Model Categories by Example

Lecture 2

Scott Balchin

MPIM Bonn

???

⇤

BIFIBRANT

Recap

A model category is a category f with 3 classes ofMorpheus 1) small limits took

Ms

• weak Equus2) 2-out

. of-3 forweak Eglin

* fibration s

↳ Cogitations 3) retracts

Satisfied me , - Meg4) lifting prop

5) factorization

Recap

Introduced a homotopy relation XY .

8g. seeXny Xyy

£ Hoke ) fg.lv - Ecw")bijibrat

objects

↳ Top when

Cat Nat

A non-invertible weak equivalence

The pseudocircle is the topological space S whose underlying set is the quadruple

{x1, x2, x3, x4} whose open subsets are

{{x1, x2, x3, x4}, {x1, x2, x3}, {x1, x2, x4}, {x1, x2}, {x1}, {x2}, ∆}

x Ty

•

s. :S'-sis f y

x. Y

•

o? I.£,

Eu,

g.is continuous tweak hmpty Equiv . f : s

'

egg in Top@when

But the only Cts function g: 5→ s

'

is constant !

(Co)limits in the homotopy category

Proposition: Let C be a model category, then Ho(C) has all small products and coproducts.

However, in general Ho(C) does not have all small limits and colimits.

(Co)limits in the homotopy category

Let S1 = {z 2 C | |z| = 1} and f : S1 ! S1such that f (z) = z2

. Consider the span

S1f//

✏✏

S1

⇤ - .. p

in Ho (Topantlers ⇒ VX, CBI]=Hom

Picks out 2- torsion

Jaggi}ftj s 's BI Ess'

→ RP? Both) → Clp?Bs' Bt,

a

⇒ exact sequence CP , s'T→ CP, IRP.

] → Cp, Epg

⇒ exact o → Ik → O &segue

Comparing homotopy theories

idea Haie 2 model categories f ,D

.

Functor onthe underlying categories f

: f→ D

Question when do we get a"derived functor

"on the hptg

Categories ?

Fi Ho CE) → Ho CD )

Quillen functors

Let C and D be model categories. An adjoint pair F : C � D : U is a Quillen adjunction if:

• , F preserves cofibrations and acyclic cofibrations;

• , U preserves fibrations and acyclic fibrations;

• , F preserves cofibrations and U preserves fibrations;

• , F preserves acyclic cofibrations and U preserves acyclic fibrations.

Ken Brown’s Lemma: Given a Quillen adjunction F : C � D : U then

• F preserves

• U preserves

weak equivalences between Cofibront objects

" "

fibroid objects,

Quillen functors

Let F : C � D : U be a Quillen adjunction, define

• The left derived functor of F to be the composite

F : Ho(C) Ho(Q)���! Ho(C) Ho(F)���! Ho(D).

• The right derived functor of U to be the composite

U : Ho(D)Ho(R)���! Ho(D)

Ho(U)���! Ho(C).

Proposition: F : Hol 8)F Ho CD) : O is an adjoint pair .

Unbounded chain complexes

Let R be a ring an Ch(R) the category of unbounded chain complexes of R-modules.

This admits a model structure

• W s quasi isosHo (ch CR)pro;) : DCR)

• Fib : degreewise Cpimorphisms• coz if

it is degree wise split inj

with projectile. Colonel Lcpcacygib)

we call this the projectile model

structure Chcrdproj

Unbounded chain complexes

Let X be a Cg-brat object in Ch (Rlproj

(Assume Ris commutative)

Hom r ( X ,

-) : Ch (Rlproj → Ch CR) pro; is a right Qu-Hou

functor .Left adjoint X Qr

-

RHom, ( X ,

- ) = Ext (X,

- )

Lil X Qr -) s Tor ( X , - I

Quillen equivalences

Let C and D be model categories equipped with a Quillen adjunction F : C � D : U, then Cand D are Quillen equivalent if the derived adjunction F : Ho(C) � Ho(D) : U is an

equivalence of categories.

Proposition: An adjoint par is a Quillen quiff forall cofibront

XEG,

and gibran YED ,

a morphism g: FX → Y is a

weak CE in D iff 4G. ) : X → Uy is a weak Guv in

8

* tact Quillen guru satisfy2-out -y -3

Stable module categories

• A ring R is Frobenius is the projective and injective R-modules coincide.

• Maps f , g : M ! N in R-modules are stably equivalent if f � g factors through a

projective module.

Diop Ra Frobenius ring .

There is a model Strite on R -mod

a W s stable Equis Ho ( R - mod se ) is StMod (R)• Fibs surjection

← Cogs injectionsAll objects are bifibrmt.

A non-Quillen equivalence (Exotic models)

Proposition: Let p be an odd prime, R = Z/p2and S = (Z/p)[#]/(#2).

Schlichting

Ho ( R - modest) = Ho CS- modst )

But They are not Quillen Equivalent .

Model structures on Set

Fact: There are exactly nine model structures on the category Set of sets and functions

between them.

bij = bijections

inj = injections

surj = surjections

all = all morphisms

inj∆ = injections w/ empty domain

inj 6=∆ = injections w/ non-empty domain

all 6=∆ = morphisms w/ non-empty domain

Seta(�2)= (all, all, bij)

Setb(�2)= (all, inj, surj)

Setc(�2)= (all, surj[ inj∆, inj 6=∆ [{id∆})

Setd(�2)= (all, surj[ bij∆, all 6=∆ [{id∆})

Sete(�2)= (all, surj, inj)

Set f(�2)= (all, bij, all)

Seta(�1)= (all 6=∆ [{id∆}, surj[ inj∆, inj)

Setb(�1)= (all 6=∆ [{id∆}, bij[ inj∆, all)

Seta(0)= (bij, all, all)

Model structures on Set

Fact: There are exactly nine model structures on the category Set of sets and functions

between them.

• bij = bijections

• inj = injections

• surj = surjections

• all = all morphisms

• inj∆ = injections w/ empty domain

• inj 6=∆ = injections w/ non-empty domain

• all 6=∆ = morphisms w/ non-empty domain

Seta(�2)= (all, all, bij)

Setb(�2)= (all, inj, surj)

Setc(�2)= (all, surj[ inj∆, inj 6=∆ [{id∆})

Setd(�2)= (all, surj[ bij∆, all 6=∆ [{id∆})

Sete(�2)= (all, surj, inj)

Set f(�2)= (all, bij, all)

Seta(�1)= (all 6=∆ [{id∆}, surj[ inj∆, inj)

Setb(�1)= (all 6=∆ [{id∆}, bij[ inj∆, all)

Seta(0)= (bij, all, all)

Model structures on Set

Fact: There are exactly nine model structures on the category Set of sets and functions

between them.

• bij = bijections

• inj = injections

• surj = surjections

• all = all morphisms

• inj∆ = injections w/ empty domain

• inj 6=∆ = injections w/ non-empty domain

• all 6=∆ = morphisms w/ non-empty domain

(1) Seta(�2)= (all, all, bij)

(2) Setb(�2)= (all, inj, surj)

(3) Setc(�2)= (all, surj[ inj∆, inj 6=∆ [{id∆})

(4) Setd(�2)= (all, surj[ bij∆, all 6=∆ [{id∆})

(5) Sete(�2)= (all, surj, inj)

(6) Set f(�2)= (all, bij, all)

(7) Seta(�1)= (all 6=∆ [{id∆}, surj[ inj∆, inj)

(8) Setb(�1)= (all 6=∆ [{id∆}, bij[ inj∆, all)

(9) Seta(0)= (bij, all, all)

(w ,Fib

,( of)

⇐ into (8) left Quillen

Ho ( Seti,) = * *

Model structures on Set

b

⌅⌅

a

::

✏✏

⇢⇢

//

))

c

✏✏uu

⌅⌅

f doo

e

dd

•

① Setc. a,

→ s idfunctorbeing left Qukn.

0

There is no direct Quiller equivalence !

b If I esometimes we need a

Zig-Zag

Simplicial sets

D categorywith objects Cnt. {och - - - en}

.

morphisms ae order preserving maps"simplex category

"

Dej? A simplicial set is a functor X.i D°P→ Set

sset : category g simp . setst natural truss . set

's"

f a categorySf is f

""

face mapsdi : Xn → Xn . ,

X. essetthis the data of degeneracy maps siixn

→ Xm ,

Xnsxlcns) Osian

Simplicial sets

↳ BECat define its nervy NCE) esset.

Si inserts identity at posi

(NE ) . sob ( f) d, composes its CitDst arrows

(NE) , s Mor ( E )

( E) , :{ strings gk composite arrows in E)

T,:SSet F' Cat : N

.

Simplicial sets

By Yoneda, have representable objects Dcn)

Demons Hom, ( Cms , cm).

DCDso

,

osksn A''

Cn) C DEN)

( kn) -" horn

"

m) detente fate

opposite vertex K.

A' CDs.. ,

A''

Cn ) tha O

X is a Kancomptesc it f .

.

.

-

Osian

Dkny -

-

Simplicial sets

Prof sset has a model structure where :

- fibrat objects are the Kan complexes

- Cogitation are the monomorphisms .

sset Kan

Simplicial sets

Proposition: There is a model structure on sSet where:

• The fibrant objects are

• The cofibrations are

Simplicial sets as spaces

Let X c-Top . The singular complex gX to be the

simp . set SC x).

SH )n ' Homage X)

topological n-simplex

is a QuellerThy SC -) : Top when ssetkan : I - Iequivalence .

⇒ Ho (Top owner) = Hocssetkm)

equiv ijy HI : txt→ HI are

g : X ay in ssetkan is a weakweak ga

in%Pautler

Kan’s Ex•

functor

The barycentric subdivision is a left adjoint sd : sSet ! sSet.

sd(D[2]) =

Iv: sd Dons →Dcn)↳ohEx G) no Hom# ( sd Dcns ,

X) the

right adjoint to Sd.

X ⇐ Cx) → Ex-Cx) → .

. . .

colimty this system is Ex-(x)

props c.⇐Cx) is a Kon complex , X→ Ex.

CX) is an

acyclic Cofib ration .