Orthogonality Logic Lurdes Sousa Center for Mathematics of the University of Coimbra School of Technology of Viseu joint work with Jiˇ rí Adámek and Michel Hébert CT2006 W HITE P OINT J UNE 25 – J ULY 1 – p. 1/25
Orthogonal Subcategory Problem and Orthogonality Logic category A , H ⊆ Mor ( A ) H ⊥ := full subcategory of A -objects orthogonal to H – p. 2/25
Orthogonal Subcategory Problem and Orthogonality Logic category A , H ⊆ Mor ( A ) H ⊥ := full subcategory of A -objects orthogonal to H r A � A the construction of the reflection in- A volves categorical " rules " ( composition, limits, colimits, factorization, ...) – p. 2/25
Orthogonal Subcategory Problem and Orthogonality Logic category A , H ⊆ Mor ( A ) H ⊥ := full subcategory of A -objects orthogonal to H r A � A the construction of the reflection in- A volves categorical " rules " ( composition, limits, colimits, factorization, ...) H ⊥ � � r A ∈ ⊥ – p. 2/25
Orthogonal Subcategory Problem and Orthogonality Logic category A , H ⊆ Mor ( A ) H ⊥ := full subcategory of A -objects orthogonal to H r A � A the construction of the reflection in- A volves categorical " rules " ( composition, limits, colimits, factorization, ...) H ⊥ � � r A ∈ ⊥ Question: When are these " rules " part of a sound and complete deduction system for orthogonality? – p. 2/25
Orthogonal Subcategory Problem and Orthogonality Logic Find a Deduction System of RULES such that h is deducible from H by succes- � H ⊥ � h ∈ ⊥ ⇔ sively applying the RULES – p. 3/25
Orthogonal Subcategory Problem and Orthogonality Logic Find a Deduction System of RULES such that h is deducible from H by succes- � H ⊥ � h ∈ ⊥ ⇔ sively applying the RULES H | = h ⇔ H ⊢ h – p. 3/25
Orthogonal Subcategory Problem and Orthogonality Logic Find a Deduction System of RULES such that h is deducible from H by succes- � H ⊥ � h ∈ ⊥ ⇔ sively applying the RULES H | = h ⇔ H ⊢ h H | = h := ( A ⊥ H ⇒ A ⊥ h ) , for all objects A H ⊢ h := there is a formal proof of h from H by using the De- duction System – p. 3/25
The Finitary Case: Sentences versus Morphisms e ≡ ( u = v ) q e : FX → FX/ ∼ e u and v terms in X algebras satisfying algebras orthogonal to E ′ = { q e i , i ∈ I } E = { e i , i ∈ I } , e i ≡ ( u i = v i ) – p. 4/25
The Finitary Case: Sentences versus Morphisms e ≡ ( u = v ) q e : FX → FX/ ∼ e u and v terms in X algebras satisfying algebras orthogonal to E ′ = { q e i , i ∈ I } E = { e i , i ∈ I } , e i ≡ ( u i = v i ) Analogously for implications and regular sentences – p. 4/25
The Finitary Case: Sentences versus Morphisms A satisfies A is orthogonal to equations epimorphisms with projective domain ∀ x E ( x ) (orthogonality=inject.) implications epimorphisms (orthogonality=inject.) ∀ x ( E ( x ) → F ( x )) limit sentences morphisms ∀ x ( E ( x ) → ∃ ! y F ( x , y )) E ( x ) and F ( x ) involving a finitary morphisms, i.e., finite number of variables with finitely presentable and equations domain and codomain – p. 5/25
G. Ro¸ su, Complete Categorical Equational Deduction (2001): A sound and complete deduction system for finitary epimor- phisms with projective domains Adámek, Sobral, Sousa, Logic of implications (2005): A sound and complete deduction system for finitary epimorphisms – p. 6/25
Finitary Logic A a finitely presentable category Formulas: finitary morphisms, i.e., morphisms of A fp Formal proofs have only a finite number of steps – p. 7/25
If F is a set of finitary morphisms admitting a left calculus of fractions (in A fp ) then F ⊥ is reflective in A . Hébert, Adámek, Rosický, More on orthogonality in l.p.c., Cah. Topol. Géom. Différ. Catég. 42 (2001) – p. 8/25
� � � � � � � � sound rules IDENTITY id A h 1 h 2 COMPOSITION h 2 · h 1 h h if PUSHOUT h ′ h ′ f h ′ h h if COEQUALIZER g h ′ f · h = g · h h ′ = coeq ( f, g ) – p. 9/25
� � � � h Soundness of COEQUALIZER h ′ f h ′ h g – p. 10/25
� � � � � h Soundness of COEQUALIZER h ′ f h ′ h g x X – p. 10/25
� � � � � h Soundness of COEQUALIZER h ′ f h ′ h g x X ( xf ) h = ( xg ) h ⇒ xf = xg – p. 10/25
� � � � � h Soundness of COEQUALIZER h ′ f h ′ h g � � � � � � � � � x X ( xf ) h = ( xg ) h ⇒ xf = xg – p. 10/25
CANCELLATION is not sound f g � { 0 , 1 } � { 0 } { 0 } � � g · f = id { 0 } �| = f because { 0 , 1 } | = id { 0 } but { 0 , 1 } �| = f ) – p. 11/25
� � � � � � f · h ∇ h ∇ - CANCELLATION h h � B A u h 1 B B C v � ∇ h � � � 1 B B – p. 12/25
� � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: h � B A u h 1 B B C v � ∇ h � � � 1 B B – p. 12/25
� � � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: h � B A � � � � � � � u h � � � 1 B � � � � B C � � v � � � ∇ h � � � � � � � 1 B � B � � � k � � � � � � � � X – p. 12/25
� � � � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: f h � B A � � � ������������������������������ � � � � � � u h � � � � 1 B � � � � B C � v � � � ∇ h � � � � � � � � 1 B � B � � k � � � � � � � � X – p. 12/25
� � � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: h � B A u h 1 B p B C v � ∇ h � � � 1 B B q � X – p. 12/25
� � � � � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: h � B A u h 1 B p B C v � ∇ h � � � 1 B t B q X – p. 12/25 X
� � � � � � � � � � f · h ∇ h ∇ - CANCELLATION h is sound: h � B A u h 1 B p B C v � ∇ h � � � 1 B t B � � � q t ′ � � � X p = t ′ · ∇ h · u = t ′ · ∇ h · v = q – p. 12/25 X
� � � � � � � Finitary Orthogonality Deduction System IDENTITY id A h 1 h 2 COMPOSITION h 2 · h 1 h h if PUSHOUT h ′ h ′ f h h m � h if COEQUALIZER h ′ g fh = gh, h ′ = coeq ( f, g ) f · h ∇ h ∇ - CANCELLATION h – p. 13/25
The Finitary Orthogonality Deduction System is sound and complete, that is, H | = h iff H ⊢ h – p. 14/25
� � � � � � � � Finitary Orthogonality Deduction System IDENTITY id A h 2 h 1 COMPOSITION h 2 · h 1 h h PUSHOUT h ′ h ′ h f COEQUALIZER h ′ h h ′ g f · h ∇ h ∇ - CANCELLATION h – p. 15/25
� � � � � � � � � � � � � � � � � � � � � � � Finitary Orthogonality Deduction System � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � IDENTITY id A h 2 h 1 COMPOSITION h 2 · h 1 h h PUSHOUT h ′ h ′ h f COEQUALIZER h ′ h h ′ g f · h ∇ h ∇ - CANCELLATION h – p. 16/25
� � � � � � � � � � � � � � � � � � � � � � � � Finitary Orthogonality Deduction System � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � IDENTITY id A h i , i ∈ α h 1 � h 2 � . . . TRANSFINITE h COMPOSITION h h h PUSHOUT h ′ h ′ h f COEQUALIZER h ′ h h ′ g f · h ∇ h ∇ - CANCELLATION h – p. 17/25
� � � � � � � � � � � � � � � � � � � � � � � � Finitary Orthogonality Deduction System � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � IDENTITY id A h i , i ∈ α h 1 � h 2 � . . . TRANSFINITE h COMPOSITION h h h PUSHOUT h ′ h ′ h f COEQUALIZER h ′ h h ′ g f · h ∇ h ∇ - CANCELLATION h – p. 18/25
� � � � � � � � � Orthogonality Deduction System h i , i ∈ α TRANSFINITE h 1 � h 2 � . . . h COMPOSITION h h h PUSHOUT h ′ h ′ h f COEQUALIZER h ′ h h ′ g f · h ∇ h ∇ - CANCELLATION h – p. 19/25
The Orthogonality Deduction System is sound and complete. That is, H | = h iff H ⊢ h – p. 20/25
Incompleteness Example: a cocomplete category where the Orthogonality Logic is not complete Objects: ( X, ≤ , α ) , where ( X, ≤ ) is a CPO with a least element, and α : X → X CPO ⊥ ( 1 ) Morphisms: continuous maps preserving the least element and the unary operation – p. 21/25
Recommend
More recommend
