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An Elementary Approach to Elementary ToposTheory
Todd Trimble
Western Connecticut State UniversityDepartment of Mathematics
October 26, 2019
Back Story
I Tierney’s approach: private communication.
I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.
I Tierney’s approach: constructions are more natively”set-theoretical”.
Back Story
I Tierney’s approach: private communication.
I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.
I Tierney’s approach: constructions are more natively”set-theoretical”.
Back Story
I Tierney’s approach: private communication.
I Standard approach: forbiddingly technical (monadicitycriteria, Beck-Chevalley conditions, ...) for those who grew upon naive set theory.
I Tierney’s approach: constructions are more natively”set-theoretical”.
Back Story
I Standard approach to deduce existence of colimits:P : E op → E is monadic.
I Construction of coproducts: X + Y is an equalizer:
X+Y P(PX×PY )
uP(PX×PY )−→−→P〈P(PπPX uX ),P(PπPY uY )〉
PPP(PX×PY )
u : 1E → PP is unit uX (x) = A : PX |x ∈ A
Back Story
I Standard approach to deduce existence of colimits:P : E op → E is monadic.
I Construction of coproducts: X + Y is an equalizer:
X+Y P(PX×PY )
uP(PX×PY )−→−→P〈P(PπPX uX ),P(PπPY uY )〉
PPP(PX×PY )
u : 1E → PP is unit uX (x) = A : PX |x ∈ A
Notation and Preliminaries
I Power-object definition of topos: finite limits, universalrelations 3X → PX × X .
R → X × Y
X →χR
PY
R 3Y
X × Y PY × Y
i
y
χi×1Y
I singX : X → PX classifies δX : X → X × X .
Notation and Preliminaries
I Power-object definition of topos: finite limits, universalrelations 3X → PX × X .
R → X × Y
X →χR
PY
R 3Y
X × Y PY × Y
i
y
χi×1Y
I singX : X → PX classifies δX : X → X × X .
Notation and Preliminaries
I 31= 1→ P1× 1, aka t : 1→ Ω.
I All monos are regular:
A X Ω
1
i χi
!t
I Epi-mono factorizations are unique when they exist.
I Toposes are balanced.
Notation and Preliminaries
I 31= 1→ P1× 1, aka t : 1→ Ω.
I All monos are regular:
A X Ω
1
i χi
!t
I Epi-mono factorizations are unique when they exist.
I Toposes are balanced.
Cartesian closure
I Exponentials PZY exist, namely P(Y × Z ) ∼= (PZ )Y :
X → P(Y × Z )
R → X × Y × Z
X × Y → PZ
X → PZY
I
X 1 XY 1Y
PX P1 PXY P1Y
singXy
ty
tY
τ τY
Slice theorem
I If E is a topos, then for any object X , the category E/X isalso a topos. The change of base X ∗ : E → E/X is logicaland has left and right adjoints.
I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical.
I Colimits in E/Y , when they exist, are stable under pullbackf ∗ : E/Y → E/X .
Slice theorem
I If E is a topos, then for any object X , the category E/X isalso a topos. The change of base X ∗ : E → E/X is logicaland has left and right adjoints.
I f ∗ : E/Y → (E/Y )/f ' E/X , for f : X → Y , is logical.
I Colimits in E/Y , when they exist, are stable under pullbackf ∗ : E/Y → E/X .
Internal logic
1× 1t×t→ Ω× Ω
∧ = χt×t : Ω× Ω→ Ω
[≤] → Ω× Ω
⇒ = χ[≤] : Ω× Ω→ Ω
Internal logic
1× 1t×t→ Ω× Ω
∧ = χt×t : Ω× Ω→ Ω
[≤] → Ω× Ω
⇒ = χ[≤] : Ω× Ω→ Ω
Internal logic
X!→ 1
t→ Ω
tX : 1→ ΩX = PX
∀X = χtX : PX → Ω
Define⋂
X : PPX → PX by⋂F = x : X | ∀A:PX A ∈PX F ⇒ x ∈X A
Internal logic
X!→ 1
t→ Ω
tX : 1→ ΩX = PX
∀X = χtX : PX → Ω
Define⋂
X : PPX → PX by⋂F = x : X | ∀A:PX A ∈PX F ⇒ x ∈X A
Construction of coproducts
I Initial object: define 0 → 1 to be “intersection all subobjectsof 1”, classified by
1tP1→ PP1
⋂→ P1
I Lemma: 0 is initial.
I Uniqueness: if f , g : 0⇒ X , then Eq(f , g) 0 is an equality,by minimality of 0 in Sub(1).
I Existence: consider
P X
0 1 PX
ysingX
tX
Coproducts
I 0 is strict by cartesian closure, so 0→ X is monic.
I Given X ,Y , disjointly embed them into PX × PY :
X × 1 PX × PY 1× Y PX × PYχδ×χ0 χ0×χδ
X t Y is the “disjoint union”: the intersection of thedefinable family of subobjects of PX × PY containing theseembeddings.
Coproducts
I 0 is strict by cartesian closure, so 0→ X is monic.
I Given X ,Y , disjointly embed them into PX × PY :
X × 1 PX × PY 1× Y PX × PYχδ×χ0 χ0×χδ
X t Y is the “disjoint union”: the intersection of thedefinable family of subobjects of PX × PY containing theseembeddings.
Coproducts
I Lemma: Any two disjoint unions of X ,Y are isomorphic.
I Proof: If Z = X ∪Y via i : X → Z and j : Y → Z , then mapZ into PX × PY via
X〈1X ,i〉→ X × Z Y
〈1Y ,j〉→ Y × Z
Z → PX Z → PY
Then Z → PX × PY is monic. Both Z and X t Y are leastupper bounds of X and Y in Sub(PX × PY ).
Coproducts
I Theorem: X t Y is the coproduct.
I Proof: Given f : X → B and g : Y → B, form
X〈1X ,f 〉→ X × B, Y
〈1Y ,g〉→ Y × B.
Then (X t Y )× B ∼= (X × B) t (Y × B). So both X ,Yembed disjointly in (X t Y )× B. Obtain
X t Y → (X t Y )× B.
Image factorization
I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.
IB X B
X Y X Y
1Xy
f f
I im(f ) =⋂
Y B : PY | f ∗B = X
Image factorization
I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.
IB X B
X Y X Y
1Xy
f f
I im(f ) =⋂
Y B : PY | f ∗B = X
Image factorization
I For f : X → Y , define im(f ) to be the intersection of the(definable) family of subobjects through which f factors.
IB X B
X Y X Y
1Xy
f f
I im(f ) =⋂
Y B : PY | f ∗B = X
Image factorization
I Lemma: f : X → Y indeed factors through im(f ) : I → Y .
I Proof: We must show f ∗(im(f )) = X . But
f ∗
( ⋂B | f ∗B=X
B
)=
⋂B | f ∗B=X
f ∗B [E/Yf ∗→ E/X is logical]
=⋂
B | f ∗B=X
X
= X
Image factorization
I Lemma: X → im(f ) → Y is the epi-mono factorization off : X → Y .
Proof: Put I = im(f ); suppose X → I equalizes g , h : I ⇒ Z .Then
X → Eq(g , h) I → Y
makes Eq(g , h) a subobject through which f factors. HenceEq(g , h) = I and g = h.
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
CoequalizersLet f , g : X → Y be maps. Construct coequalizer X ⇒ Y → Q:
I Form the image factorization of 〈f , g〉 : X → Y × Y :
X → R Y × Y
I The equivalence relation E on Y generated by R : P(Y × Y )is the intersection of the definable family of equivalencerelations containing R.
I Form the classifying map χE : Y → PY of E → Y × Y(mapping y : Y to its E -equivalence class).
I Form the image factorization of χE :
Y Q PY
I Theorem: Y → Q is the coequalizer of f , g .
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