Model Category TheoryWolfson LecturesJanuary 4-6, 2006

University of Manchester

Brooke ShipleyUniversity of Illinois at Chicago

This material comes from many sources (in particu-lar: Quillen, Dwyer-Spalinski, Hovey, Goerss).

These slides are full of small lies, some of which areintentional.

1

Topic I: Definitions and Examples

First Example: Homological AlgebraAn R-module P is projective if and only if givenf : A → B surjective and g : P → B there is a liftl : P → A such that fl = g:

0 //

��

Af����

P

l>>

g// B

We say 0→ P has the left lifting property withrespect to all surjections f .

Similarly, f is surjective if and only if for everyprojective P (and g : P → B) a lift l exists suchthat fl = g.

We say f has the right lifting property (RLP)with respect to any map 0→ P with P projective.

Chain complexes: ChR, non-negatively gradedchain complexes of R-modules.

Lifting Property: Assume i : A. → B. is amonomorphism such that each Bk/Ak is projectiveand p is surjective in each degree k > 0:

A.f

//��

i��

X.p

����

B.

l<<

g// Y.

then a lift l exists (with li = f and pl = g) if eitheri or p is a quasi-isomorphism (i.e., induces anisomorphism in homology).

Factorization: Any map in ChR, f : X. → Y.factors in two ways:

(1) X.i

� Y.′p� Y. where i is a monomorphism with

projective cokernels and p is a quasi-isomorphism andsurjective.

(2) X.j

� X.′q

� Y. where j is a monomorphismwith projective cokernels and a quasi-isomorphismand q is surjective.

Second Example: Topological Spaces, Top

The analogues of projective objects and surjectivemaps are CW complexes and fibrations.

Definition: A map of spaces p : X → Y is a Serrefibration if and only if p has the right lifting prop-erty with respect to inclusions i : A× 0→ A× [0, 1]for each CW complex A.

A× 0 //

i��

Xp

����

A× [0, 1]

l99

// Y

Definition: Suppose given a direct system of in-clusions of spaces X0 → X1 → · · · → Xn → · · ·such that each pair (Xn+1, Xn) is a relative CW pair.Then we say the map X0 → colimn Xn is a gener-alized relative CW inclusion.

Lifting Property: Assume i : A→ B is a retractof a generalized relative CW inclusion and p : X →Y is a Serre fibration.

A //��

i��

Xp

����

B

l>>

// Y

then a lift l exists if either i or p is a weak equiv-alence (induces an isomorphism on homotopy).

Factorization: Any map in Top, f : X → Y fac-tors in two ways:

(1) Xi

� Y ′p� Y where i is a generalized relative

CW inclusion and p is a weak equivalence and a Serrefibration.

(2) X.j

� X ′q

� Y. where j is a generalized relativeCW inclusion and a weak equivalence and q is a Serrefibration.

Definition: A model category is a category Cwith three distinguished classes of maps:(1) weak equivalences (

∼−→)(2) cofibrations (�)(3) fibrations (�)

each closed under composition and containing theidentity maps and subject to the following axioms.

An acyclic cofibration (∼�) is a cofibration which

is a weak equivalence.

An acyclic fibration (∼−�) is a fibration which is

a weak equivalence.

Axioms:

M1. C is closed under finite limits and colimits;

M2. (2 out of 3) If f and g are composable mapssuch that any two of the three maps f, g, gf are weakequivalences, then so is the third;

M3. (Retracts) The three distinguished classes ofmaps are closed under retracts;

M4. (Lifting) A lift l exists in every diagram

A //

i��

Xp

����

B

l>>

// Ywhere i is a cofibration, p is a fibration and i or p isa weak equivalence.

M5. (Factorization) Any map f can be factored intwo ways:(1) f = pi, where i is a cofibration and p is an acyclicfibration, and(2) f = qj where j is an acyclic cofibration and q is afibration.

Remarks:

M4 and M5 both have two parts.

M1 and M5 have variations.

Cop is also a model category.

By MC1, any model category has an initial object ∅and a terminal object ∗.

Definitions:

An object X is cofibrant if ∅ → X is a cofibration;Y is fibrant if Y → ∗ is a fibration.

A cofibrant replacement cX exists for any Xby the factorization axiom: ∅ → X factors as ∅ �cX

∼−� X .

Similarly a fibrant replacement exists for any Y :

due to the factorization Y∼� fY � ∗.

Lemma: The three classes of maps are not inde-pendent:

A map is a cofibration if and only if it has the LLP(left lifting property) with respect to any acyclic fi-bration.

A map is an acyclic cofibration if and only if it hasthe LLP with respect to any fibration.

Similarly, the fibrations and acyclic fibrations can bedefined using RLPs (right lifting properties).

Examples:

Projective model category: ChprojR

(1) The weak equivalences are the quasi-isomorphisms.(2) The fibrations are the maps which are surjectivein positive degrees.(3) The cofibrations are the monomorphisms withlevelwise projective cokernels.

All complexes are fibrant here. The cofibrant replace-ment of a module is a projective resolution.

Injective model category: ChinjR

(1) The weak equivalences are the quasi-isomorphisms.(2) The fibrations are the maps which are surjectivewith levelwise injective kernels.(3) The cofibrations are the monomorphisms.

All complexes are cofibrant here. The fibrant replace-ment of a module is an injective resolution.

Two different model structures on the same underly-ing category with the same weak equivalences.

Next, two different model structures on the same cat-egory with different weak equivalences.

Weak equiv. model structure: Topw.e.

(1) Weak equivalences the maps inducing isomor-phisms in homotopy.(2) Fibrations the Serre fibrations.(3) Cofibrations the retracts of generalized relativeCW inclusions.

Homotopy equiv. model structure: Toph.e.

(1) Weak equivalences the homotopy equivalences.(2) Fibrations the Hurewicz fibrations.(3) Cofibrations the closed Hurewicz cofibrations.

A map p : X → Y is a Hurewicz fibration if phas the LLP with respect to A× 0→ A× [0, 1] forevery space A.

An inclusion of a closed subspace i : A → B is aclosed Hurewicz cofibration if a lift exists inevery diagram below for every space Y :

B × 0 ∪ A× [0, 1] //

i��

Yp

��B × [0, 1]

l66mmmmmmmmmmmmmmmm

//∗

Topic II: Brief introduction to sSet

A simplicial set X. is a sequence of sets Xn withface maps di : Xn → Xn−1 and degeneracy mapssj : Xn → Xn+1 for 0 ≤ i, j ≤ n such that certainsimplicial identities hold among composites of thesemaps.

Example: The standard n-simplex ∆[n] has(∆[n])q = {(a0, · · · , aq)|0 ≤ a0 ≤ · · · ≤ aq ≤ n}with di(a0, · · · , aq) = (a0, · · · , ai−1, ai+1, · · · aq)and sj(a0, · · · aq) = (a0, · · · , aj, aj, · · · aq).

∆[n] has exactly one non-degenerate n-simplex:(0, 1, · · · , n) = ιn. Every other simplex is the im-age of ιn under some composite of di and sj maps.

Each n-simplex of a simplicial set X. corresponds toa map from ∆[n]: Xn = sSet(∆[n], X.).

Formally, the category of simplicial sets is the cate-gory of contravariant functors from a category ∆ tosets.

sSet = (Set)∆op

∆ has objects [n] = {0, 1, · · · , n} and morphismsthe (weakly) order preserving maps (which are allcomposites of maps di, which skips i, and sj, whichrepeats j).

Examples: We see that ∆[n] = ∆(−, [n]) and di

in ∆ induces a map di : ∆[n− 1]→ ∆[n].

Define the boundary ∂∆[n] ⊆ ∆[n]:

∂∆[n] = ∪0≤i≤ndi∆[n− 1]

and the horn ∆k[n] ⊆ ∆[n]:

∆k[n] = ∪i 6=kdi∆[n− 1]

Adjoint functors: | − | : sSet � Top : Sing

There are topological standard n-simplices σn withmaps di : σn−1 → σn and sj : σn+1 → σn whichsatisfy the dual of the simplicial identities.

Define the geometric realization of a simplicialset X. by |X.| = (∪nXn × σn)/(dix, u) ∼ (x, diu).

|X.| is a CW-complex with one n-cell for each non-degenerate n-simplex of X . (E.g. |∆[n]| = σn.)

The right adjoint of | − | : sSet → Top is the sin-gular set functor Sing : Top→ sSet.

(Sing(X))n = Top(σn, X)

Model category for sSet:(1) A map f is a weak equivalence if |f | is a weakequivalence of spaces.(2) The cofibrations are the monomorphisms.(3) The fibrations are the maps with the RLP withrespect to ∆k[n]→ ∆[n], for all k, n.

In fact, i is a cofibration in sSet if and only if |i| is acofibration in Top.

Dually, Sing p is a fibration in sSet if and only if p isa Serre fibration in Top.

The acyclic fibrations are the maps with the RLPwith respect to ∂∆[n]→ ∆[n] for all n.

Topic III: The Homotopy Category

Definition: For the homotopy category Ho(C)

objects(Ho(C)) = objects(C)

Ho(C)(X, Y ) = {X → X1∼←− X2 → · · ·

∼←− Y }/ ∼

Example:Ho(Chproj

R )(M, N) ∼= ChR(P.(M), N)/ ∼ (chain homotopy)

where P.(M) is a projective resolution of M (or acofibrant replacement).

Chain homotopy: Let I = R[1] ⊕ R[0] ⊕ R[0]with boundary ∂(x, a, b) = (0, x,−x).(P. ⊗ I) = ΣP. ⊕ P. ⊕ P. with a natural inclusioni : P.⊕ P.→ P.⊗ I .

A chain homotopy between f, g : P. → Y. is amap H which completes the following diagram:

P.⊕ P. //

f⊕g ''OOOOOOOOOOOOO P.⊗ IH

��

Y.

Definition: Let A be an object in a model categoryC. A cylinder object for A is a factorization ofthe fold map ∇ = qi

A⊕ A // i //

∇ &&LLLLLLLLLLLLC(A)

q∼��

Asuch that i is a cofibration and q is a weak equiva-lence.

Example: In ChR, for P. cofibrant C(P.) = P.⊗I .In Top, for A a CW complex, C(A) = A× [0, 1].

Definition: Assume A is cofibrant. A left homo-topy between f, g : A→ Y is a diagram:

A∐

A //

f∐

g &&MMMMMMMMMMMC(A)

H��

Y.where C(A) is a cylinder object for A.

Definition: Let X be an object in a model categoryC. A path object for X is a factorization of thediagonal ∆ = pj

X ∼j

//

∆ ((PPPPPPPPPPPPPP XI

p����

X ×Xsuch that j is a weak equivalence and p is a fibration.

Examples: In ChR, (X.)I = HomChR(I, X.).

In Top, XI = X [0,1] is a path object.

Definition: Assume X is fibrant. A right homo-topy between f, g : B → X is a diagram:

XI

p����

B f×g//

H77

X ×X

where XI is a path object for X .

Lemma: If A is cofibrant and X is fibrant, thenf, g are left homotopic if and only if they are righthomotopic. We write f ∼ g.

For each object X in a model category C fix acofibrant and a fibrant replacement: cX and fX .

Theorem: The homotopy category of a model cat-egory C is the category with the same objects as Cand with

Ho(C)(X,Y ) ∼= [fcX, fcY ]C ∼= [cX, fY ]Cwhere [X,Y ]C = C(X,Y )/ ∼.

Lemma: (Dwyer-Kan)

Ho(C)(X,Y ) ∼= {X ∼←− X ′ → Y ′∼←− Y }/ ∼

where the equivalence relation is generated by dia-grams of the form

X1∼}}{{

{{{{

{

//

∼

��

Y1

∼

��

X Y

∼``@@@@@@@

∼~~~~~~

~~~

X2

∼aaCCCCCCC

// Y2

Topic IV: Derived Functors

A Quillen functor from C to D (two model cate-gories) is an adjoint pair of functors:

F : C � D : G

such that the left adjoint F preserves cofibrationsand acyclic cofibrations. (It follows that G preservesfibrations and acyclic fibrations.)

Ken Brown’s Lemma: Any left Quillen functorF takes weak equivalences between cofibrant objectsto weak equivalences. Dually, G takes weak equiva-lences between fibrant objects to weak equivalences.

Any Quillen functor F : C � D : G induces adjointtotal derived functors

LF : Ho(C) � Ho(D) : RG

defined by LF (X) = F (cX) and RG(Y ) = G(fY ).

Example: − ⊗ N is a left Quillen functor.TorRi (M, N) = Hi(L(−⊗R N)) = Hi(P.(M)⊗R N)

Example: Homotopy colimits, homotopy limits.

Definition: A Quillen functor F is a Quillenequivalence if and only if the induced adjoint pairLF : Ho(C) � Ho(D) : RG is an equivalence ofcategories.

Equivalently, F is a Quillen equivalence if for eachcofibrant A in C and each fibrant X in D a mapf : FA

∼−→ X is a weak equivalence in D if and onlyif its adjoint f ′ : A

∼−→ GX is a weak equivalence inC.

Example: There is a Quillen functor

id : ChprojR � Chinj

R : id

which is a Quillen equivalence.

Ho(ChprojR ) ∼= Ho(Chinj

R ) ∼= D(R)

Example: There is a Quillen functor

id : Topw.e. � Toph.e. : id

which is not a Quillen equivalence.

Ho(Topw.e.) = homotopy category of CW-complexes.Ho(Toph.e.) = homotopy category of all topologicalspaces.

Theorem: |−| : sSet � Topw.e. : Sing is a Quillenequivalence.

Theorem: ([Quillen], [B-G])

Ho(sSet1,Q,f)op ∼= Ho(CDGA1,Q,f)

CDGA1,Q = simply connected, finite type,commutative dg Q-algebras

sSet1,Q = simply connected, finite type, weakequivalences are H∗(−, Q) isomorphisms.

Finite type = each Hk(−, Q) is finitely generated.

Theorem: [Mandell ’01]

Ho(Top1,p−complete,f.p−type)op ∼=full subcat.Ho(E∞DGAFp

)

Theorem: [Mandell ’03]X

∼−→ Y if and only if C∗(X)∼−→ C∗(Y )

(for X , Y finite type, simply-connected.)

Ho(Top1,f)op faithful−−−−−→ Ho(E∞DGAZ)

Topic V: Cofibrantly generated model cat-egories

Example: Topw.e. is cofibrantly generated.(1) f is a fibration if and only if it has the RLP withrespect to J = {Dn × 0→ Dn × [0, 1]}.(2)f is an acyclic fibration if and only if it has theRLP with respect to I = {Sn−1 → Dn}.(3) Any (acyclic) cofibration is in I-cof (J -cof).

I-cell denotes the maps built from I using pushoutsand possibly infinite compositions (colimits).I-cof denotes the retracts of maps in I-cell.

Example: In ChprojR , let Sn = R[n] and Dn =

R[n+1]⊕R[n] with dn+1 = id. Then (1)-(3) abovehold with I = {Sn−1 → Dn} and J = {0→ Dn}.

A is sequentially small if there is a bijection.

colimn

C(A, Bn)→ C(A, colimn

Bn)

In ChR, Sn and Dn are sequentially small. In Top,Sn and Dn are sequentially small with respect tocofibrations (each Bn � Bn+1 is a cofibration).

Small object argument:Suppose the domains ofK are sequentially small with respect to maps in

K-cell. Given any map Xf−→ Y , there is a functorial

factorization Xi∞(f)−−−→ Z

p∞(f)−−−→ Y such that i∞(f )is in K-cell and p∞(f ) has the RLP with respect toK.

Proof: We’ll inductively construct a sequence

X = Z0i0 //

f=p0 %%KKKKKKKKKKKZ1

i1 //

p1��

Z2//

p2}}{{{{

{{{

· · · // Z = colim Zn

p∞ssffffffffffffffffffffffffffff

YAssume Zn, pn have been constructed. Let Sn be theset of commutative squares with k ∈ K:

A //

k��

Znpn

��

B // YDefine Zn+1 to be the pushout in the digram below∐

s∈SnAs //

ks ��

Zn

in��∐

s∈SnBs // Zn+1

The map pn+1 is induced by the map pn.

Define i∞(f ) : X → Z to be the composition of themaps in and p∞(f ) = colim pn. It follows that i∞(f )is in K-cell.

Check that p∞(f ) has the RLP w.r.t. K: Given

Ag

//

k��

Zp∞(f)

��

B h// Y

Since A is sequentially small w.r.t. K-cell, g factorsthrough some Zj

Ag′

//

k��

Zj//

pj��

Zj+1//

{{xxxxxxxxZ

p∞uulllllllllllllllllllll

B h// Y

Since the square on the left is in Sj, by constructionthere is a map l : B → Zj+1. Composing l withZj+1 → Z provides a lift in the original square above.�

Remark: This is used to verify the factorizationaxiom, and to set-up inductive proofs for cofibrantobjects.

Definition: A model category C is cofibrantlygenerated if there are sets I and J such that:(1) The domains of I are small w.r.t. I-cell.(2) The domains of J are small w.r.t. J -cell.(3) The fibrations are the maps with the RLP w.r.t.J .(3) The acyclic fibrations are the maps with the RLPw.r.t. I .

Proposition: If C is cofibrantly generated withgenerating sets I and J , then:(1) The cofibrations are the maps in I-cof.(2) The acyclic cofibrations are maps in J -cof.

Example: For C = sSet,I = {∂∆[n]→ ∆[n]}n and J = {∆k[n]→ ∆[n]}k,n.

Example: As above, C = ChprojR and C = Topw.e..

Recognition Theorem: Let C be a category thatis closed under all limits and colimits and let W bea subcategory of C that is closed under retracts andsatisfies the “two out of three” axiom. If I and J aresets of maps in C, then C has a cofibrantly generatedmodel structure determined by W , I and J if andonly if the following are satisfied:(1) The domains of I are small w.r.t. I-cell;(2) The domains of J are small w.r.t. J -cell;(3)J -cofibrations are I-cofibrations and in W ;(4)Every map with the RLP w.r.t. I is in W and hasthe RLP w.r.t. J ; and(5)Either (a) any I-cofibration in W is a J -cofibration,or (b)any map in W with the RLP w.r.t. J has theRLP w.r.t. I .

A map is K-injective if it has the RLP w.r.t. K.

Define fibrations to be the J -injectives.Define cofibrations to be I-cofibrations.

Proof:

(3) and (5a) showJ -cof = I-cof ∩W= acyclic cofibrations.

(4) and (5b) showI-inj = W∩ J -inj = acyclic fibrations.

(1) and (2) give factorizations intoI-cofibration and I-injective andJ -cofibration and J -injective.

I-cofibrations have LLP w.r.t. I-injectives.

J -cofibrations have LLP w.r.t. J -injectives.

Localizations: One can use this theorem andthe Bousfield-Smith cardinality argument to verifythat certain localization model structures exist. (leftproper, combinatorial; left proper cellular)

Left localization enlarges W , cofibrations stay thesame, so fibrations decrease. Local objects are thenew fibrant objects.

For example, sSeth∗ (or sSetQ) is a cofibrantly gen-erated model category with W the h∗-equivalences(maps which induce isomorphisms in h∗ (or H∗(,Q))).Here I = IsSet = {∂∆[n] → ∆[n]}. Then J isa set of representatives of the isomorphism classes ofmonomorphisms f : A→ B that are h∗-equivalenceswhere A and B are of “size” less than some fixed car-dinal γ.

Motivic homotopy theory: (A1-homotopy the-ory) Morel and Voevodsky start with a model cat-egory on simplicial sheaves where the weak equiva-lences are the maps which induce weak equivalenceson all stalks. Then they localize with respect to mapsX × A1 → X .

Lifting model structures:Assume given an adjoint pair F : C � D : G with Fthe left adjoint and C a cofibrantly generated modelcategory. Make the following definitions:f in D is a fibration iff G(f ) is a fibration in C.f in D is a weak equivalence iff G(f ) is so in C.f in D is a cofibration iff it has the LLP w.r.t. theacyclic fibrations.

Lifting Lemma:[Crans; Schwede-Shipley; Berger-Moerdijk]This defines a lifted model structure on D if(1)F preserves small objects and(2)any map in F (JC)-cell is a weak equivalence in D.

Moreover, D is cofibrantly generated with ID =F (IC) and JD = F (JC).

Also, F and G are Quillen functors.

(1) holds if G preserves filtered colimits.

Applications: rings, algebras and modules oversymmetric spectra or gamma spaces

Quillen’s path object argument:[Quillen, II.4; Schwede-Shipley, A.3; Rezk 7.6]Recall: a path object for X is a factorization of thediagonal X

∼−→ XI � X ×X .

If (a) D has a fibrant replacement functor and(b) D has functorial path objects for fibrant objects,then condition (2) above holds (F (JC)-cell ⊂ W ).

If all objects are fibrant in D, then (a) automaticallyholds.

Applications: simplicial algebras, differential gradedalgebras, dg-modules, operads, algebras over operads

Differential graded algebras:Lifted model structure on DGA

T : ChprojR � DGA : U

(1) T (C) = R⊕ C ⊕ (C ⊗ C)⊕ · · · ⊕ C⊗n · · ·(2) U preserves filtered colimits.(3) All objects are fibrant.(4) A path object for A in DGA is given byHomChR

(I, A).

I = R〈ι1, [0]0, [1]0〉 with ∂ι = [1]− [0].I is a coassociative, counital coalgebra.(∆[0] = [0]⊗[0], ∆[1] = [1]⊗[1], ∆ι = [0]⊗ι+ι⊗[1].)Counit η : I

∼−→ R and two inclusions i0, i1 : R→ Iinduce maps A

∼−→ HomChR(I, A) � A× A.

Also DGAs over fixed commutative DGA C,DG-modules over a fixed DGA A

Not commutative DGAs. (I is not cocommutative.)Let S : Chproj

R → CDGA be the free symmetric al-gebra functor. If char R 6= 0, then S(0 → Dn)is not a quasi-isomorphism, so no such lifted modelcategory exists.

Topic VI: Simplicial model categories

Simplicial Categories:

Definition: The ordinal number category ∆has objects [n] = {0, 1, . . . , n} and morphisms theweakly order preserving maps φ : [n] → [m]. Anymap φ is a composition of maps di : [n − 1] → [n]which skips i and sj : [n + 1] → [n] which doublesup j.

Definition: The category of simplicial objectsin C, sC = C∆op

is the category of contravariant func-tors from ∆ to C.

(1) Action (or Tensor):sSet× sC→ sC

(K, X)→ K ⊗X

Simplicial sets acts on sC:K ∈ sSet, X ∈ sC, then (K ⊗X)n =

∐Kn

Xn.Associative: (K × L)⊗X ∼= K ⊗ (L⊗X)Unital: ∆[0]⊗X ∼= X

Example: If sC = sSet, (K ⊗X)n = Kn ×Xn.

(2) Cotensor:

sC× sSetop → sC

(Y, K)→ Y K

K ∈ sSet, Y ∈ sC, then Y K is determined by

sC(K ⊗X,Y ) ∼= sC(X, Y K).

(3) Enrichment:

sC× sCop → sSet

(X,Y )→ mapsC(X,Y )

For X,Y ∈ sC, define mapsC(X,Y ) ∈ sSet bymapsC(X,Y )n = sC(∆[n]⊗X,Y ).

Associative composition:mapsC(Y, Z)×mapsC(X, Y )→ mapsC(X,Z)

Since ∆[0]⊗X ∼= X , mapsC(X,Y )0 ∼= sC(X,Y ).

(4) Adjoint isomorphisms:

mapsC(K ⊗X,Y ) ∼= mapsC(X, Y K)∼= mapsSet(K, mapsC(X,Y )).

(1) - (4) together give sC the structure of a sim-plicial category (compatible tensor, cotensor andenrichment).

Examples: If sC = sSet, (K ⊗ X)n = Kn × Xn

and Y K = mapsSet(K, Y ).

If sC = sModR, (K ⊗ X)n = R[Kn] ⊗R Xn and(Y K) = mapsSet(K, Y ).

Top is also a simplicial category: define K ⊗ X =|K| × X and Y K = Y |K| the topological mappingspace. Then

mapTop(X,Y )n = Y ∆[n]⊗X

Model categories for simplicial categoriesIf C has an underlying set functor U , then we definethe following.

L : sSet � sC : U

f in sC is a fibration iff U(f ) is a fibration in sSet.f in sC is a weak equivalence iff U(f ) is so in sSet.f in sC is a cofibration iff it has the LLP w.r.t. theacyclic fibrations.

Proposition: This forms a cofibrantly generatedmodel category for sC when sC is the category of:(1) simplicial associative algebras(2) simplicial Lie algebras(3) simplicial groups(4) simplicial commutative algebras(5) simplicial R-modules

Proof: All objects are fibrant in each of these ex-amples. Also, X∆[1] forms a functorial path object.This is because of the interaction between the modelstructures and the simplicial structures.

X∆[0] = X∼−→ X∆[1] � X ×X = X∆[0]

∐∆[0]

p : ∆[1]∼−→ ∆[0] (between cofibrant objects)

and i : ∆[0]∐

∆[0] = ∂∆[1] � ∆[1]

Simplicial Model Categories:A model category C which is also a simplicial cate-gory is a simplicial model category if the fol-lowing equivalent axioms hold.

SM7 Axiom: Suppose j : A � B is a cofibrationand q : X � Y is a fibration in C, then

mapC(B, X) � mapC(A, X)×mapC(A,Y ) mapC(B, Y )

is a fibration of simplicial sets which is acyclic ifeither j or q is.

Pushout product axiom: SM7 holds if and onlyif for any cofibration i : K � L in sSet and anycofibration j : A � B in C the map

i � j : (A⊗ L) ∪(A⊗K) (B ⊗K) � B ⊗ L

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

Third equivalent axiom: SM7 holds if and onlyif for any cofibration i : K � L in sSet and anyfibration q : X � Y in C the map

XL � (XK)×(Y K) (Y L)

is a fibration which is acyclic if either i or q is.

All of the above listed simplicial categories are simi-plicial model categories.

Corollary: Let C be a simplicial model category.If X in C is fibrant, then X∆[1] is a natural pathobject. If X in C is cofibrant, then X ⊗ ∆[1] is anatural cylinder object.

X∐

X = ∂∆[1]⊗X � ∆[1]⊗X∼−→ ∆[0]⊗X ∼= X.

Corollary: If X is cofibrant and Y is fibrant in asimplicial model category C, then

π0mapC(X,Y ) = Ho(C)(X,Y ) = [X,Y ]C

Simplicial vs. Differential:

sC replaces differential graded objects in C as a placeto do homological algebra.

Theorem: (Dold-Kan equivalence)N : sModR � ChR : Γ

(N, Γ) is an equivalence of categories as well as aQuillen equivalence of model categories.

For X ∈ sModR, the normalized chain complexNX is given by moding out by the degneracies:

(NX)n = Xn/s0Xn−1 + · · · + sn−1Xn−1

with∂n = Σn

0(−1)ndi.

Then the inverse functor Γ just adds degeneraciesback in. For example,

Γ(C.)2 = C2 ⊕ s0C1 ⊕ s1C1 ⊕ s0s0C0.

There is another chain complex N ′X :

(N ′X)n = ∩ni ker di : Xn → Xn−1

with differential d0 such that N ′Xn → Xn → NXn

is an isomorphism (which takes d0 to ∂n above).

One can check (since X is fibrant):

πnX = [Sn, |X|]Topw.e.

∼= π0mapsSet(∆[n]/∂∆[n], X)∼= Hn(N

′X) ∼= Hn(NX)

Thus, N takes weak equivalences in sModR to quasi-isomorphisms in ChR.

N : sRing → DGA is a Quillen equivalence (butnot an equivalence of categories).

For simplicial commutative rings,N : sComm→ CDGA is not a Quillen equivalence(except in characteristic zero).

Quillen developed a notion of homology for anymodel category (given by derived functors of abelian-ization).

For sComm this produces Andre-Quillen coho-mology which is an important invariant.

Approximating simplicial model categories:

The Dwyer-Kan hammock localization (’80)defines for any model category C a category L(C, W )enriched over simplicial sets such that

π0mapL(C,W )(X,Y ) = Ho(C)(X,Y ) = [X,Y ]CIf C is a simplicial model category, then

mapL(C,W )(X,Y ) ∼ mapC(X,Y ).

More recently, for model categories with functorialfactorizations (e.g., cofibrantly generated model cat-egories), one can use “framings” to construct tensors,cotensors and enrichments with nice properties. (SeeHovey.)

Theorem: [Dugger ’01] If C is either left properand combinatorial or left proper and cellular, then Cis Quillen equivalent to a simplicial model categorystructure on sC.

Basic summary: One can usually assume one hasa simplicial model category.

Topic VII: Monoidal model categories

Definition: A monoidal model category is a

model category C with a monoidal product C×C⊗−→

C and unit I such that:

(1) Pushout product axiom:If i : A → B and j : X → Y are cofibrations in Cthen

i � j : (A⊗ Y ) ∪(A⊗X) (B ⊗X) � B ⊗ Y

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

(2) Let q : cI ∼−→ I be a cofibrant replacement forthe unit I. If X is cofibrant, then q⊗id : cI⊗X

∼−→I⊗X = X and id⊗ q : X ⊗ cI ∼−→ X ⊗ I = X areweak equivalences.

Examples:(1) sSet is a monoidal model category.(2) Chproj

R is a monoidal model category (R comm.).

(ChinjR is not an example.)

(3) SpΣ is a monoidal model category.(4) MS is a monoidal model category.

Proposition: If C is a monoidal model category,then Ho(C) is a monoidal category under the leftderived product ⊗L.

Lemma: Let C be cofibrantly generated with gen-erating sets I and J . If i � i′ is a cofibration for anyi, i′ ∈ I and if i � j is an acyclic cofibration for anyi ∈ I and j ∈ J , then the pushout product axiomholds in general.

Definition: Let C be a monodial model category.A C-model category is a model category D with an

action by C, D× C⊗−→ D, such that

(1) If i : A→ B is a cofibration in D and j : X → Yis a cofibration in C then

i � j : (A⊗ Y ) ∪(A⊗X) (B ⊗X) � B ⊗ Y

is a cofibration in D which is acyclic if either i or jis.(2) Let q : cI ∼−→ I be a cofibrant replacement for theunit I in C. If X is cofibrant, then id⊗q : X⊗cI ∼−→X ⊗ I = X is a weak equivalence.

Examples:(1) sSet-model categories = simplicial model cate-gories.(2) DG-ModA is a Chproj

R -model category for anyDGA A over R.(3) Chinj

R is a ChprojR -model category.

(4) ModR is a SpΣ-model category (or MS-modelcategory) for R a ring spectrum in SpΣ (or MS).

One can define rings, modules and algebras for anymonoidal category C. (For example: ChR, SpΣ, MS)

Quillen’s path-object argument doesn’t apply for lift-ing model categories over SpΣ.

SpΣ: no monoidal fibrant replacement functorFunctorial path objects do exist. (weak equivalencesand fibrations are lifted from SpΣ.)

Recall the lifting lemma: F : C � D : G ..... if anymap in F (JC)-cell is a weak equivalence in D.

Analyze pushouts over F (JC) in categories of mod-ules and rings.

Let J ∧ C denote the class of maps A∧Z → B ∧Zwith A→ B in J and Z any object in C.

Monoid axiom: A cofibrantly generated, monoidalmodel category satisfies the monoid axiom if anymap in (JC ∧ C)-cell is a weak equivalence.

Theorem: [Schwede-Shipley ‘00]Let C be a cofibrantly generated, monoidal modelcategory which satisfies the monoid axiom. (Assumeall objects in C are small.) Then there are cofibrantlygenerated lifted model structures for R-modules (Ra monoid) and for R-algebras (R a commuatativemonoid).

Moreover, if IC is cofibrant, then every cofibrant R-algebra is also cofibrant as an R-module.

Theorem: [SS ‘00]Assume IC is cofibrant and − ∧R N preserves weakequivalences for any cofibrant left R-module N .

(1)If f : R∼−→ S is a weak equivalence of monoids,

then−∧RS : ModR � ModS : U is a Quillen equiv-alence.(2)If f : R

∼−→ S is a weak equivalence of commu-tative monoids, then − ∧R S : AlgR � AlgS : U isa Quillen equivalence.

Verifying the monoid axiom:

To show that the monoid axiom holds in ChprojR we

use the fact that ChinjR is a Chproj

R -model category.(See proof below.)

The argument for SpΣ is similar, one uses the actionof a projective model structure on an injective modelstructure.

Proof: In ChinjR all objects are cofibrant. So, the

pushout product axiom implies that (A∼� B) �

(0 � Z) = A∧Z∼� B∧Z is an acylcic cofibration

in ChinjR whenever A

∼� B is an acyclic cofibration

in ChprojR . If J ∧ C consists of acyclic cofibrations

(in ChinjR ), then pushouts and colimits will also be

acyclic cofibrations (in ChinjR ) and hence also weak

equivalences.

Equivalences of monoidal model categories:[SS ‘03] If C and D are Quillen equivalent monoidalmodel categories with lifted model structures on theircategories of rings (monoids), when are RingC andRingD Quillen equivalent?

Definitions: R : (D,⊗) → (C,∧) is laxmonoidal if there is a map ν : ID → R(IC) andnatural associative and unital maps

φ : RX ∧RY → R(X ⊗ Y ).

If R has a left adjoint L then the lax monoidalstructure on R induces a lax comonoidal (op-lax monoidal) structure on L. Namely, there is amap ν : LID → IC and natural associative and uni-tal maps:

φ : L(A ∧B)→ LA⊗ LB

L is strong monoidal if φ and ν are isomorphisms.If L is strong monoidal, then R is lax monoidal.

Definitions: Let L : C � D : R be Quillen func-tors between monoidal model categories. If L isstrong monoidal and ID is cofibrant, L and R arecalled strong monoidal Quillen functors.

L and R are weak monoidal Quillen functorsif R is lax monoidal and the following two conditionshold:(1)φ : L(A∧B)

∼−→ LA⊗LB is a weak equivalencein C whenever A and B are cofibrant in D and(2) if cID

∼−→ ID is a cofibrant replacement, then

L(cID)→ L(ID)ν−→ IC is a weak equivalence in C.

Assume C and D are monoidal model categories suchthat there are lifted model structures on RingC andRingD.

Theorem:[SS ‘03] If L : C � D : R is a weak (orstrong) monoidal Quillen equivalence and IC and ID

are cofibrant, then L′ : RingC � RingD : R is aQuillen equivalence on the lifted model structures.

One also gets Quillen equivalences between respectivecategories of modules over C and over D (assumingthe lifted model structures exist).

Proof when L is strong monoidal (so L =L′):Since the model categories are lifted, weak equiva-lences and fibrations are determined on the underly-ing categories. Thus, R preserves weak equivalencesand fibrations and is a right Quillen functor.

To check that L and R form a Quillen equivalence, letA be cofibrant in RingC and B be fibrant in RingD

(and thus also fibrant in D). We need to show that

LA∼−→ B if and only if A

∼−→ RB.

Since IC is cofibrant, A is also cofibrant in C. SinceC and D are Quillen equivalent, this follows. �

For a weak monoidal Quillen pair, one shows thatLA

∼−→ L′A is a weak equivalence for A cofibrantin RingC.

Strong monoidal examples: There are strongmonoidal Quillen equivalences connecting all of thenew monoidal model categories of spectra: (sim-plicial and topological) symmetric spectra [HSS,MMSS], orthogonal spectra [MMSS], simplicial func-tors [Lyd98] and W-spaces [MMSS] S-modules [EKMM].

Weak monoidal example:

N : sModR � ChR : Γ

(N, Γ) is a weak monoidal Quillen equivalence whichis not strong. (N, Γ) are not monoidally adjoint(id→ ΓN is not monoidal).

Corollary: There is a Quillen equivalence betweenthe categories of connective differential graded k-algebras and simplicial k-algebras.

DGAk � s(Algk)

Stable extension:

CH � SpΣ(Ch) � SpΣ(s(Ab)) � HZ -Mod

a string of weak monoidal Quillen equivalences whichinduces a Quillen equivalence

DGA � HZ -Alg

University of Illinois at Chicago

E-mail address: bshipley@math.uic.edu