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 / 31

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

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

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

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

� � � � � (Orthogonal) factorization system ( E , M ) in C u · · � ! w � e m e ⊥ m � � � · · v E = ⊥ M , M = E ⊥ (FS*1&2) (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

� � � � � � � � � Alternative characterization (FS1) Iso ⊆ E ∩ M (FS2) E · E ⊆ E , M · M ⊆ M (FS3) C = M · E (FS3!) · � � e � m � � � � � � � � � · · ! ∼ = � � � � � � � � � � � e ′ m ′ � · Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 8 / 31

� � � � � � � � � Strict factorization system ( E 0 , M 0 ) in C (M. Grandis) (SFS1) Id ⊆ E 0 ∩ M 0 (SFS2) E 0 · E 0 ⊆ E 0 , M 0 · M 0 ⊆ M 0 (SFS3) C = M 0 · E 0 (SFS3!) · � � e � m � � � � � � � � � · · 1 � � � � � � � � � � � e ′ m ′ � · Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 9 / 31

� � � � � � � “Higher” Justification: u u · · · · e f � e g F ( u,v ) � � F ( f ) g F ( g ) f m f � m g � · � · · · v v F : C 2 → 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

� � � � � � � Free structure on C 2 1 u u � · · · · · g = g f f d � · � · � · · · v v 1 Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 11 / 31

� � � � � � � � � � � � � � Mac Lane again: (BC1) Id ⊆ E 0 ∩ M 0 (BC2) E 0 · E 0 ⊆ E 0 , M 0 · M 0 ⊆ M 0 (BC3) C = M 0 · Iso · E 0 j � · (BC3!) · � e � � m � � � � � � � � � · · 1 1 � � � � � � � � � � � e ′ � · m ′ � · j ′ (BC4) E 0 · Iso ⊆ Iso · E 0 , Iso · M 0 ⊆ M 0 · Iso � ≤ 1 � � (BC5) � M 0 · E 0 ∩ C ( A, B ) Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 12 / 31

� � � ∼ � im φ � � G/ ker φ � � � � � � � � � � � � � � � � � G H φ epimorphisms from G ⇐ ⇒ congruences on G Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 13 / 31

Set ∼ objects: 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 ) Z ⊆ X, Z ∼ = { x ∈ X | ∃ z ∈ Z : x ∼ X z } closure: compare: Freyd completion! Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 14 / 31

� � ∼ � f ( X ) ∼ X f � � � � � [1 X ] � � � � � � � � � � � � � � � Y X [ f ] x ∼ f x ′ f ( x ) ∼ Y f ( x ′ ) ⇐ ⇒ { [1 X ] : X → X ′ | ∼ X ⊆∼ X ′ } E 0 = → Y ] | Z ∼ = Z } M 0 = { [ Z ֒ [ f ] mono ⇐ ⇒ ∼ X = ∼ f f ( X ) ∼ = Y [ f ] epi ⇐ ⇒ Epi ∩ Mono = Iso ⇐ ⇒ AC ⇐ ⇒ Epi = SplitEpi Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 15 / 31

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

� � � � Top ∼ ⇒ U = U ∼ U ⊆ X open = bifibration Set ∼ � f ( X ) ∼ X f � � � � � � � � � � � � � � � � � � � [ f ] X Y ⇒ ∃ V ⊆ Y open : U = f − 1 ( V ) Mac Lane: U ⊆ X f open ⇐ ⇒ ∃ V = V ∼ ⊆ Y : U = f − 1 ( V ) open Better: U ⊆ X f open ⇐ Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 17 / 31

� � � � � � � Double factorization system ( E 0 , J , M 0 ) in C u . · � ! w e � � k � ( e, j ) ⊥ ( k, m ) · · ! z � m j � � � · · v (DFS*1) Iso · E 0 ⊆ E 0 , Iso · J · Iso ⊆ J , M 0 · Iso ⊆ M 0 (DFS*2) ( E 0 , J ) ⊥ ( J , M 0 ) (DFS*3) C = M 0 · J · E 0 ( E , M ) fs ⇐ ⇒ ( E , Iso , M ) dfs Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 18 / 31

� � � � � � � � � � � � � � Alternative characterization (DFS1) Iso ⊆ E 0 ∩ J ∩ M 0 (DFS2) E 0 · E 0 ⊆ E 0 , J · J ⊆ J , M 0 · M 0 ⊆ M 0 (DFS3) C = M 0 · J · E 0 j � · (DFS3!) · � e � � m � � � � � � � � · � · ! ∼ ! ∼ = = � � � � � � � � � � � � · e ′ � m ′ · j ′ (DFS4) J · M 0 ⊆ M 0 · J , E 0 · J ⊆ J · E 0 ( E 0 , J , M 0 ) dfs ⇐ ⇒ ( E 0 , M 0 · J ) , ( J · E 0 , M 0 ) fs J = J · E 0 ∩ M 0 · J Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 19 / 31

� � � � � � � � � � � � � � � Free structure on C 3 : 1 1 u � · u · · · · · g 1 g 1 f 1 f 1 f 1 vf 1 1 1 v � · � · v � · � · · = · g 2 g 2 g 2 f 2 f 2 wf 2 � · � · � · � · · · w w 1 1 Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 20 / 31

( E 0 , J , M 0 ) ↔ ( E , W , M ) E 0 = E ∩ W E = J · E 0 J = E ∩ M W = M 0 · E 0 M 0 = M ∩ W M = M 0 · J 0 W is closed under retracts in C 3 . When does W have the 2-out-of-3 property? Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 21 / 31

Double factorization systems “Quillen factorization ( E 0 , J , M 0 ): systems” ( E , W , M ): ( E 0 , M 0 · J ) , ( J · E 0 , M 0 ) fs, ( E ∩ W , M ) , ( E , M ∩ W ) fs, E 0 · M 0 ⊆ M 0 · E 0 , W has 2-out-of-3 property. ej ∈ E 0 , e ∈ E 0 , j ∈ J = ⇒ j iso, jm ∈ M 0 , m ∈ M 0 , j ∈ J = ⇒ j iso . (Pultr-Tholen 2002) Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 22 / 31

� � � � � � Weak factorization system ( E , M ) in C u · · � w � e m e � m � � v � · · E = � M , M = E � (WFS*1&2) (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

(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

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

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

� � F reflective in finitely complete C with reflection ρ : 1 → R ∀ f : A → B : � R − 1 (Iso) , Cart( R, ρ ) � ( E , M ) = fs ⇐ ⇒ ( ρ A ,f ) � � A − − − − → RA × RB B ∈ E ⇒ F = F ( M ) semilocalization E stable under pb along M ⇐ E stable under pullback F = F ( M ) localization ⇐ ⇒ Walter Tholen (York University, Toronto) Mac Lane and Factorization CT2006 27 / 31

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