We construct models for the motivic homotopy category based on simplicial
functors from smooth schemes over a field to simplicial sets. These spaces are
homotopy invariant and therefore one does not have to invert the affine line in
order to get a model for the motivic homotopy category

Herrmann, Philip

Strunk, Florian

English

arXiv

AbstractWe construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category

Elsevier - Publisher Connector 

Journal of Pure and Applied Algebra 215 (2011) 1663–1668
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra
journal homepage: www.elsevier.com/locate/jpaa
Enriched simplicial presheaves and the motivic homotopy category
Philip Herrmann ∗, Florian Strunk
University of Osnabrück, Mathematics Institute, 49069 Osnabrück, Germany
a r t i c l e i n f o
Article history:
Received 22 July 2010
Received in revised form 29 September
2010
Available online 30 October 2010
Communicated by C.A. Weibel
MSC: 14F42
a b s t r a c t
We construct models for themotivic homotopy category based on simplicial functors from
smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and
therefore one does not have to invert the affine line in order to get a model for the motivic
homotopy category.
© 2010 Elsevier B.V. All rights reserved.
0. Introduction
In this paper, we study certain simplicial functors as an alternative for simplicial presheaves in the construction of the
motivic homotopy category. An enriched simplicial presheaf is a simplicial functor from a category of schemes enriched
over simplicial sets to the category of simplicial sets enriched over itself. Considering enriched simplicial presheaves instead
of simplicial presheaves seems to be quite natural in the spirit of motivic homotopy theory. For example there is a naive
homotopy contracting the affine line in the category of schemes. More precisely, for any constant map c there exists a
morphism H of smooth schemes over a field, such that the diagram
A1
i0 #G
GG
GG
GG
GG c
&
A1 × A1 H / A1
A1
id
9
i1
;wwwwwwwww
commutes. The simplicial presheaf represented by A1 resists from being weakly equivalent to the point until it is finally
forced to be weakly contractible by Bousfield localization. In contrast to this the enriched simplicial presheaf represented
byA1 is objectwise contractible (cf. Corollary 1.5). Hence themotivic models based on these spaces can be obtained without
the A1-contracting Bousfield localization.
Conventions
Throughout this paper let k be a field and Sm/k the category of smooth schemes of finite type over k. The category of
simplicial (Set-valued) presheaves on Sm/k is denoted by sPre. The results of this paper apply also for Sm/S with a more
general base scheme S, e.g. a noetherian scheme of finite Krull dimension.
∗ Corresponding author.
E-mail addresses: pherrman@math.uos.de (P. Herrmann), fstrunk@math.uos.de (F. Strunk).
0022-4049/$ – see front matter© 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jpaa.2010.10.002
1664 P. Herrmann, F. Strunk / Journal of Pure and Applied Algebra 215 (2011) 1663–1668
1. The category of enriched simplicial presheaves
In this section we introduce the category SPre of enriched simplicial presheaves as an alternative for the category sPre of
simplicial presheaves. The construction of SPre is based on categories enriched over simplicial sets. In a simplicial category
C there are hom-simplicial sets sSetC(A, B) instead of just hom-sets associated with any two objects, in a way compatible
with an associative and unital composition. The 0-simplices of sSetC(A, B) can be thought of as morphisms A → B. The
relation of being connected by a zig-zag of 1-simplices, models a notation of naive homotopy depending on the enrichment.
In the following we consider the category sSet of simplicial sets as a simplicial category by
sSetsSet(A, B)n = homsSet(A×∆n, B).
The naive homotopy relation turns out to be pretty sensible in the sense that it coincides with a notation of left homotopy
in the usual model structure on simplicial sets. This enrichment is natural in many aspects, for example it is given by the
Yoneda embedding and the following straightforward lemma.
Lemma 1.1. Let C be a category with finite products. Any cosimplicial object c : ∆ → C with c0 the terminal object of C gives
rise to a simplicial category, which we also denote by C, with underlying category C and
sSetC(A, B)n = homC(A× cn, B).
Proof. A map σ : [m] → [n] in∆ induces a map sSetC(A, B)n → sSetC(A, B)m by assigning the composite
A× c([m]) (pr1,c(σ )◦pr2)−−−−−−−→ A× c([n]) f−→ B
to f ∈ sSetC(A, B)n. Clearly sSetC(A, B)(id[n]) = idsSetC (A,B)n and one observes that for composable morphisms σ and τ in∆
the identity
sSetC(A, B)(τ ◦ σ)(f ) = f ◦ (pr1, (c(τ ◦ σ) ◦ pr2))
= sSetC(A, B)(σ ) ◦ sSetC(A, B)(τ )(f )
holds and hence sSetC(A, B) is in fact a simplicial set. The composition maps
cABC : sSetC(B, C)× sSetC(A, B)→ sSetC(A, B), (g, f ) → g ◦ (f , pr2)
are maps of simplicial sets and satisfy the relevant coherence diagrams [1, 6.9,6.10]. The underlying category UC has by
definition the same objects as C and the hom-sets are given by
homUC(A, B) := homsSet(∆[0], sSetC(A, B))
∼= sSetC(A, B)0
∼= homC(A, B).
The composition in UC is the same as the composition in simplicial dimension 0 of the enriched category and therefore
UC ∼= C. 
By applying this lemma to the algebraic cosimplicial object ∆∆(-) given by
∆∆ p = Spec k[X0, . . . , Xp]/

1−
−
Xi

one obtains Sm/k as a simplicial category.
Definition 1.2. The category SPre of enriched simplicial presheaves is the category of simplicial functors from Sm/kop to sSet,
i.e. functors X assigning a simplicial set XU to any smooth k-scheme U and a morphism
sSetSm/k(U, V )→ sSetsSet(XV , XU)
of simplicial sets to any pair of objects U, V compatible with composition.
Lemma 1.3 (Adjunction Lemma). Let D be an essentially small category, C a cocomplete category and c : D → C a functor.
There exists a commutative diagram
D
Yoneda /
c
#F
FF
FF
FF
FF
Pre(D)
|−|

C
and an adjunction |−| : Pre(D)  C : Sing with Sing(X) = hom(c(−), X).
Proof. This is a standard fact about left Kan extensions [7, Theorem I.5.2]. 
P. Herrmann, F. Strunk / Journal of Pure and Applied Algebra 215 (2011) 1663–1668 1665
The Adjunction Lemma 1.3 applied to the functor
c : Sm/k×∆→ SPre, (U, [n]) → sSetSm/k(−,U)×∆n
provides an adjunction
L : sPre  SPre : R. (1.1)
The composite functor RL is well known and was already studied in [9] as a functor called Sing, defined by
Sing(X)(U)m = homPre(U ×∆∆m, Xm).
Lemma 1.4. The functors RL and Sing coincide.
Proof. Since the functors R, L and Sing preserve colimits we only need to check their behavior on representable objects.
RL(U ×∆n)(V , [m]) = homSPre(sSetSm/k(−, V )×∆m, sSetSm/k(−,U)×∆n)
∼= sSetSPre(sSetSm/k(−, V ), sSetSm/k(−,U)×∆n)m
∼= homSm/k(V ×∆∆m,U)×∆nm∼= U(V ×∆∆m)m ×∆nm∼= homPre(V ×∆∆m,Um)×∆nm∼= Sing(U ×∆n)(V )m. 
Corollary 1.5. The enriched simplicial presheaf represented by the affine line is objectwise contractible.
Proof. As a corollary of Lemma 1.4 we obtain
A1(U) = sSetSm/k(U,A1) = LA1(U)
= RLA1(U) = Sing(A1)(U)
which is contractible by [9, Corollary 3.5]. 
Lemma 1.6. The category of enriched simplicial presheaves is bicomplete and colimits and limits can be computed objectwise.
Proof. The category SPre is the underlying category of a sSet-category in which all weighted sSet colimits and limits exist
[1, Proposition 6.6.17], so SPre is bicomplete by [1, Proposition 6.6.16]. 
We use the conventional terminology and say that a set I of morphisms in a category permits the small object argument,
if the domains of the elements of I are small relative to transfinite compositions of pushouts of elements in I .
Lemma 1.7. Let I be a set of morphisms in sPre. Then the set LI of morphisms in SPre permits the small object argument.
Proof. We make use of the fact that all objects in the locally presentable category sPre are small. So there exists a cardinal
κ , such that for all κ-filtered ordinals λ and any λ-sequence S : λ→ SPre the following diagram commutes.
colim
β<λ
homSPre(LX, Fβ)
∼=

Φ / homSPre(LX, colim
β<λ
Fβ)
∼=

colim
β<λ
homsPre(X, RFβ)
∼= / homsPre(X, colim
β<λ
RFβ)
Hence LX is small and LI permits the small object argument. 
2. Model structures for enriched simplicial presheaves
In this sectionwe construct model structures on the category SPre of enriched simplicial presheaves. Theorem 2.4 shows
that these models are Quillen equivalent to model structures for the motivic homotopy category on simplicial presheaves.
Subsequently, Corollary 2.10 gives a characterization of the fibrant objects.
Definition 2.1. Let C andD be a model categories and L : C  D : R an adjunction. The model structure onD is called
R-lifted if amorphism f ofD is aweak equivalence (resp. a fibration) if and only ifR(f ) is aweak equivalence (resp. a fibration)
of C. A cofibrantly generated model category C is called (I, J)-cofibrantly generated if I is a set of generating cofibrations and
J is a set of generating acyclic cofibrations for the model structure on C.
Remark 2.2. IfC is amodel category, L : C  D : R an adjunction andD is equippedwith theR-liftedmodel structure, then
the adjunction (L, R) is necessarily a Quillen adjunction since the right adjoint R preserves fibrations and acyclic fibrations.
The lifted model structure onD is right proper if and only if C is a right proper model category.
1666 P. Herrmann, F. Strunk / Journal of Pure and Applied Algebra 215 (2011) 1663–1668
Lemma 2.3 (Lifting Lemma). LetC be a (I, J)-cofibrantly generatedmodel category,D a bicomplete category and L : C  D : R
an adjunction such that the right adjoint R commutes with colimits and LI and LJ permit the small object argument. Then there
exists a unique (LI, LJ)-cofibrantly generated R-liftedmodel structure onD if and only if for every j ∈ J and every pushout diagram
L(A) /
L(j)

X
p

L(B) / Y
the morphism R(p) is a weak equivalence of C.
Proof. This is a standard lifting argument [5, Theorem 11.3.2]. 
Theorem 2.4. Consider the adjunction
L : sPre  SPre : R
constructed in Eq. (1.1). Let sPre be equipped with a cofibrantly generated model structure with A1-local weak equivalences as
weak equivalences andwith the property that every cofibration is in particular amonomorphism. Then the R-liftedmodel structure
on SPre exists and the adjunction (L, R) is a Quillen equivalence.
Proof. Let I be a set of generating cofibrations and J be a set of generating acyclic cofibrations for the model structure on
sPre, j an element of J and
L(A) /
L(j)

X
p

L(B) / Y
be a pushout diagram in SPre. Since R commutes with colimits, the diagram
RL(A) /
RL(j)

R(X)
R(p)

RL(B) / R(Y )
is also a pushout. The morphism j is an acyclic cofibration of sPre and therefore in particular an acyclic cofibration in the
A1-local injective model structure on sPre, that is a A1-local weak equivalence and a monomorphism. Lemma 1.4 identifies
the functor RLwith the singular functor Sing. The singular functor respectsmonomorphisms andA1-localweak equivalences
by [9, Corollary 3.8]. Therefore RL(j) is an acyclic cofibration in the A1-injective model structure on sPre. The class of acyclic
cofibrations of a model category is closed under pushouts and hence R(p) is a A1-local weak equivalence. The category SPre
is bicomplete by Lemmas 1.6 and 1.7 provided that LI and LJ permit the small object argument. Hence the category SPre
can be equipped with the R-lifted model structure by Lemma 2.3. To prove that (L, R) is a Quillen equivalence, let η be the
unit of the adjunction (L, R) and let X be a simplicial presheaf. Lemma 1.4 identifies η(X) with the canonical morphism
X → Sing(X)which is a A1-local weak equivalence by [9, Corollary 3.8]. The diagram
X
η(X) /
f ♯ !C
CC
CC
CC
C RL(X)
R(f )

R(Y )
shows that a morphism f : LX → Y is a weak equivalence if and only if its adjoint f ♯ is a weak equivalence. Therefore (L, R)
is a Quillen equivalence. 
Remark 2.5. The assumptions on the model structure on sPre of Theorem 2.4 are fulfilled by all intermediate model
structures, e.g. the projective, flasque and injective model structures.
Lemma 2.6. Consider the adjunction L : sPre  SPre : R and let (sPre,×) be equipped with a monoidal model structure. If the
category (SPre,×) is endowed with the R-lifted model structure, then it is a monoidal model category.
Proof. General results on enriched category theory imply that SPre is cartesian closed [2]. Let i : A → B and j : C → D be
cofibrations. One has to show that the pushout product
i j : (B× C)

(A×C)
(A× D)→ B× D
P. Herrmann, F. Strunk / Journal of Pure and Applied Algebra 215 (2011) 1663–1668 1667
is a cofibration and an acyclic cofibration if i or j is a weak equivalence. This follows from the property of L being a left Quillen
functor and from the relation L(i j) ∼= L(i)L(j) holding as the functor L is strong monoidal, which is the case since
L(X × Y ) = L(colim(hom(−,U)×∆n)× colim(hom(−, V )×∆m))
= L(colim(hom(−,U × V )×∆n ×∆m))
= colim(sSet(−,U × V )×∆n ×∆m)
= colim(sSet(−,U)×∆n)× colim(sSet(−, V )×∆m)
= L(X)× L(Y ). 
Lemma 2.7. Consider the adjunction L : sPre  SPre : R and let sPre be equipped with a simplicial model structure. If the
category of enriched simplicial presheaves is endowed with the R-lifted model structure, then it is a simplicial model category.
Proof. The category SPre is naturally enriched over the category of simplicial sets by
sSetSPre(X, Y ) = homSPre(X ×∆(-), Y ).
It is tensored with X ⊗ A = X(−)× A and cotensored with XA = homsSet(A×∆(-), X(−)). Let i : A → B be a cofibration in
SPre and j : C → D a cofibration of simplicial sets. It is equivalent to the (SM7) axiom [4, II.3.11] to show that
(B⊗ C)

(A⊗C)
(A⊗ D)→ B⊗ D
is a cofibration and an acyclic cofibration if i or j is a weak equivalence. This has already been observed in the proof of
Lemma 2.6. 
Lemma 2.8. Every enriched simplicial presheaf X is homotopy invariant, that is the map
X(U)→ X(U × A1)
induced by the projection is a weak equivalence of sSet for all objects U of Sm/k.
Proof. An enriched simplicial presheaf X maps a morphism f : U → V of Sm/k to a 0-simplex of the simplicial set
sSet(XV , XU) and it maps a naive homotopyH : U×∆∆ 1 → V of Sm/k to a 1-simplex of sSet(XV , XU), which is a homotopy
equivalence of the simplicial sets XV and XU with respect to the cylinder object ∆1. Therefore X takes naive homotopy
equivalences in Sm/k to weak equivalences in sSet. The assertion is obtained from the fact that the affine line A1 is naive
homotopy equivalent to the point Spec (k) in Sm/k where a homotopy equivalence is given by the map k[X] → k[X, Y ],
X → XY of k-algebras. 
Corollary 2.9. Let SPre be equipped with a simplicial model structure in which every object of Sm/k is cofibrant. Then the class
C = {U × A1 pr−→ U | U ∈ Sm/k}
consists of weak equivalences.
Proof. Lemma 2.8 provides that sSet(U, X)→ sSet(U × A1, X) is a weak equivalence of simplicial sets for every enriched
simplicial presheaf X by an enriched version of the Yoneda Lemma. Weak equivalences in a simplicial model category
are detected by the property of the above morphism being a weak equivalence of simplicial sets for all fibrant objects X
[5, Corollary 9.7.5]. 
Corollary 2.10. Consider the adjunction L : sPre  SPre : R and the class
C = {U × A1 pr−→ U | U ∈ Sm/k}
of morphisms of simplicial presheaves. Let sPre be equipped with a Bousfield localized model structure LC (sPre) in which every
object of Sm/k is cofibrant. Suppose that the R-lifted model structure on SPre exists. Then an object X of SPre is fibrant if and
only if the object R(X) is fibrant in sPre before localizing.
Lemma 2.11. Consider the adjunction L : sPre  SPre : R and let sPre be equipped with a left proper cofibrantly generated
model structurewithA1-local weak equivalences asweak equivalences andwith the property that every cofibration is in particular
a monomorphism. If the category of enriched simplicial presheaves is endowed with the R-lifted model structure, then it is a left
proper model category.
Proof. It is sufficient to show that the R-lifted A1-local injective model structure is left proper. The injective model
structure on SPre is left proper and it is the R-lifted model of the injective structure on sPre [6, Proposition B.1]. Let B
be a class of cofibrations in sPre, such that the localization at B is the local injective model structure. Then (L, R) is a Quillen
adjunction between the local injective model on sPre and the localizationM of the injective model structure on SPre at L(B)
[5, Theorem 3.3.20].We show thatM coincides with the R-liftedA1-local injectivemodel structure on SPre. Let the injective
model structure on sPre be (I, J)-cofibrantly generated, then the injective model structure on SPre is (LI, LJ)-cofibrantly
1668 P. Herrmann, F. Strunk / Journal of Pure and Applied Algebra 215 (2011) 1663–1668
generated and so is its left Bousfield localizationM . By the same arguments, the R-lifted A1-local injective model structure
on SPre is also (LI, LJ)-cofibrantly generated. Hence both model structures have the same cofibrations. Moreover, their
fibrant objects coincide by Corollary 2.10 and the fact that an object X is fibrant in the Bousfield localizationM if and only if
sSet(−, X)maps B toweak equivalences. Therefore themodel structures are the same since amodel structure is determined
by its cofibrations and its fibrant objects. 
Acknowledgements
We would like to thank Prof. Oliver Röndigs for his support and many helpful discussions. Enriched objects in a stable
motivic setting were considered by Dundas et al. in [3]. The work was supported by the Graduate School ‘Combinatorial
Structures in Algebra and Topology’ at the University of Osnabrück.
References
[1] Francis Borceaux, Handbook of categorial algebra 2 - categories and structures, in: Encyclopedia ofMathematics and its Applications, vol. 51, Cambridge
University Press, 1994.
[2] Brian Day, On closed categories of functors, in: Reports of the Midwest Category Seminar, IV, in: Lecture Notes in Mathematics, vol. 137, Springer,
Berlin, 1970, pp. 1–38.
[3] Bjørn Ian Dundas, Oliver Röndigs, Paul Arne Østvær, Motivic functors, Documenta Mathematica 8 (2003) 489–525.
[4] Paul Gregory Goerss, John Frederick Jardine, Simplicial homotopy theory, in: Progress in Mathematics, vol. 174, Birkhäuser, 1999.
[5] Philip Steven Hirschhorn, Model categories and their localizations, in: Mathematical Surveys and Monographs, American Mathematical Society, 2003.
[6] Jacob Lurie, Higher Topos Theory, in: Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009.
[7] Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic. Universitext, Springer, 1992.
[8] Fabien Morel, On the motivic π0 of the sphere spectrum, Axiomatic, enriched and motivic homotopy theory (2004) 219–260.
[9] Fabien Morel, Vladimir Voevodsky, A1-homotopy theory of schemes, Publications Mathématiques de l’Institut des Hautes Études, Scientifiques 90
(1999) 45–143.


Enriched Simplicial Presheaves and the Motivic Homotopy Category

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector