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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 .