A Completeness Theorem for Injectivity Logic J. Ad amek, M. H - PDF document

A Completeness Theorem for Injectivity Logic J. Ad´ amek, M. H´ ebert and L. Sousa CT06 White Point, June 2006 1

� 2 C is h-injective is written C | = h h � B A � � � � � � � � � ∀ g ∃ g ′ C C ∈ H △ is written C | = H (= ∀ h ∈ H ( C | = h )) f ∈ ( H △ ) ▽ is written H | = f ∀ C ( C | = H ⇒ C | = f )

� � 3 EXAMPLE: In Alg( Σ ) (Σ a signature), any h can be “presented by generators and relations”: h A = < x ; E ( x ) > < x , y ; E ( x ) ∧ F ( x , y ) > = B � � � � � � � � � � � � � � � � � � � � � � � � � ∀ g ∃ g ′ C ( E, F sets of equations (i.e., ∈ ∧ Atomic )) C | = h means C | = ∀ x ( E ( x ) → ∃ y F ( x , y )) If A and B are finitely presentable, h “is” a (regu- lar) finitary sentence. Conversely, any regular sentence “is” a morphism. C | = H means ∀ h ∈ H ( C | = h ) H | = f means ∀ C ( C | = H ⇒ C | = f )

4 CONTEXT: A can be locally presentable, or Top , or... QUESTIONS: Given H | = f , (1) Can we “deduce” (= construct) f from H ? (2) If H and f are “finitary”, is there a “finitary” proof? ANSWERS: (1) Yes for all sets H of morphisms: this fol- lows directly from the ”Small-Object Argument” ([Quillen, 67], [Ad-Her-Ros-Tho, 02]) (see below) (2) Yes (our main result). This will give in par- ticular a Compactness Theorem: = f ⇒ H ′ | H | = f for some finite H ′ ⊂ H (will extend to a λ -ary version)

� � � � 5 (1) Proof. Note first: (a) Mod( H ) (= H △ ) is weakly reflective in A . (b) the reflectors r A : A → A are cellularly gener- ated by H : r A ∈ cell ( H ) = Comp(P.O.( H )) i.e., r A is the colimit of a smooth chain of pushouts of members of H (i.e., all r α : A α → A α +1 below are in P.O.( H )) Hence, given H | = f : A → B , we have r A � A α +1 � . . . A = A 0 A β = A A 1 . . . A α r α � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∃ u f � � � � � � � � � � � � � � � � � � B � (since A | = H | = f ). Hence f is “deduced” from H using the rules:

� � � � � � � � � 6 r = comp( r α ) α<β � ∗ h ∈H ∗ � A α +1 � . . . A = A 0 A 1 . . . A α r α A β = A � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � u � � � � f � � � � � � � � � � � � � � � � � � � � � � � � � � � B Injectivity Deduction System ( ⊢ ∞ ) transfinite r α ( α < β ) if r = comp( r α ) α<β , composition r β is any ordinal h h � pushout if r α r α cancellation u · f if u · f is defined f

7 We write this as H ⊢ ∞ f Soundness ( H ⊢ ∞ f ⇒ H | = f ) is straightforward, hence: H | = f iff H ⊢ ∞ f for every set H and every f

� � � � � � � 8 [ λ -ary] Injectivity Deduction System ( \\\ ⊢ ∞ ) [ ⊢ λ ] h 1 � h 2 � [ λ -ary] h α ( α < β ) transfinite h β is any ordinal h composition [ β < λ ] h � h pushout if h ′ h ′ u · f cancellation u · f � � � � f � � � � � f ( u ) � �

� � � � � � � � � 9 (2) Definitions : Finitary proof ( H ⊢ ω f ): if f can be obtained from H by a finite number of applications of the rules: Finitary Injectivity Deduction System ( ⊢ ω ) identity id A h 1 h 2 � h 1 h 2 � � � � composition � � � � � � � � h 2 · h 1 � h 2 · h 1 h h � pushout h ′ h ′ u · f u · f cancellation � � � � f � � � � � f ( u ) � � f : A → B is finitary if A and B are finitely pre- sentable ( � = “ f is finitely presentable”).

� � � � � � � � � � � � � � 10 Theorem (When f and all h ∈ H finitary) H | = f iff H ⊢ ω f Proof. (Assume A locally finitely presentable) As before, A | = H | = f : A → B gives: r A � A α +1 � . . . A = A 0 A 1 . . . A α r α A β = A � � � � � � � � � � � � � � � � � � � � � � � � � � � f ∃ u � � � � � � � � � � � � � � � � � B This time A and B are finitely presentable, so: r 0 ,α � A α +1 � . . . A = A 0 A 1 . . . A α A β = A � � � � � � � � � � � � � � � � � � � � � � � � ∃ v � � � � � � � � u � � � � � f � � � � � � � � � � � � � � � � � � � � � � � � B for some α However H �⊢ ω r 0 ,α ! The wanted deduction is not (quite) part of this diagram.

� � � � � � � � � � � 11 We know that the class of ordinals S = { α | some α -chain in P.O.( H ) factorizes through f } is not empty, hence it has a first element σ . We show that σ is finite: Suppose σ is infinite. Then σ = τ + k for τ limit ordinal and k finite. - k � = 0 (because A , B are finitely presentable) - We can assume k = 1. D h ∈H � D ′ p . . . � A i +1 � A τ +1 = A σ . . . A = A 0 A 1 A i A τ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � u � � f � � � � � � � � � � � � � � � � � � � � � � B � Then p factorizes through the chain by some q (be- cause D is finitely presentable) Let ( h i , q i ) = Pushout( h, q ): h D ′ D � ��������������������������� � ������������������������������������������ q � A 1 . . . . . . � A τ +1 = A σ A = A 0 A i A i +1 A τ h i � � � � � q i P i

� � � � � � � � 12 Then take successive pushouts, and their colimits, etc.: h D ′ D � ��������������������������� q p � A 1 . . . = A σ . . . A = A 0 A i +1 A τ +1 A i A τ � � � � � s � � � � � h i � h i +1 h τ � . . . � P i +1 P i P τ Then there exists an isomorphism s making the triangle commute, since h τ (= colim( h j ) j ≥ i ) is also the pushout of h by p ! But then the smooth τ -chain in P.O.( H ) h i A → A 1 → · · · A i − → P i → P i +1 → · · · P τ factorizes through f , contradicting the minimality of σ .

13 EXAMPLES AND COUNTEREXAMPLES 1) The Finitary Completeness Theorem H | = f ⇔ H ⊢ ω f holds in all weakly locally ranked categories (the proof is more involved). 2) In locally finitely presentable categories, H | = ω f ⇒ / H ⊢ ω f. in general (Here H | = ω f means H | = f in A fp ) 3) In CPO(1) (= continuous posets with an extra binary relation), H | = f ⇒ / H ⊢ ∞ f ( H a set) in general. 4) In locally finitely presentable categories, the ( ∞ - ary) Completeness Theorem H | = f ⇔ H ⊢ ∞ f does NOT hold for CLASSES H in general. However it holds for classes H made of (a) epimorphisms (easy), or of (b) finitely presentable morphisms (less easy).