cofibrant so are our maps into the coproduct; thus by closure under composition we get cofibra-

tions A→ Cf ,B → Cf . The whole diagram commutes, which proves the lemma. Functoriality is

due to our assumption of functoriality in our factorization axiom.

The dual statement is immediate from our discussion ofMop.

Corollary 1.3.11. A functor between model categories F ∶ M → N which preserves trivial

cofibrations sends weak equivalences of cofibrant objects to weak equivalences.

Proof. In the diagram of Lemma 1.3.10, assuming f is a weak equivalence gives if a weak

24



equivalence; by our assumption, and the two out of three axiom, we then get all resulting di-

agonal arrows have to be weak equivalences, and hence the map F (A) F (f)ÐÐ→ F (B) is a weak

equivalence.

We also want a suitable way of classifying model structures (model categories) as equiva-

lent. This leads us to the following definition

Definition 1.3.12. A (left) Quillen functor between model categories is a functor F ∶M → N

which is a left adjoint which preserves both cofibrations and acyclic cofibrations.

Definition 1.3.13. A Quillen Adjunction is a trio (F,G,φ) where F ∶M → N a Quillen left

functor, G ∶ N →M a functor, and φ ∶ N (Fa, b)→M(a,Gb) expresses F ⊣ G

We note now that the choice of F a left Quillen functor is arbitrary; we could instead

characterize in terms of G a right Quillen functor. We take a novel approach to this proof, by

inserting a (purely categorical) lemma which is not present in any reference on model categories.

It is stated as a lemma [Rie17] without proof.

Lemma 1.3.14. Consider F ∶ C D ∶ G
F

G
with isomorphism φ ∶ D(Fc, d) ≅ C(c,Gd).

The naturality of the isomorphism in c and d is equivalent to the assertion that, for the two

diagrams below, the left hand side commutes if and only if the right hand side commutes.

Fc d c Gd

Fc′ d′ c′ Gd′

Fh

f ♯

k

f ♭

h Gk

g♯ g♭

Where the naturality conditions for an adjunction are stated as follows. For k ∶ d→ d′, for a map

25



f ♯ ∶ Fc→ d corresponding to f ♭ ∶ c→ Gd under φ, we have a commutative diagram

c Gd

Gd′

f ♭

(kf ♯)♭
Gk

and equivalently, naturality in the second argument means for k ∶ c→ c′, we have a commutative

diagram
c

c′ Gd

h
(f ♯Fk)♭

f ♭

Proof. This lemma tells us that naturality in both arguments is equivalent to saying the left dia-

gram commutes if and only if the right diagram commutes.

First, assume we have naturality. We wish to show the left diagram commutes if and only

if the right diagram commutes. We will show left-hand commutativity implies right-hand com-

mutativity – the other direction follows the same argument, mirrored. The proof boils down to

the following diagram

c Gd

c′ Gd′

(kf ♯)♭

(g♯Fh)♭

f ♭

h Gk

g♭

where the dashed arrows come from our assumption of naturality, and the two diagrams of dif-

ferent colors commute. So, we wish to show equality of the dashed arrows. By assumption

of commutativity of the left diagram, we have that kf ♯ = g♯Fh, and hence we must have that

the transposes are the same. Thus, our right diagram is commutative. The argument assuming

naturality and showing the right diagram commutes implies the left diagram commutes is similar.

We now show the other direction; that if we have the diagrams commuting if and only if

26



the other commutes, we must have naturality in the two arguments of our natural isomorphism.

By assumption, we have that g♯h = Gkf ♭ ⇐⇒ g♯Fh = kf ♯. We will show naturality in our

first argument (c), and the other argument is similar. We have g♯ ∶ Fc′ → d′ corresponding to

g♭ ∶ c′ → Gd′ by our isomorphism, and h ∶ c→ c′. We (cleverly) set up a diagram on the left

Fc d′ c Gd′

Fc′ d′ c′ Gd′

Fh

g♯Fh

idd′

(g♯Fh)♭

h idGd′

g♯ g♭

where our implication is a direct application of our assumption that the left hand diagram of the

lemma commuting implies the right hand diagram commutes. As such, associated to g♯ we have

that our diagram
c

c′ Gd

h
(g♯Fk)♭

g♭

commutes, giving naturality in c. Naturality in d follows from a similar argument.

Armed with this lemma, we now state our desired result.

Corollary 1.3.15. (F,G,φ) is a Quillen Adjunction if and only if G is a right Quillen functor.

Proof. As a result of our lemma, since we have an adjunction, we have the following pair of

commutative diagrams

Fc d c Gd

Fc′ d′ c′ Gd′

Fi ∼ p i ∼ Gp

so Fi has the left lifting property with respect to p if and only if i has the left lifting property

with respect to Gp. Thus if (F,G,φ) is a Quillen adjunction then G is a right Quillen functor,

27



and if G is a right Quillen functor our diagram shows us we have a Quillen adjunction as defined

above.

We have a notion of left and right derived functors for Quillen functors. The following is

the specific result needed, which comes as a consequence of more general constructions discussed

in [GJ99, II.8]

Theorem 1.3.16 (Quillen’s total derived functor theorem). For a pair of Quillen functors F ∶M N ∶ G ,

we have a left derived functor LF ∶ Ho(M) → Ho(N) and a right derived functor RG ∶

Ho(N) → Ho(M), which take weak equivalences to isomorphisms, and which are an adjoint

pair.

Proof. The proof of this can be found in [Qui67] or, better, [GJ99, Theorem 8.7, p. 124]. The

proof uses our Lemma 1.3.10.

Definition 1.3.17. A Quillen functor F ⊣ G is a Quillen equivalence if for A ∈ M cofibrant

and X ∈ N fibrant, our M(A,GX) ≅ N (FA,X) descends to a bijection on the sets of weak

equivalences wM(A,GX) ≅ wN (FA,X).

Proposition 1.3.18. A Quillen adjunction (F,G,φ) is a Quillen equivalence if and only if the

derived adjunction (LF,RG,Rφ) is an adjoint equivalence of categories.

Proof. This is from the definitions, and can again be found in [Qui67] or [GJ99]. This is some-

times taken to be the definition of a Quillen equivalence.

With our background on model categories set, we now wish to demonstrate a model cat-

egory structure on S . However, to do so in full detail would require too many pages and does

not serve in the overarching goal of this thesis – we will have to take the result on a bit of blind

28



faith. A full account of the model structure on S is contained through multiple chapters of either

[GJ99] or [Hov07]; the original argument, which both references are modeled on, is given in

[Qui67]

Theorem 1.3.19. The category of simplicial sets S is given the structure of a model category. In

this structure, we have

1. cofibrations i ∶X ↪ Y monomorphisms.

2. fibrations p ∶ X → Y maps which have the right lifting property with respect to inclusions

of horns Λn
k →∆n, 0 ≤ k ≤ n.

3. weak equivalences maps whose geometric realizations are weak equivalences of topologi-

cal spaces

We prove (part of) (M5) in the following, giving an example of a small object argument.

We will not give a full proof of the Quillen model structure.

Theorem 1.3.20. For any map of simplicial sets f ∈ homS(X,Y ) we have a functorial factor-

ization

X Z Yi p

∼

where i is inclusion (a cofibration, by our above theorem) followed by a trivial fibration.

Proof. First, consider the set of all diagrams of the following form

∂∆n X

∆n Y

i f

Call this set D1; that way, d ∈D1 is a diagram of the above form, for some nd. If we consider the

29



diagram of the form

∆nd ∂∆nd X
i

we can take the pushout to get a new object; taking the pushouts over all such diagram, we get

the pushout X1 in the following diagram

∐d∈D1
∂∆nd X

∐d∈D1
∆nd X1

Y

We claim the right map is an inclusion; this is an immediate consequence of Theorem 1.3.19

combined with Lemma 1.3.9. This is a pushout, so we have a factorization.

Now we have the factorization of X ↪X1 → Y , and we can repeat the same process letting

D2 be the set of diagrams
∂∆n X1

∆n Y

i f

so by the same argument repeated, we get a chain X = X0 ⊆ X1 ⊆ X2 ⊆ . . .. Then we claim that

our functorial factorization is of the form

X limÐ→Xi Y

We now claim that limÐ→Xi Y is an acyclic fibration. We can explicitly construct a lifting.

30



The map ∂∆n → limÐ→Xi factors through Xi for some i, we have a diagram

∂∆n Xi

∆n Xi+1 limÐ→Xi

Y

which, collapsing compositions, gives us a lifting

∂∆n limÐ→Xi

∆n Y

so by Theorem 1.3.19 our claim is true, giving a functorial factorization.

We feel now is the best time to establish the following corollary:

Corollary 1.3.21. In S with the model structure of Theorem 1.3.20, every object is (vacuously)

cofibrant. We also note that the fibrant objects are exactly Kan complexes.

Proof. The first statement is vacuously true. For fibrant objects – we have the diagram:

Λn
k X

∆n ∗
i

so X is a Kan complex.

We won’t comment any further on the Quillen model structure. There are many model

structures which we can endow on S , and another one that will be mentioned is that of the Joyal

model structure, which is a result of modern work involving∞−categories.

31



With the concept of a model category now thoroughly understood, we now define simplicial

model categories. These will be our starting point for the A1−homotopy category, and this is most

natural place to cover them.

Definition 1.3.22. For C a category and W a class of morphisms in C, the localization of C by

W is a category C[W −1] together with a functor L ∶ C → C[W −1] which satisfies the following

properties;

1. L(w) is an isomorphism for every w ∈W

2. If F ∶ C →D such that F (w) is an isomorphism for all w ∈W , then we have a factorization

C D M D

C[W −1]

F

L

F

⇓≅
G○LG

3. The induced functor between functor categories Fun(C[W −1],D) Fun(C,D)−○L is

fully faithful.

Such a category C[W −1] need not exist.

As Quillen said in his original work, model category is short for “a category of models for

a homotopy theory” – as such, we should hope to use model categories as a basic building block

for a homotopy category, where we have an inversion of weak equivalences. Quillen made this

connection formal with the following theorem.

Theorem 1.3.23. [Qui67, Theorem 1]. If C a model category with weak equivalences W , then

the homotopy category Ho(C) ∶= C[W −1] exists.

Proof. The original proof can be found in the cited work. However, there is a more constructive

32



(and modern) treatment in [GJ99, p.75].

We now give the definition for a simplicial model category, following our standard refer-

ence

Definition 1.3.24 ([GJ99, Definition 2.1]). A category C is a simplicial category if there is a

mapping space functor

HomC(−,−) ∶ Cop ×C → S

which satisfies the following three properties for A,B objects in C:

1. HomC(A,B)0 = homC(A,B),

2. the functor HomC(A,−) ∶ C → S has a left adjoint

A⊗ − ∶ S → C

for which we have associativity A ⊗ (K × L) ≅ (A ⊗K) ⊗ L which is natural in A ∈ C,

K,L ∈ S ,

3. The functor HomC(−,B) ∶ Cop → S has a left adjoint

homC(−,B) ∶ S → Cop.

Notation. We often call the adjoint 2 the action of S on C, and the adjoint of 3 the exponential.

They are also sometimes called the simplicial tensor and simplicial cotensor.

The definition of a mapping space functor will sound similar to Definition 3.1.13 encoun-

33



tered later; that definition gives such a functor in the case of simplicial sets. Directly from Def-

inition 1.3.24 we can derive a few results, here we present a couple; the full proofs are left to

[GJ99].

Lemma 1.3.25 ([GJ99, Lemma 2.2]).

1. For K ∈ S , there are a pair of adjoint functors − ⊗K ∶ C → C and

homC(K,−) ∶ C → C,

2. For all K,L ∈ S and B ∈ C, there are natural isomorphisms homC(K×L,B) ≅ homC(K,homC(L,B)),

3. for all n ≥ 0, we have HomC(A,B)n ≅ homC(A⊗∆n,B).

Lemma 1.3.26 ([GJ99, Lemma 2.3]). The adjointness of Definition 1.3.24 can be stated as

HomS(K,HomC(A,B)) ≅ HomC(A⊗K,B)

and

HomS(K,HomC(A,B) ≅ HomC(A,hom(K,B))

But any simplicial model category will satisfy the stronger adjointness condition:

HomS(K,HomC(A,B)) ≅ HomC(A⊗K,B)

HomS(K,HomC(A,B) ≅ HomC(A,hom(K,B))

We now come to our definition of a simplicial model category:

Definition 1.3.27 ([GJ99, 3.1, Axiom SM7]). Let C be a closed model category and a simplicial

34



category. Suppose that j ∶ A→ B is a cofibration and q ∶X → Y a fibration. Then

HomC(B,X) HomC(A,X) ×HomC(A,Y ) HomC(B,Y )(j∗,q∗)

is a fibration of simplicial sets, which is acyclic either if j or q is acyclic.

A category in which the above statement holds is said to satisfy Axiom SM7, and is called

a simplicial model category.

The above material can be a bit hard to keep track of at first sight, and most references rush

past – so we will pause to provide the following diagram:

HomC(B,X)

HomC(A,X) ×HomC(A,Y ) HomC(B,Y ) HomC(A,X)

HomC(B,Y ) HomC(A,Y )

∃!(j∗,q∗)

q∗

j∗

There are equivalent ways to formulate axiom SM7; one such way gives us a useful way to

recognize when the axiom holds.

Proposition 1.3.28. Let C be a model category and a simplicial category. Then axiom SM7 holds

if and only if for all cofibrations i ∶K → L in S and cofibrations j ∶ A→ B in C, the map

(j ⊗ 1)∐(1⊗ i) ∶ (A⊗L) ∐
(A⊗K)

(B ⊗K)→ B ⊗L

is a cofibration which is trivial if either j or i is.

Proof. This comes from the adjointness of our tensor and mapping space functor. Explicitly,

35



applying our adjoint isomorphisms, we have an equivalence of dotted arrow diagrams

K HomC(B,X) A⊗L∐A⊗K B ⊗K X

⇐⇒

L HomC(A,X) ×HomC(A,Y ) HomC(B,Y ) B ⊗L Y

directly from the definition of a simplicial category which yields, as a corollary, our proposition

(as the diagram tells us that the left diagrams right map is a fibration if and only if the right

diagrams left map is a cofibration, which is really what we wish to prove).

Remark 1.3.29. One could rephrase the above discussions in the language of Quillen adjunc-

tions.

Simplicial model categories end up being a ‘better’ setting for doing homotopy, in the sense

that we have an easier way to define homotopy in a clearer way. The full details are in [GJ99]

and [Qui67] for model categories; the result is the following.

Remark 1.3.30. If C is a simplicial model category and A ∈ C is a cofibrant object, then two

maps out of A, f, g ∶ A→X are homotopic if and only if there is a factorization

A∐A A⊗∆1

X

d1∐d0

f∐ g
H

Then H is our homotopy, and we write f ∼ g or g ∼ f . This will, of course, be an equivalence

relation on HomC(A,X). The fact that this gives the homotopy classes of maps in Ho(C) is

checked in [GJ99, p. 96].

It is then shown that CGHaus can be given a simplicial model structure. Since our overar-

ching goal is to construct the stable A1−homotopy category, we won’t say more.

36



We now present some more details on model categories which will be useful in the next

section.

Definition 1.3.31. A model category M is left proper if for any pushout

A B

C D

i

w ∼ w′

i′

with i a cofibration and w a weak equivalence, then w′ is also a weak equivalence. Stated more

concisely, pushouts of weak equivalences along cofibrations should be weak equivalences.

The dual definition for a pullback square, with a fibration is known as a right proper model

category. We will not use these in practice.

We now wish to show that our standard model category structure on S is left proper. We

can actually prove a slightly stronger result and derive our desired fact as a corollary.

Proposition 1.3.32. Let M be a model category (not necessarily left proper) with a pushout

diagram satisfying the same conditions as in Definition 1.3.31

A B

C D

i

w ∼ w′

i′

i.e. that i a cofibration, w a weak equivalence. If A and C are cofibrant, then w′ is a weak

equivalence.

Proof. This is a result originally found in an unpublished paper of Reedy, [Ree74, Theorem

B].

Corollary 1.3.33. If M is a model category in which every object is cofibrant, then M is left

37



proper.

Proof. Immediate from Proposition 1.3.32 since all objects are cofibrant.

Corollary 1.3.34. The simplicial model structure from Theorem 1.3.19 is left proper.

Proof. Immediate from Corollary 1.3.21 and the above result, which shows (retroactively) why

we chose to establish Corollary 1.3.21 explicitly.

We now collect some results about some special types of simplicial model categories, which

will be used in our construction of the (un)stable motivic homotopy category. A thorough treat-

ment of combinatorial model categories can be found in the appendix of [Lur09]. We give some

definitions.

Definition 1.3.35. [Lur09, Definition A.1.1.2] A category C is presentable if it satisfies the

following four conditions:

1. C is cocomplete.

2. There is a set S of objects of C which generate C under colimits. That is, every object in

C is the colimit of a small diagram in S.

3. Every object in C is small, or equivalently, every object in S is small.

4. For all X,Y ∈ C, HomC(X,Y ) is a (small) set.

There are, of course, a wide array of results and properties of presentable categories. They

will not be discussed here.

Definition 1.3.36. [Lur09, Definition A.2.6.1] A model category M is said is combinatorial if it

satisfies the following conditions:

1. M is presentable

38



2. There is a set I of generating cofibrations, such that the collection of all cofibrations of M

is the smallest weakly saturated class of morphisms containing I .

3. There exists a set J of generating trivial cofibrations such that the collection of all acyclic

cofibrations in M is the smallest weakly saturated class of morphisms containing J .

The definition of weakly saturated is given [Lur09, Definition A.1.2.2.]. However, for a

more thorough and easier to read treatment, albeit in a different context, one can consult [Rez22,

Chapter 3]. This also contains a discussion of anodyne maps, a topic which we have chosen to

omit. Lurie goes on to give a conditions under which a presentable category can be endowed with

a left proper combinatorial model structure, see [Lur09, Proposition A.2.6.13].

We will use (twice) the left Bousfield localization of a (left proper combinatorial simpli-

cial) model category, stated below. In what follows, we give the definition for a general model

category, but we will be working exclusively with simplicial model categories from now on.

Thus, instead of considering classes of maps, we will be considering sets of maps. This is a very

important technicality.

Definition 1.3.37. Let I be a class of maps in a model category M . An object X ∈M is said to

be I-local if X is fibrant, and for every f ∶ A → B in I , the induced map f∗ ∶ HomM(B,X) →

HomM(A,X) is a weak equivalence.

A map f ∶W → Y in M is a I-local equivalence if for every I-local object X the induced

map f∗ ∶ HomM(Y,X)→ HomM(W,X) is a weak equivalence.

Definition 1.3.38. Let M a model category and I a class of maps in M . The left Bousfield

localization of M with respect to I , if it exists, is the model category structure LIM on the

underlying category M satisfying the following three conditions:

1. The class of weak equivalences of LIM is the class of I−local equivalences of M ,

39



2. The class of cofibrations of LIM is the same as the initial class for M,

3. The class of fibrations of LIM is defined, as earlier, to be the maps with right lifting

property with respect to cofibrations that are I−local equivalences.

Bousfield localization is a ubiquitous construction within homotopy theory, and is ripe for

discussion. It is so ripe, in fact, that the entire book [Hir03] is centered around it. We cannot

hope to cover them wholly, so we instead present the key results needed for our purposes. One

of the main ingredients of this construction is the following result of Lurie.

Theorem 1.3.39. [Lur09, A.3.7.3] Let C be a left proper combinatorial simplicial model category

and let S be a (small) set of morphisms in C. Then, the left Bousfield localization LSC exists and

is a left proper combinatorial simplicial model structure on C.

Proof. A full proof is provided in [Lur09, A.3.7.3]. We will provide our own perspective.

Note that because the underlying category is the same, it is immediately a simplicial cate-

gory – that is, the simplicial tensor, simplicial cotensor, and simplicial mapping functor all satisfy

the required adjointness by virtue of them being defined independently of the model structure.

The assumptions can be weakened while arriving at a similar result, and we have not found

it mentioned in the reference. Under only the assumption that M a simplicial model category, we

can get that LSC is also a simplicial model structure from some of our discussion last sections.

By our remarks above, we only need to comment on the actual properties of the adjoint func-

tors, mainly that we satisfy axiom SM7 – that the model structure plays well with our inherited

simplicial structure.

By our proof of Proposition 1.3.28, however, we have a way of recognizing that the axiom

is satisfied, and by the remark below Proposition 1.3.28 it suffices to check that the (mapping,

40



tensor, cotensor) adjunctions are Quillen adjunctions. That is, it is enough to be seen that

A⊗ − ∶ S LSC ∶ HomC(A,−)

− ⊗K ∶ LSC LSC ∶ homC(K,−)

form Quillen adjunctions. So, it suffices to check where (acyclic) cofibrations are sent by our

Corollary 1.3.15 from the previous section, which is straightforward.

Theorem 1.3.40. [Hir03, Theorem 4.1.1] Let M be a left proper cellular model category, and S

a set of maps in M . Then, the left Bousfield localization LSM exists and is a left proper cellular

model category. The fibrant objects of LSM are the S−local objects of M . Under the additional

assumption that M is a simplicial model category, LSM is also a simplicial model category.

We wish to strengthen the conclusion of Theorem 1.3.39 similar to our above result, and

we do so with the following proposition:

Proposition 1.3.41. If M is a left proper (combinatorial simplicial) model category and I a set of

maps, with left Bousfield localization LIM , the fibrant objects of LIM are precisely the I−local

objects of M .

Proof. This is immediate from [Hir03, Proposition 3.4.1]; we have just provided a weaker state-

ment.

So, we know precisely what the fibrant objects in our left Bousfield localization are now.

The next result is never stated explicitly in references ([Hir03], [GJ99],[AE16]), but we insert it

to be abundantly clear

Proposition 1.3.42. The morphisms of I are weak equivalences in LIM

41



Proof. This is by construction.

We provide one more result, which is typically given as an exercise.

Proposition 1.3.43. Let M a simplicial model category and I a set of morphisms in M (so that

the left Bousfield localization exists – alternatively assume the left Bousfield localization exists).

The identity functors id ∶M LIM ∶ id form a Quillen adjunction.

Proof. The proof is just an immediate application of Corollary 1.3.15 by construction of the left

Bousfield localization. That is, we preserve (acyclic) cofibrations going ‘to the right’, and by

definition again we preserve (acyclic) fibrations going ‘to the left.’

We will apply a left Bousfield localization more than once. Our first use will be applying

a more general construction which we apply to the category of motivic spaces, involving hyper-

covers. This requires the notion of a Grothendieck pretoplogy, which can be found in Section

2.1.

42



Chapter 2: Motivic Homotopy Theory

2.1 Grothendieck Topologies and Simplicial Presheaves

Here we will present the relevant constructions to define motivic spaces, as well as some

background needed specifically for synthetic spectra. In what follows, we will fix a (finite dimen-

sional) Noetherian base scheme S and consider SmS the full subcategory of smooth schemes of

finite type over S.

Definition 2.1.1. For a category C, a presheaf on C is a functor F ∶ Cop → Set. We use PshC to

denote the category of presheaves on C; the morphisms are natural transformations of functors.

We will need the notion of a Grothendieck pretopology, as below.

Definition 2.1.2. Let C a category with all pullbacks. A Grothendieck pretopology τ , some-

times called a basis for a Grothendieck topology on C is an assignment for each object X ∈ C

a collection of covering families Covτ(X). A covering family is a set of morphisms {fα ∶ Uα →

X ∣α ∈ A} in C. The set of covering families Cov(X) (we drop the τ from the notation where it

is understood) are required to satisfy the following axioms:

1. Given any isomorphism X ′ →X then the one-element family {X ′ →X} is in Cov(X),

2. coverings are closed under pullbacks; that is, if {fi ∶Xi →X} is in Cov(X) and g ∶ Y →X ,

then the pullback {g × fi ∶ Y ×X Xi → Y } are in Cov(Y ), and,

3. coverings are closed under refinements; that is, if {fi ∶Xi →X} is in Cov(X) and for each

43



i we have {fij ∶ Xij → Xi} in Cov(Xi), then the composite family {fi ○ fij ∶ Xij → X} is

in Cov(X).

Remark 2.1.3. This may seem a bit abstract at first glance, but we are really just trying to gen-

eralize our axioms for a topology and apply it to a category. The first axiom is analogous to

requiring that the whole space is open for a topology. Fiber products are our categorical analogue

of intersections, so our second condition is like requiring that finite intersections of opens are

open. The final condition is like requiring that arbitrary unions of open sets are open.

Remark 2.1.4. The definition above and the following discussion of sheaves generalizes to the

setting of∞−categories, which we haven’t yet touched on.

Definition 2.1.5. A category C together with a Grothendieck pretopology τ is called a site. We

will often refer to C as a site, meaning it comes equipped with a Grothendieck pre-topology.

We will never explicitly work with a Grothendieck topology – it is enough to know that

they are generated by Grothendieck pretopologies.

Definition 2.1.6. Let C a site. A sheaf on C is a presheaf F on C such that for all X ∈ C and all

{fi ∶Xi →X} ∈ Cov(X), the following is an equalizer diagram

F ∏i∈I F(Xi) ∏(i,j)∈I2 F(Xi ×X Xj)

Where, of course, the two maps are given by restriction.

Definition 2.1.7. For a site C we denote Sh(C) the category of sheaves. This is the full subcat-

egory consisting of objects the sheaves for our pretopology on C.

We would like, as we have in algebraic geometry, the notion of sheafification.

Theorem 2.1.8. Let C a site. The inclusion functor ι ∶ Sh(C) → Psh(C) has a left adjoint

44



a ∶ Psh(C) → Sh(C), which we will call the sheafification functor. It commutes with finite limits

and is exact.

Proof. While far from the original reference, a proof can be found in [DLØ+07, II.9.2], over the

course of a few pages.

We now wish to put a topology on SmS . The correct choice is the Nisnevich topology,

which falls between the Zariski and étale topologies. It will provide the right starting point, from

which we will take successive Bousfield localizations (on simplicial presheaves on SmS , really,

but that will be addressed soon). A discussion of why this is the correct starting point requires a

lot of buildup of scheme-theoretic concepts; [MV99] gives a discussion, and further discussion

can be found in [DLØ+07].

Fact 2.1.9. The category SmS is essentially small. That is, the class of isomorphism classes is

small (i.e. they form a set). SmS is also not (co)complete, so ‘doing homotopy theory’ directly on

SmS is not possible.

We will recall the definitions and facts relevant to our Nisnevich topology. We first recall

one possible characterization of étale maps.

Definition 2.1.10. A map f ∶ X → Y in SmS is étale if for every x ∈ X , the induced map of

tangent spaces is an isomorphism.

Definition 2.1.11. A morphism of schemes f ∶X → Y is Nisnevich if it is an étale morphism, and

for all y ∈ Y with f−1(y) ≠ ∅, we have some x ∈ f−1(y) such that the induced map k(x) → k(y)

of residue fields is an isomorphism.

This gives the appropriate definition for a Nisnevich morphism in SmS .

We are concerned with covering families (per the definition of a Grothendieck pretopol-

45



ogy). It extends the above definition in the obvious way, but we state it regardless:

Definition 2.1.12. A family of morphisms {fi ∶ Xi → X} in SmS is a Nisnevich cover if each

fi is étale, and for any x ∈ X , there is some i ∈ I and x′ ∈ f−1i (x) such that the induced map

k(x′)→ k(x) is an isomorphism.

There is an equivalent definition, which is formulated in [Lur18, Appendix B]

Definition 2.1.13. The Nisnevich topology on SmS is the topology generated by finite families

of étale morphisms {fi ∶ Yi → X} such that there is a finite sequence ∅ = Zn ⊆ Zn−1 ⊆ ⋅ ⋅ ⋅ ⊆ Z1 ⊆

Z0 = X of finitely presented closed subschemes of X such that for each Um = Zm ∖ Zm+1, there

is some fi ∶ Yi →X such that the pullback

fi∣Ui
∶ Ui ×X Y → Ui

admits a section. That is, there is a map g ∶ Ui → Ui ×X Y for which fi∣Ui×XY ○ g = idUi
.

We have commented that the Nisnevich topology falls between the Zariski and étale topolo-

gies, yet we haven’t defined a Zariski covering.

Definition 2.1.14. A covering {fi ∶Xi →X} is a Zariski covering if the fi are open immersions

and it is jointly surjective, in that each x ∈X is in the image of some fi.

Proposition 2.1.15. A Zariski covering is a Nisnevich covering, and a Nisnevich covering is an

étale covering.

Proof. This fact is ubiquitous in the literature. While it is mostly by definition, none of the

references show this explicitly, so we will.

Recall that a morphism of schemes is called an open immersion if it’s an open immersion of

locally ringed spaces; that is, the map of spaces is a homeomorphism and the map of sheaves is an

46



isomorphism. It then follows (perhaps with the assistance of an exercise from [Har13, II.2]) that

an open immersion is an étale morphism. Further, an open immersion satisfies our condition of a

Nisnevich map by the isomorphism of local rings, so a Zariski covering is a Nisnevich covering

by our first definition. The fact that a Nisnevich covering is an étale covering is immediate.

Nisnevich (Zariski, étale) coverings define a Grothendieck pretopology on SmS , so we can

refer to the Nisnevich site, sometimes denoted SmS,Nis.

Definition 2.1.16. A presheaf F on SmS which satisfies the sheaf condition for Nisnevich (or

Zariski, étale) coverings is said to be a Nisnevich sheaf (resp. Zariski, étale sheaf). We denote

ShNis(SmS) the collection of Nisnevich sheaves on SmS .

We want to show that all representable presheaves are sheaves in the Nisnevich topology;

we can show this for the étale topology, for a stronger result. In other terminology, we want to

show that the étale topology is subcanonical. So, by the Yoneda embedding, ShNis(SmS) (or

really Shét(SmS)) contains SmS as a full subcategory.

Proposition 2.1.17. It suffices to check that a presheaf F ∈ Psh(SmS) satisfies the sheaf condi-

tion on étale covers consisting of one morphism. That is, if F satisfies the sheaf condition for all

one map {f ∶ Y →X} étale covers, then it is a Nisnevich sheaf.

Proof. This relies on some properties of étale morphisms and étale coverings. In particular, if we

have an étale covering fi ∶ Xi → X which satisfies the sheaf condition, then it still satisfies the

condition when more étale morphisms are added.

So, to check that a covering {fi ∶ Xi → X} satisfies descent, first take a finite subcover (X

is compact, topologically). It suffices to check on this collection (as adding more maps won’t

change things, since our finite collection is a covering). Take U =∐iXi the finite coproduct and

47



f =∐i fi, and we have a single étale map f ∶ U → X which we must check satisfies descent (the

sheaf condition).

This reduction applies to any étale cover, so to check that a presheaf is an étale sheaf, it

suffices to show it satisfies descent with respect one-morphism étale coverings.

Corollary 2.1.18. If F ∈ Psh(SmS) satisfies Nisnevich descent for all surjective Nisnevich maps,

then F ∈ ShNis(SmS).

Proof. By Proposition 2.1.15 and the above result.

With this corollary, we can now provide the following result, which was the goal of this

discussion.

Corollary 2.1.19. For X ∈ SmS , the presheaf represented by X is a sheaf in the étale (and hence

Nisnevich, Zariski) topology.

Proof. From the above result, with some properties of étale morphisms.

Which establishes that the étale topology is subcanonical, as stated before. We have a

common technique for detecting when a presheaf is a sheaf in the Nisnevich topology, using what

are known as elementary distinguished squares. The intuitive idea is that, since the Nisnevich

topology falls between the Zariski and étale topologies, a diagram which contains information

from the Zariski and étale maps should combine to tell us when we satisfy Nisnevich descent.

We make this formal.

Definition 2.1.20. An elementary distinguished Nisnevich square in SmS is a pullback square

(sometimes called cartesian)
U ×X V V

U X

p

i

48



where p is étale, i an open immersion, and p−1((X ∖U)red) → (X ∖U)red is an isomorphism of

schemes ( X ∖U is endowed with the reduced induced scheme structure).

Such a cover {p ∶ U → X, i ∶ V → X} is a Nisnevich cover. The reason for introducing

elementary distinguished squares is the following result:

Theorem 2.1.21. [MV99] A presheaf F ∈ Psh(SmS) is a Nisnevich sheaf if and only if it sends

every elementary distinguished Nisnevich square into a pullback square.

Proof. This can be found in [MV99, p. 97]

We won’t work directly with ShNis(SmS), but rather simplicial Nisnevich presheaves. We

record some background information first.

Definition 2.1.22. A simplicial presheaf X on a category C is just a functor X ∶ Cop → S to

simplicial sets. Alternatively, a simplicial presheaf is a simplicial object in Psh(C).

We denote sPsh(C) the category of simplicial presheaves, and sSh(C) the subcategory

of simplicial sheaves when C is a site.

Remark 2.1.23. We can of course embed Psh(C)→ sPsh(C), and similarly Sh(C)→ sSh(C).

We also can embed S → sPsh(C) by sending X to the constant presheaf.

We give one construction for simplicial presheaves of any small (or essentially small) site.

The main reference is [DHI03].

Definition 2.1.24. For C a site with τ a Grothendieck topology and X● ∈ sPsh(C)and X → Y a

map with Y a representable simplicial presheaf, we say that X → Y a hypercover if we have the

following three conditions:

1. Xn =∐Wni
where each Wni

is a representable,

2. the map X0 → Y is a cover in τ ,

49



3. X∆n →X∂∆n is a cover in degree 0

Here, we write for X ∈ sPsh(C),K ∈ S,XK is the presheaf defined by XK(U) = homS(K,X(U))

The more concise way to state the conditions is that Xn is a coproduct of representables

and X → Y is a local acyclic fibration. This will not be used except in the following construction,

however.

Theorem 2.1.25. For C small site with Grothendieck (pre)topology τ , the left Bousfield localiza-

tion of sPsh(C) with respect to any class of hypercovers, denoted Lτ sPsh(C), exists. Further-

more, if C is a (left) proper combinatorial simplicial model category, then so is Lτ sPsh(C)

Proof. By Theorem 1.3.39 we just note that the collection of all hypercovers in τ forms a set,

since C small.

Remark 2.1.26. The category sPsh(SmS) is (co)complete, and S represents a terminal object of

sPsh(SmS).

2.2 Unstable Motivic Homotopy

We now construct our category of spaces and the unstable motivic homotopy category.

While SpcS has some nice properties, we wish to enforce our goal that A1 should be contractible

(i.e. play the role of the interval). A lot of the discussion can be done more abstractly thanks to

the work of [Dug00] and [Lur09]. The first requirement for a suitable homotopy category would

be establishing a left proper combinatorial simplicial model category structure on our category

of spaces. So, let’s do that.

Definition 2.2.1. For C an essentially small site, the category of simplicial presheaves sPsh(C)

can be endowed with the structure of a model category, using the model category structure on

50



simplicial sets, called the projective model category structure. It is constructed as follows

1. Weak equivalences are the objectwise weak equivalences. That is, maps f ∶ X → Y such

that f(U) ∶X(U)→ Y (U) is a weak equivalence of simplicial sets,

2. fibrations are the projective fibrations, also called objectwise fibrations. That is, p ∶ X →

Y a fibration if p(U) ∶X(U)→ Y (U) is a fibration of simplicial sets,

3. cofibrations are projective cofibrations, i.e. those with the left lifting property with respect

to acyclic objectwise fibrations

This is also sometimes called the global projective model structure

Proposition 2.2.2. For an (essentially) small site C category sPsh(C) with the projective model

category structure is a left proper combinatorial simplicial model category.

Proof. This is just a special case of [Lur09, A.2.8.2], which is stated for functor categories into a

combinatorial model categories; properness comes from [Lur09, A.2.8.4]

Definition 2.2.3. There is another left proper combinatorial simplicial model category, namely

the injective model category structure, where we have the same weak equivalences but instead

define our injective cofibrations to be objectwise cofibrations of simplicial sets, i.e. monomor-

phisms (hence injective cofibrations) and then we define our injective fibrations to have the

right lifting property with respect to acyclic injective cofibrations. Again, this is often called the

global injective model category structure

Proposition 2.2.4. For C a (small) site the global injective and projective model structures are

Quillen equivalent.

Proposition 2.2.5. The left Bousfield localization LNis sPsh(SmS) with respect to Nisnevich

hypercovers of sPsh(SmS) with the global projective model structure exists, giving a left proper

51



combinatorial simplicial model category.

Proof. This is an immediate result of Theorem 2.1.25.

While existence is helpful, we wish to know what this model structure consists of. We

make the following definition:

Definition 2.2.6. Let C, τ be a (small) site, and let f ∶ X → Y a map of simplicial (pre)sheaves

sPsh(C). We say that f is a local weak equivalence if f∗ ∶ π0X → π0Y is an isomorphism

of sheaves, and for n > 0, U ∈ C and x0 ∈ X(U) the induced morphism f∗ ∶ πn(X ∣U , x0) →

πn(Y ∣U , f(x0)) is an isomorphism.

Definition 2.2.7. For C, τ a small site, the local projective model structure on sPsh(C) has:

1. Weak equivalences are the local weak equivalences described above,

2. fibrations are projective fibrations, i.e. those taken objectwise.

3. cofibrations the maps with the left lifting property with respect to acyclic fibrations.

We can define the local injective model structure in the analagous way.

The next results are essential to the“larger picture.” We connect the different (modern)

ways of constructing the unstable motivic category (really, the category of spaces), and shows the

equivalence of all the difference approaches one might find in the literature. These differences

are rarely addressed in expository works, so we will draw special attention.

Theorem 2.2.8. The left Bousfield localization LNis sPsh(SmS) (the Nisnevich local model cat-

egory) has the local projective model structure.

Proof. This result was first proved in [DHI03] (see the introduction, or section 6 for more explicit

details).

Taking the left Bousfield localizations with respect to hypercovers is a less common choice,

52



as the local projective (or injective) model structures are easy to work with, and its more imme-

diately clear what the model structure looks like.

Proposition 2.2.9. The local injective model structure and local projective model structure on

sPsh(C) (and sSh(C)) are Quillen equivalent.

Proof. This is from the identity functor applied the underlying category which are, of course, ad-

joint. Quillen equivalence comes from the “symmetry” in the construction of projective (co)fibrations

and injective (co)fibrations. This is discussed in [DHI03].

Remark 2.2.10. It is actually true, more generally, that the global projective model structure

and the global injective model structure are Quillen equivalent, due to [DHI03]. So, our choice

to using the projective structure is somewhat arbitrary, in the sense that our resulting homotopy

(and stable homotopy) categories will be the same.

The local projective structure actually ends up being a lot nicer than what we’ve described

here, which is due to the work of Blander. We present the full statement below.

Theorem 2.2.11 ([Bla01, Theorem 1.6]). Let T be any essentially small site. Then the category

of simplicial presheaves on T sPsh(T ) with the local projective model structure is a proper

simplicial cellular (so cofibrantly generated) model category.

Remark 2.2.12. By remarks later, we will get some other results of [Bla01], namely an equiv-

alence in the model category of simplicial sheaves and simplicial presheaves under the local

projective structure. These also apply to the local injective structure.

We are now ready to establish the category which is worth of the name “motivic spaces.”

Simplicial (pre)sheaves provide a natural starting point for constructing a homotopy category,

so we have an appropriate category of spaces now. However, a choice has to be made – do we

53



consider (simplicial) presheaves or sheaves? This is a choice that varies among authors, but we

will rectify this.

Lemma 2.2.13. For (C, τ) a (small) site, the inclusion functor ι ∶ sSh(C) → sPsh(C) has a left

adjoint a ∶ sPsh(C) → sSh(C), which we will sometimes call the sheafification functor, or the

associated sheaf functor.

Proof. This is an immediate consequence of Theorem 2.1.8

Theorem 2.2.14. For a small site C, τ and the local projective model structure on sPsh(C) and

sSh(C), the adjoint pair

sSh(C) sPsh(C)ι

a

form a Quillen equivalence. The same holds for the local injective structure.

Proof. This is a result initially from [Bla01]. A longer discussion (in textbook format) can be

found in [Jar15].

So, to summarize all of the above results, we state our following result:

Theorem 2.2.15. For C a small site, we have Quillen equivalences

sShl.inj(C) ≃Q sPshl.inj(C) ≃Q sPshl.proj(C) ≃Q sShl.proj(C)

where l.inj denotes the local injective model structure, and l.proj denotes the local projective

model structure. As such, we have equivalence of categories

Ho(sShl.inj(C)) ≃ Ho(sPshl.inj(C)) ≃ Ho(sPshl.proj(C)) ≃ Ho(sShl.proj(C))

54



Proof. All of the above results.

This is a general result for any site; we have our following corollary

Corollary 2.2.16. There are Quillen equivalences

LNis sShl.inj(SmS) ≃Q LNis sPshl.inj(SmS) ≃Q LNis sPshl.proj(SmS) ≃Q LNis sShl.proj(SmS)

and thus equivalences on the corresponding homotopy categories.

Proof. Immediate.

Definition 2.2.17. We call the category LNis sPsh(SmS) with the (local) projective model struc-

ture the category of motivic spaces, often denoted SpcS .

Notation. We will sometimes refer to SpcS as sShNis(SmS) instead of LNis sPsh(SmS). This is

understood to be the category sPsh(SmS) with the (proper, cellular, combinatorial simplicial)

local projective model structure.

Remark 2.2.18. The way we have constructed the category sShNis(SmS) has guaranteed it con-

tains both SmS and S . That is, every representable presheaf is a Nisnevich sheaf, and so by

Remark 2.1.23 we have for each X ∈ SmS the associated representable sheaf, a motivic space.

Similarly, sShNis(SmS) contains every K ∈ S , in terms of the constant sheaf taking the value K.

That is, the simplicial sheaf sending X ↦K for all X ∈ SmS .

The category SpcS is (co)complete, and of course has a simplicial structure.

Now that we have a new model category of interest, its helpful to know something about

the (co)fibrant objects.

Proposition 2.2.19. A presheaf F ∈ sShNis(SmS) is fibrant (also called Nisnevich-fibrant) if and

55



only if for every elementary distinguished square

U ×X V V

U X

p

i

the natural map F(X) → F(V ) ×F(U×XV ) F(U) is a weak equivalence of simplicial sets and

F(∅) is a final object.

Proof. This is equivalent to saying that the diagram

F(X) F(V )

F(U) F(U ×X V )

F(p)

F(i)

is homotopy cartesian. A definition can be found in [GJ99]. This is also commented on in [AE16]

where they refer to multiple other sources for a proof.

This is actually a less general version of a result prove in [Bla01, Lemma 4.1] which re-

quires the use of cd-structures and proves the result for any site. We present the characterization

here:

Lemma 2.2.20. [Bla01] A simplicial presheaf X ∈ sShNis(SmS) is fibrant with respect to the

local projective model structure if it takes values in Kan complexes and X takes distinguished

squares to homotopy cartesian squares.

Definition 2.2.21. Presheaves that satisfy this condition are said to satisfy the Brown-Gersten

property.

There are a lot of results relating to this condition, and we cannot detail all the theory.

Definition 2.2.22. The category of pointed motivic spaces, SpcS,∗, is obtained from the above

56



discussion but instead using pointed simplicial presheaves, i.e. presheaves taking values in

pointed simplicial sets S∗.

The category of pointed motivic spaces is also symmetric monoidal with the construction

of the smash product, which we will construct.

Remark 2.2.23. We have a pair of adjoint functors between SpcS and SpcS,∗, where ‘going to

the right’ provides a disjoint basepoint, and the other functor forgets the basepoint.

Definition 2.2.24. We define the wedge product (which is really the coproduct in SpcS,∗) of two

pointed spaces (X,x) ∨ (Y, y) to be the pushout of (X,x) ∗ (Y, y) . In terms of pointed

simplicial presheaves, this sends a space to the wedge product in pointed simplicial sets of that

space.

We can also define the smash product (X,x) ∧ (Y, y) = (X,x) × (Y, y)/(X,x) ∨ (Y, y)

So now we have a left proper combinatorial simplicial model category structure on SpcS ,

but it does not have the A1−local properties we want it to have. Most importantly, A1 won’t play

the role of the unit interval yet, as A1×SX →X is not guaranteed to be a weak equivalence. This

suggests that we need to take another left Bousfield localization.

Before proceeding, it would be useful to review Definition 1.3.37. We want to define our

A1−local spaces, and then the A1−homotopy category, otherwise known as our unstable motivic

category.

Definition 2.2.25. Let I be the class of maps in LNis sPsh(SmS) of the form A1 ×S X →X as X

ranges over the objects of SmS . Using the fact that SmS is essentially small, pick a subset J ⊂ I

where we only range over a representative from each class of isomorphic objects X .

The left Bousfield localization with respect to the set J , denoted LA1LNis sPsh(SmS) gives

us the A1−local model structure. The homotopy category Ho(LA1LNis sPsh(SmS)) is the A1−homotopy

57



category of S.

Notation. Recall that we denote QX,RY as the cofibrant replacement of X and the fibrant re-

placement of Y , respectively.

Definition 2.2.26. An object X ∈ LA1 LNis sPsh(SmS) is said to be A1−local if it is locally

(projective) fibrant and, for any U ∈ SmS we have

Map(U × Y,Y ) Map(U,Y )≃loc

a local weak equivalence.

We say that a map f ∶ X → Y in LA1 LNis sPsh(SmS) is an A1−local weak equivalence if

for all A1-local Z, the map

Map(QY,Z) Map(U,Z)Qf∗

≃

is a local weak equivalence.

Notation. We also denote SpcA
1

S for LA1LNis sPsh(SmS).

Theorem 2.2.27. The category SpcA
1

S is a left proper combinatorial simplicial model category.

The weak equivalences of this category are the A1−local weak equivalences, and fibrations are

local projective fibrations.

Proof. This is immediate from Theorem 1.3.39 and Definition 1.3.38.

Notation. For X,Y ∈ sPsh(SmS), the notation [X,Y ]A1 is often used to denote the set of A1-

homotopy classes of maps, i.e. maps in Ho(LA1 LNis sPsh(SmS)).

Remark 2.2.28. We have a convenient way of characterizing the A1−local objects. Of course, for

58



X ∈ sPsh(SmS) to be A1−local it should satisfy the conditions of being fibrant in sPsh(SmS),

SpcS , and SpcA
1

S . We require

1. X(U) is a Kan complex for all U ,

2. takes distinguished squares to homotopy cartesian squares, or alternatively satisfies Nis-

nevich hyperdescent,

3. X(U)→X(U ×S U) is a local weak equivalence

Remark 2.2.29. We now have that A1 is, indeed, contractible in our homotopy category, since

X × A1 → X is an A1−weak equivalence. This also gives that An is contractible for all n ≥ 1,

since we can just take it as the (fiber) product of contractible spaces.

Notation. We often denote H(S) ∶= Ho(LA1 LNis sPsh(SmS)) for the A1homotopy category.

Now that we have our A1-homotopy category defined, we should comment on what an

A1−homotopy is.

Definition 2.2.30. As in classical (unstable) homotopy theory, the motivic spheres allow us to

define the bigraded abelian motivic homotopy groups for X ∈ H(S) to be

πs,t(X) ∶= [Ss,t,X]H(S).

We can do one better than abelian groups, however, with the following construction.

Definition 2.2.31. For X ∈ H(S) and U ∈ SmS we can define the (bigraded) motivic homo-

topy sheaf of abelian groups πs,t(X) to be the sheafification (in the Nisnevich topology) of the

presheaf

U ↦ [Ss,t ∧U+,X]H(S),

59



where U+ is ‘given a disjoint basepoint.’ We then recover the unstable homotopy groups as global

sections.

Remark 2.2.32. It is not uncommon to see the notation [X,Y ]A1 used to signify that we are

considering maps up to A1−local homotopy rather than [X,Y ]H(S). More common, however, is

suppressing any notation denoting where the equivalence class of maps is taken, as it is usually

clear from context.

We should comment on the notion of naive, or elementary A1-homotopy.

Definition 2.2.33. For f, g ∶ X → Y in SpcS . A naive A1−homotopy from f to g is a map

H ∶X ×A1 → Y where we have

X × 0

X ×A1 Y

X × 1

f

H

g

Remark 2.2.34. Of course, any two maps which are naively A1−homotopic will induce the same

morphism in H(S). However, there are far more classes of maps in H(S) than those that arise

from a Naive A1−homotopy. This concept is included for the sole purpose of ensuring the reader

does not make the misconception that naive A1−homotopies are the correct way of thinking about

H(S)

2.3 Stable Motivic Homotopy

We wish to construct a stable motivic homotopy category. Before we do so, we need

to comment on the sphere objects in motivic homotopy. One of the interesting (and unique)

parts of motivic homotopy theory is that we get two spheres, which will end up resulting in a

60



bigraded theory. We make these notions precise over the coming section. The original reference

is [Voe98]

Definition 2.3.1. We let Gm ∶= A1 ∖ {0} pointed at 1 ∈ Gm. This is called the Tate circle, and it

is one of our two motivic circles.

The (pre)sheaf represented by the simplicial sphere S1, i.e. the (pre)sheaf represented by

∆1/∂∆1, is called the simplicial circle. Of course, our category has all simplicial n−spheres

representing Sn ∶=∆n/∂∆n.

With the smash product we remarked in the last section, the natural thing to ask is: what

happens when we smash these two motivic circles? The answer to that question is as fol-

lows

Proposition 2.3.2. P1 is A1-homotopy equivalent to S1 ∧Gm.

Proof. P1 is the pushout of A1 Gm A1 . But, as A1 is contractible, we have the homo-

topy pushout diagram
Gm ∗

∗ P1

where the homotopy colimit is the suspension S1 ∧Gm. This identifies P1 ≃ S1 ∧Gm.

Notation. With a view towards our (bigraded) spectra we will often denote the simplicial sphere

as S1
s and the Tate circle as S1

t . We can then simplify notation and list our spheres with a bigrad-

ing. That is,

Sp,q = (S1
s ∧ S1

s ∧ ⋅ ⋅ ⋅ ∧ S1
s)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
p−q times

∧ (S1
t ∧ S1

t ∧ ⋅ ⋅ ⋅ ∧ S1
t )

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
q times

More compactly, Sp,q = (S1
s)∧p−q ∧ (Gm)∧q.

This means that P1 ≃ S2,1 in H(S).

61



We will now give the construction of [DLØ+07], which avoids some of the difficulties asso-

ciated to the more popular construction of the stable homotopy category. This uses (s, t)−bispectra,

where we are essentially inverting both Ss and St consecutively. We will comment, after showing

the alternative construction (P1−spectra), on their equivalence.

Definition 2.3.3. An s−spectrum E is a sequence of pointed spaces En ∈ SpcS,∗ with structure

maps

S1
s ∧En → En+1

A map of s−spectra f ∶ E → F is defined degreewise, fn ∶ En → Fn so that we have a commuta-

tive diagram

En Fn

En+1 Fn+1

fn

σn σn

fn+1

Definition 2.3.4. The above construction defines the category of s−spectra, denoted Sps(S).

Definition 2.3.5. We have an s−suspension spectrum associated to a pointed space X ∈ SpcS,∗,

denoted Σ∞s X , defined by (Σ∞s X)n = (S1
s)∧n ∧X , and structure maps identity. This assignment

is functorial.

We want a proper simplicial stable model category structure on Sps(S). Although, up until

this point we have avoided giving a definition for a stable model category, so we should do so

here.

Definition 2.3.6. Taking the smash product S1
s ∧X is functorial; that is, we have a functor Σs ∶

SpcS,∗ → SpcS,∗ which sends X to S1
s ∧X . This functor is left adjoint to the loop functor, where

ΩsX = homSpcS,∗(S1
s ,X)

Definition 2.3.7. We say that a model category C is stable if the suspension functor Σs ∶

62



Ho(C)→ Ho(C) is an equivalence of categories.

Alternatively, we can say that the adjunction

Σs ∶ Ho(SpcS,∗) Ho(SpcS,∗) ∶ Ωs≅

≅

form an inverse equivalence between the homotopy categories.

Definition 2.3.8. For E ∈ Sps(S), we define the nth sheaf of abelian stable homotopy groups

to be the sheaf associated to the presheaf of abelian groups

πn(E) ∶= colimm>n πm+n(Em)

Remark 2.3.9. This is really the colimit

colimm ( . . . π∗+m(E∗) π∗+m+1(ΣE
∗ ) π∗+m+1(E∗+1) . . .

Σs σ∗ )

Definition 2.3.10. We say that a morphism of s−spectra f ∶ E → F is an s-stable weak equiva-

lence if the induced map f∗ ∶ πn(E)→ πn(F ) is an isomorphism for all n.

A morphism of s−spectra is called a stable cofibration if the morphism f0 ∶ E0 → F0 and

En+1∐S1∧En
S1 ∧ Fn → Fn+1 for n > 0 are both cofibrations.

With our notion of weak equivalences and cofibrations, we now wish to define a (proper)

simplicial model category structure on Sps(S).

Proposition 2.3.11. [Mor03] The category Sps(S) together with s-stable weak equivalences and

stable cofibrations (and fibrations defined to have the right lifting property with respect to acyclic

cofibrations) forms a proper combinatorial simplicial model category structure. We call this the

63



s-stable model structure.

Proof. This can be found in any of [Mor03],[DLØ+07], [MV99]

Remark 2.3.12. There is an associated stable homotopy category, denoted SHs, which is ob-

tained by taking the left Bousfield localization (with respect to s-stable weak equivalences).

We can comment on suspension-loop space adjunction on the stable category

Remark 2.3.13. We have the induced adjunction

SHs Sps(S)
Σ∞
Ω∞

Where the infinite loop space is defined Ω∞(E) ∶= colimn≥0 hom∗(Sn,En) where the notation

denotes pointed morphisms.

Recall briefly from Definition 2.3.6 we have an adjunction between the suspension and loop

space functors on SpcS,∗. This can be extended to an actual adjunction on spectra; our structure

maps σn having adjoints ωn giving ωn ∶ En → ΩEn+1.

Definition 2.3.14. We say that an S1-spectrum E is an Ω−spectrum if the adjoint ωn ∶ En →

ΩEn+1 is a (stable) weak equivalence.

We don’t have any of the A1−local properties we would like for our stable category though

– mainly, we need A1 to be contractible. Seeing this, we take the only natural next step: localizing

in the exact same way as we did with the unstable category.

Definition 2.3.15. We say that an s-spectrum E is A1−local if for all U ∈ SmS and n ∈ Z the

projection U ×A1 → U defines a bijection

HomSHs(Σ∞s U,Σn
sF )→ HomSHs(Σ∞s (U ×A1),Σn

sF )

64



A map of s-spectra f ∶ E → F is an A1−local stable weak equivalence if it satisfies the expected

condition of Definition 1.3.37. That is, if for any A1−local s-spectrum H , there is a canonical

bijection

HomSHs(F,H)→ HomSHs(E,H)

Proposition 2.3.16. There is a model structure, called the s-stable A1−local model structure

on Sps(S) which is obtained by applying Definition 1.3.38, the left Bousfield localization on

A1−local weak equivalences to the model structure of Proposition 2.3.11. We denote the category

with this model structure as SpA1

s (S), if needed.

Proof. This is entirely analogous to our construction of SpcA
1

S .

Definition 2.3.17. We have a stable homotopy category associated to our A1-local model struc-

ture, which we will simply call the motivic s-stable homotopy category SHA1

s (S).

Just inverting with respect to S1,0 isn’t good enough though - we want to invert with respect

to Gm as well. This requires us to use the theory of bispectra.

Definition 2.3.18. We denote τ ∶ S1
s ∧ S1

t → S1
t ∧ S1

s the isomorphism given by the fact that the

smash product is symmetric monoidal on SpcS,∗.

Definition 2.3.19. An (s,t)-bispectrum E is a sequence of pointed spaces Ep,q ∈ SpcS,∗ with

structure maps

σs ∶ S1
s ∧Ep,q → Ep+1,q, σt ∶ S1

t ∧Ep,q → Ep,q+1

65



which are compatible in the sense of the following commutative diagram

S1
s ∧ S1

t ∧Ep,q S1
t ∧ S1

s ∧Ep,q

S1
s ∧Ep,q+1 Ep+1,q+1 S1

t ∧Ep+1,q

id∧σt

τ∧id

id∧σs

σs

σt

A morphism of bispectra f ∶ E → F is defined level-wise fp,q ∶ Ep,q → Fp,q which commute

with structure maps in the obvious way. Bispectra of motivic spaces form a category which we

will denote Sps,t(S).

Definition 2.3.20. We have a suspension functor Σp,q ∶ Sps,t(S) → Sps,t(S) which acts on E

level-wise, sending Er,s to Sp,q ∧ Er,s. It acts on morphisms in the obvious way, i.e. taking the

smash with identity.

Definition 2.3.21. There is a functor Σ∞s,t ∶ SpcS → Sps,t(S) for which Σ∞Xp,q = (S1
s)∧p ∧

(S1
t )∧q ∧X with structure maps being the identities. We have the expected right adjoint Ω∞s,t

The goal with (s, t)−bispectra is to localize at both spheres (using our above construction

of s−spectra), so we need a bigraded notion of homotopy sheaves.

Definition 2.3.22. For E ∈ Sps,t(S), U ∈ SpcS , and (p, q) ∈ Z2 we define the bigraded stable

homotopy sheaves πp,q to be the sheaf associated to the presheaf

U ↦ colimmHom
SHA1

s (S)
(Sp−q

s ∧ Sq+m
t ∧Σ∞s U, Em)

These are sheaves of abelian groups (the category SpA1

s (S) is additive, so the Hom-set has the

structure of an abelian group). We assume that q+m ≥ 0. A note on the choice of indices follows

below.

66



Remark 2.3.23. As in [DLØ+07], we note that the above definition makes sense when p < q as

there is an equivalence of categories SHA1

s (S) SHA1

s (S)
S1
s∧− . We also note that, while at

first glance the smash Sp−q
s ∧ Sq+m

t may seem rather unnatural, in our bigraded convention this is

just Sp+m,q+m. So, we have emulating our normal definition for stable homotopy.

We will again point out that having stable homotopy sheaves is a powerful tool here, and the

extra coherence data gives more insight than just stable homotopy groups. We have the obvious

notion of a stable weak equivalence.

Definition 2.3.24. We say that a map f ∶ E → F in Sps,t(S) is a stable weak equivalence if the

induced map f∗ ∶ πp,q(E)→ πp,q(F ) is an isomorphism of sheaves. We say that a morphism f is a

stable cofibration if fp,q ∶ Ep,q → Fp,q is a cofibration in the sense of SpcA
1

S,∗, i.e. component-wise.

Proposition 2.3.25. The category Sps,t(S) endowed with:

1. weak equivalences the (A1-local) stable weak equivalences,

2. cofibrations the stable cofibrations,

3. fibrations those maps with the right lifting property with respect to the acyclic cofibrations

forms a proper stable simplicial model category.

Proof. This is [DLØ+07, Remark III.2.11] – but proving this requires no new techniques. It is

the same as arguments from before.

We are now ready to define our (first version of) the motivic stable homotopy category.

Definition 2.3.26. The motivic stable homotopy category SH(S) is the homotopy category of

Sps,t(S) with the above stable A1−model structure,

SH(S) ∶= Sps,t(S)[sWE−1A1]

67



Remark 2.3.27. The above construction is really just two successive localizations. We could

easily write SH(S) = SHA1

s (S)[(Gm)−1], or Sps,t(S)[(S1
s)−1][G−1m ]. The point we are trying to

make here is that we can really think of our construction on Sps,t(S) as looking at Gm−spectra

of S1
s−spectra.

This category is actually symmetric monoidal, but to prove so requires a lot of work –

according to Voevodsky, it “takes Adams thirty pages to verify that nothing goes wrong and it

is terrible.” [Voe98, p. 594]. Jardine was able to use his theory of symmetric spectra to show

that there is a symmetric monoidal structure on SH (really, on symmetric spectra, to which he

provides an equivalence) in his work [Jar00].

Proposition 2.3.28 ([Voe98, Theorem 5.6], [Jar00, Theorem 4.31]). SH(S) is a symmetric monoidal

category under the smash product of spectra ∧.

Proof. Explicitly, [Voe98, Theorem 5.6]. However, there is no explicit proof given; consult

[Jar00] for a better reference. [Jar00, Theorem 4.40] gives the required equivalence, and [Jar00,

Theorem 4.31] provides the actual construction of the symmetric monoidal stable homotopy cat-

egory.

2.4 Another Construction: P1−Spectra

Definition 2.4.1. Taking the smash product P1∧X for any space X will be called P1-suspension,

often denoted ΣP1X .

Definition 2.4.2. A P1−spectrum, also known as a motivic spectrum is a sequence of pointed

motivic spaces (E0,E1, . . . ) with structure maps σn ∶ P1 ∧En → En+1 (σn ∶ ΣP1En → En+1). The

data here consists of the spaces and the structure maps, so the notation (En, σn) is perhaps better.

68



A map of spectra f ∶ E → F is defined levelwise, i.e. it is a sequence of maps fn ∶ En → Fn

which are compatible with the structure maps; that is, we have a commutative diagram

P1 ∧En En+1

P1 ∧ F n F n+1

σE
n

id∧fn fn+1

σF
n

Of course, P1 is pointed at∞. We can use the more compact notation of (P1,∞)

Having defined objects and morphisms, we are forced to make the following definition:

Definition 2.4.3. We denote SpP1(S) for the category of P1−spectra, also called the category of

motivic spectra.

Example 2.4.4. The most basic example of a P1 spectrum is the suspension spectrum Σ∞P1(X )

for any X ∈ SpcS,∗. This is better than just an example, though:

Definition 2.4.5. Σ∞P1 ∶ SpcA
1

S,∗ → SpP1(S) defines a functor

We now want to endow P1-spectra with a proper simplicial model category structure in

order to form the stable homotopy category of motivic spaces. In his original work [Voe98], Vo-

evodsky gives a detour on (s, t)-bispectra , which eventually culminates in the definition of the

motivic stable homotopy category. There is another way of constructing the stable motivic homo-

topy category due to Jardine’s theory of motivic symmetric spectra, presented in [Jar00].

We can now define the stable motivic homotopy sheaves. This is analogous to the construc-

tion in the case of (s, t)−bispectra.

Definition 2.4.6 ([Mor03]). Let E an object in SpP1(S), U ∈ SmS , and (n,m) ∈ Z2. We define

the motivic stable homotopy sheaf, denoted πst
s,t(E) to be the (Nisnevich) sheafification of the

69



presheaf defined by

U ↦ colimrHomH(S)(Ss+r
s ∧ (U+) ∧ (P1)r−t,Er)

Definition 2.4.7. The motivic stable homotopy groups are defined to be the global sections of

the motivic stable homotopy sheaves, and give abelian groups. For X ∈ SpP1(S) and U ∈ SmS

we have,

πs,t(X)(U) ∶= colimrHomH(S)●(Ss+t
s ∧ (X+) ∧ (P1)r−t,Er)

Definition 2.4.8 ([Mor03]). A morphism f ∶ E → F in SpP1(S) is called an A1−stable weak

equivalence if for any X ∈ SmS and any (n,m) ∈ Z2 the homomorphism

f∗∶πn,m(E)→ πn,m(F )

is an isomorphism

Definition 2.4.9. In the construction of [Jar00], a map f ∶ E → F in SpP1(S) is a

1. level weak equivalence if fn ∶ En → Fn is a weak equivalence in SpcS for all n ≥ 0,

2. level fibration if fn ∶ En → Fn is a cofibration in SpcS for all n ≥ 0,

3. cofibration if it has the left lifting property with respect to acyclic levelwise fibrations.

The model structure on SpP1(S) is then defined by level weak equivalences, level fibrations, and

cofibrations.

Theorem 2.4.10 ([Jar00]). The category SpP1(S) endowed with level weak equivalences, level

fibrations, and cofibrations as in Definition 2.4.9 gives a proper simplicial model category.

Proof. This is [Jar00, Lemma 2.1].

70



Definition 2.4.11. A map of P1 spectra f ∶ E → F is said to be a motivic stable weak equiva-

lence if the induced map

f∗ ∶ πst
s,t(E)→ πst

s,t(F )

is an isomorphism of sheaves.

Proposition 2.4.12. The category SpP1(S) together with

1. Weak equivalences the motivic stable weak equivalences

2. Cofibrations the levelwise cofibrations of Definition 2.4.9

forms a proper simplicial model category.

Proof. This is [Jar00, Theorem 9.2].

Definition 2.4.13. We define the stable motivic homotopy category SH(S) to be the localiza-

tion of SpP1(S) with respect to the model structure of Proposition 2.4.12

SH(S) ∶= SpP1(S)[(P1 ∧ −)−1]

This is often treated as “the” motivic stable homotopy category; we will sometimes use the nota-

tion SHP1(S) if we need to compare to the construction of the previous section.

Proposition 2.4.14. There is an adjunction

SHA1

s (S) SHP1(S)
ω∞

σ∞

71



which gives an equivalence of stable homotopy categories,

SHA1

s (S)[G−1m ] SH(S)≃

Proof. This is a simple observation by construction of the two categories and the stated adjunc-

tion. This can be found, for example, in [ABH23].

Remark 2.4.15. With the above proposition we have shown that the two constructions give us

an equivalent motivic stable homotopy category. It is standard to work with the P1−construction.

Remark 2.4.16. In the interest of simplifying notation and presentation, it is often written that

H(S)[[P1]−1] = SH(S). By this we of course mean passing to P1−spectra, and then formally

inverting suspension. We get the standard adjunction

H(S) SH(S).
Σ∞

P1