Mac Lane and Factorization Walter Tholen York University, Toronto - - PowerPoint PPT Presentation

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Mac Lane and Factorization Walter Tholen York University, Toronto - - PowerPoint PPT Presentation

Mac Lane and Factorization Walter Tholen York University, Toronto June 15, 2006 Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 1 / 31 Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 2 /


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SLIDE 1

Mac Lane and Factorization

Walter Tholen

York University, Toronto

June 15, 2006

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 1 / 31

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SLIDE 2

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 2 / 31

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SLIDE 3

Saunders Mac Lane Duality for groups Bulletin for the American Mathematical Society 56 (1950) 485-516 Saunders Mac Lane Groups, categories and duality Bulletin of the National Academy of Sciences USA 34 (1948) 263-267)

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 3 / 31

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SLIDE 4

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 4 / 31

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Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 5 / 31

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SLIDE 6

A brief history of factorization systems

Mac Lane 1948/1950 Isbell 1957/1964 Quillen 1967 Kennison 1968 Kelly 1969 Ringel 1970/1971 Freyd-Kelly 1972 Pumpl¨ un 1972

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 6 / 31

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SLIDE 7

(Orthogonal) factorization system (E, M) in C

e⊥m

·

  • u
  • e
  • ·

m

  • ·

!w

  • v

·

(FS*1&2) E =⊥ M, M = E⊥ (FS*3) C = M · E (FS*1) Iso · E ⊆ E, M · Iso ⊆ M (FS*2) E⊥M (FS*3) C = M · E

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 7 / 31

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SLIDE 8

Alternative characterization

(FS1) Iso ⊆ E ∩ M (FS2) E · E ⊆ E, M · M ⊆ M (FS3) C = M · E (FS3!) ·

!∼ =

  • m
  • ·

e

  • e′
  • ·

·

m′

  • Walter Tholen (York University, Toronto) Mac Lane and Factorization

CT2006 8 / 31

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SLIDE 9

Strict factorization system (E0, M0) in C (M. Grandis)

(SFS1) Id ⊆ E0 ∩ M0 (SFS2) E0 · E0 ⊆ E0, M0 · M0 ⊆ M0 (SFS3) C = M0 · E0 (SFS3!) ·

1

  • m
  • ·

e

  • e′
  • ·

·

m′

  • Walter Tholen (York University, Toronto) Mac Lane and Factorization

CT2006 9 / 31

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SLIDE 10

“Higher” Justification:

·

u

  • f
  • ·

g

  • ·

u

  • ef

·

eg

  • F(f)

F(u,v) mf

F(g)

mg

  • ·

v

·

·

v

·

F : C2 → C ⇐ ⇒ Eilenberg-Moore structure w.r.t. 2 fs ⇐ ⇒ normal pseudo-algebras (Coppey, Korostenski-Tholen) sfs ⇐ ⇒ strict algebras (Rosebrugh-Wood)

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 10 / 31

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SLIDE 11

Free structure on C2

·

u

  • f
  • ·

g

  • =

·

1

  • f
  • ·

d

  • u

·

g

  • ·

v

·

·

v

·

1

·

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 11 / 31

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SLIDE 12

Mac Lane again:

(BC1) Id ⊆ E0 ∩ M0 (BC2) E0 · E0 ⊆ E0, M0 · M0 ⊆ M0 (BC3) C = M0 · Iso · E0 (BC3!) ·

1

  • j

·

1

  • m
  • ·

e

  • e′
  • ·

·

j′

·

m′

  • (BC4)

E0 · Iso ⊆ Iso · E0, Iso · M0 ⊆ M0 · Iso (BC5)

  • M0 · E0 ∩ C(A, B)
  • ≤ 1

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 12 / 31

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SLIDE 13

G/kerφ

imφ

  • G

φ

  • H

epimorphisms from G ⇐ ⇒ congruences on G

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 13 / 31

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SLIDE 14

Set∼

  • bjects:

sets X with equivalence relation ∼X morphisms: [f] : X → Y x ∼X x′ = ⇒ f(x) ∼Y f(x′) f ∼ g ⇐ ⇒ ∀x ∈ X : f(x) ∼Y g(x) closure: Z ⊆ X, Z∼ = {x ∈ X | ∃z ∈ Z : x ∼X z} compare: Freyd completion!

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 14 / 31

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SLIDE 15

Xf

∼ f(X)∼

  • X

[1X]

  • [f]

Y

x ∼f x′ ⇐ ⇒ f(x) ∼Y f(x′) E0 = {[1X] : X → X′ | ∼X⊆∼X′} M0 = {[Z ֒ → Y ] | Z∼ = Z} [f] mono ⇐ ⇒ ∼X=∼f [f] epi ⇐ ⇒ f(X)∼ = Y Epi ∩ Mono = Iso ⇐ ⇒ AC ⇐ ⇒ Epi = SplitEpi

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 15 / 31

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Grp∼ = Grp(Set∼) groups with a congruence relation homomorphisms “up to congruence” Grp∼ → Set∼ reflects isos

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 16 / 31

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Top∼

bifibration

  • Set∼

U ⊆ Xopen = ⇒ U = U∼ Xf

f(X)∼

  • X

[f]

  • Y

Mac Lane: U ⊆ Xf open ⇐ ⇒ ∃V ⊆ Y open : U = f−1(V ) Better: U ⊆ Xf open ⇐ ⇒ ∃V = V ∼ ⊆ Y : U = f−1(V ) open

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 17 / 31

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SLIDE 18

Double factorization system (E0, J , M0) in C

·

  • u
  • e

.

k

  • (e, j)⊥(k, m)

·

j !w

  • ·

m

  • ·

v

  • !z
  • ·

(DFS*1) Iso · E0 ⊆ E0, Iso · J · Iso ⊆ J , M0 · Iso ⊆ M0 (DFS*2) (E0, J )⊥(J , M0) (DFS*3) C = M0 · J · E0 (E, M) fs ⇐ ⇒ (E, Iso, M) dfs

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 18 / 31

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Alternative characterization

(DFS1) Iso ⊆ E0 ∩ J ∩ M0 (DFS2) E0 · E0 ⊆ E0, J · J ⊆ J , M0 · M0 ⊆ M0 (DFS3) C = M0 · J · E0 (DFS3!) ·

! ∼ =

  • j

·

m

  • ! ∼

=

  • ·

e

  • e′
  • ·

·

j′

·

m′

  • (DFS4)

J · M0 ⊆ M0 · J , E0 · J ⊆ J · E0 (E0, J , M0) dfs ⇐ ⇒ (E0, M0 · J ), (J · E0, M0) fs J = J · E0 ∩ M0 · J

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 19 / 31

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Free structure on C3:

·

f1

  • u

·

g1

  • ·

1

  • f1
  • ·

1

  • f1
  • ·

u

  • vf1
  • ·

g1

  • ·

f2

  • v

·

g2

  • =

·

f2

  • 1

·

wf2

  • v

·

g2

  • 1

·

g2

  • ·

w

·

·

w

·

1

·

1

·

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 20 / 31

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(E0, J , M0) ↔ (E, W, M) E0 = E ∩ W E = J · E0 J = E ∩ M W = M0 · E0 M0 = M ∩ W M = M0 · J0 W is closed under retracts in C3. When does W have the 2-out-of-3 property?

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 21 / 31

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Double factorization systems (E0, J , M0): (E0, M0 · J ), (J · E0, M0) fs, E0 · M0 ⊆ M0 · E0, ej ∈ E0, e ∈ E0, j ∈ J = ⇒ j iso, jm ∈ M0, m ∈ M0, j ∈ J = ⇒ j iso. “Quillen factorization systems” (E, W, M): (E ∩ W, M), (E, M ∩ W) fs, W has 2-out-of-3 property. (Pultr-Tholen 2002)

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 22 / 31

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SLIDE 23

Weak factorization system (E, M) in C

em

·

  • u
  • e
  • ·

m

  • ·
  • w
  • v

·

(WFS*1&2) E = M, M = E (WFS*3) C = M · E (WFS*1a) gf ∈ E, g split mono = ⇒ f ∈ E (WFS*1b) gf ∈ M, f split epi = ⇒ g ∈ M (WFS*2) E M (WFS*3) C = M · E

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 23 / 31

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(Mono,Epi) in Set

(Mono,Mono) wfs in C with binary products and enough injectives (, SplitEpi) wfs in every lextensive category C

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 24 / 31

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fs = ⇒ wfs E: closed under composition, direct products stable under pullback, intersection If C has kernelpairs, any of the following will make a wfs (E, M) an fs: M closed under any type of limit gf ∈ M, g ∈ M = ⇒ f ∈ M gf = 1, g ∈ M = ⇒ f ∈ M

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 25 / 31

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Cassidy-H´ ebert-Kelly (1985), Ringel (1970)

C finitely well-complete reflective subcategories of C (full, replete) factorization systems (E, M) with gf ∈ E, g ∈ E = ⇒ f ∈ E (E, M) → F(M) = {B ∈ C | (B → 1) ∈ M}

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 26 / 31

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F reflective in finitely complete C with reflection ρ : 1 → R (E, M) =

  • R−1(Iso), Cart(R, ρ)
  • fs

⇐ ⇒

∀f : A → B :

  • A

(ρA,f)

− − − − → RA ×RB B

  • ∈ E

E stable under pb along M

⇒ F = F(M) semilocalization

E stable under pullback

F = F(M) localization

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 27 / 31

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C with 0

(E, M) torsion theory ⇐ ⇒ (E, M) fs, E, M have 2-out-of-3 property F(M) = {B | (B → 0) ∈ M} T (E) = {A | (0 → A) ∈ E}

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 28 / 31

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C with kernels and cokernels

SKC ∼ = SC

1

  • σKC∼

=αC

  • SC
  • σC
  • KC

κC

  • C

πC

  • ρC
  • QC

βC∼ =ρQC

  • RC

1

RC ∼

= RQC C ∈ F(M) ⇐ ⇒ SC = 0 ⇐ ⇒ KC = 0 C ∈ T (E) ⇐ ⇒ RC = 0 ⇐ ⇒ QC = 0

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 29 / 31

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αC iso ⇐ ⇒ βC iso ⇐ ⇒ πCκC = 0 (E, M) simple = ⇒ (E, M) normal C homological, Cop homological: normal torsion theories (E, M) ⇐ ⇒ standard torsion theories (T , F) 0 → T → C → F → 0 C(T , F) = 0

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 30 / 31

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Example

C: abelian groups with (4x = 0 = ⇒ 2x = 0) F: abelian groups with 2x = 0

  • σ
  • Z ∼

= 2Z

κ

  • Z

ρ

  • π=1 Z

ρ

  • Z2

1

Z2

Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 31 / 31

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Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 32 / 31