Skolem Relations and Profunctors Robert Par e Aveiro, June 2015 - PowerPoint PPT Presentation

Skolem Relations and Profunctors Robert Par´ e Aveiro, June 2015

� � � � Distributivity Let � K j � be a J -family of sets and �� A j , k � k ∈ K j � j a family of families of sets A j , k ∼ � � � � = A j , s ( j ) s ∈ � K j j ∈ J k ∈ K j j ∈ J Also holds in a topos � q � E / J E / K E / K E / J E / J E / J � J E E v ∗ � Sect ( q ) E / Sect ∗ ( q ) E / Sect ∗ ( q ) E / Sect ( q ) E / Sect ( q ) E / Sect ( q ) E / Sect ( q ) � u � J is the canonical � � J • q : K j ∈ J K j • Sect ( q ) is the object of sections of q , i.e. � j ∈ J K j • Sect ∗ ( q ) is the object of pointed sections of q , i.e. J × � j ∈ J K j � K is evaluation, u : Sect ∗ ( q ) � Sect ( q ) forgetful • Sect ∗ ( q )

� � � ✤ � � � � Intersection/Union � J be a morphism in a topos E and � A k � a family of • Let q : K � K × A subobjects of A , • Consider � � � � A k = A s ( j ) j q ( k )= j s ∈ Sect ( q ) j Holds in Set (and more generally in any topos satisfying IAC ) • Internalizes as � q � Ω J Ω K Ω K Ω J Ω J Ω J � J v ∗ Ω Ω � Ω Sect ∗ ( q ) Ω Sect ∗ ( q ) Ω Sect ( q ) Ω Sect ( q ) Ω Sect ( q ) Ω Sect ( q ) Sect ( q ) � u v u � Sect ( q ) K � Sect ∗ ( q ) s s � K � K j ∈ J j ∈ J K J J K s ( j ) � �→ q q J J J J

Skolem Relations To prove � � � � A k = A s ( j ) j q ( k )= j s ∈ Sect ( q ) j • “ ⊇ ” easy • “ ⊆ ” a ∈ � � q ( k )= j A k iff for every j there is a k such that q ( k ) = j j and a ∈ A k • The k is not unique so you choose one (if you can) which gives a � K section s : J • If you can’t choose, take some or all of them. You get an entire � K relation S : J • Definition A Skolem relation for q is a relation S such that q ∗ ◦ S = Id J

� � � � � � � � � � � � Distributivity I Theorem In any topos we have � � � � A k = A k j q ( k )= j S ∈ Sk ( q ) j ∼ S k or � q � Ω J Ω K Ω K Ω J Ω J Ω J � J v ∗ Ω Ω � Ω Sk ∗ ( q ) Ω Sk ∗ ( q ) Ω Sk ( q ) Ω Sk ( q ) Ω Sk ( q ) Ω Sk ( q ) Sk ( q ) � u � Ω J × K � ∈ J × K Ω J × K Sk ( q ) Sk ( q ) Sk ∗ ( q ) Sk ∗ ( q ) ∈ J × K ∃ J × q ( w , v , u ) Ω J × J Ω J × J J × K × Ω J × K J × K × Ω J × K 1 1 � ∆ � � J × K × Sk ( q ) J × K × Sk ( q )

� � � � � Properties of Skolem Relations Proposition (i) If q has a Skolem relation, then q is epi (ii) If q is epi, then q ∗ is a Skolem relation (iii) Any Skolem relation S is an entire relation (iv) For any Skolem relation S we have S ⊆ q ∗ (v) Sk ( q ) is an upclosed subset of q ∗ (vi) Sections of q are minimal elements of Sk ( q ) Proposition S S s 2 s 1 is a Skolem relation if and only if J J K K (1) s 1 is epi (2) s 2 is mono S S s 1 s 2 (3) commutes J J K K q

� � � � � Cutting Down the Size P is internally projective if ( ) P : E � E preserves epimorphisms � � J be an internally projective cover of J . A P - section of q is Let e : P � K such that σ : P K K σ P P q � � e J J We have a morphism � Sk ( q ) φ : P - Sect ( q ) K K σ �− → Im( σ ) Im( σ ) q � � J J φ is internally initial

� � � � Cutting Down the Size Theorem � � � � A k = A σ ( p ) σ ∈ P - Sect ( q ) j p ∈ P q ( k )= j � q � Ω J Ω K Ω K Ω J Ω J Ω J � J v ∗ Ω Ω � P - Sect ( q ) Ω P - Sect ∗ ( q ) Ω P - Sect ∗ ( q ) Ω P - Sect ( q ) Ω P - Sect ( q ) Ω P - Sect ( q ) Ω P - Sect ( q ) � u Corollary If J is internally projective we have � � � � A K = A σ ( j ) j q ( h )= j σ ∈ � q K j

Limit/Colimit We would like a similar formula expressing lim lim Γ J K ← − − → J ∈ J K ∈ K J as a colimit of limits, for diagrams � Set Γ J : K J

� � Families of Diagrams Γ J should be functorial in J , so a functor � Diag J lim → K ∈ K J Γ J K should also be functorial in J , so a functor − lim − → � Set Diag A good notion of morphism of diagram, which works well for lim → is − Φ � K 2 K 1 K 1 K 2 φ ⇒ Γ 1 Γ 2 Set Set

Families of Diagrams We can put all the K ’s together in an opfibration � J Q : K and then the Γ’s and φ fit together to give a single diagram � Set Γ : K Notes: (1) Our discussion leads to split opfibrations, but general opfibrations are better (2) We could go further and take homotopy opfibrations, which are exactly the notion which makes lim → functorial − (3) We could in fact take Q to be an arbitrary functor, and take Kan extension instead of lim → , but we lose the “family of diagrams” intuition −

Limits of Colimits � J be an opfibration and Γ : K � Set a J -family of Let Q : K � Set diagrams, Γ J : K J An element of lim lim Γ K ← − − → J QK = J is a compatible family of equivalence classes � [ x J ∈ Γ K J ] K J � J • For every J we have a K J such that QK J = J • Not unique but there is a path of K ’s in K J connecting any two choices � J ′ there is a k j : K J � j ∗ K J and a path in K J ′ • For any j : J connecting Γ( k j )( x J ) with x J ′

Profunctors � K is a functor P : J op × K � Set A profunctor P : J • p � K An element of P ( J , K ) is denoted J • Composition: � K ( R ⊗ P )( J , L ) = R ( K , L ) × P ( J , K ) An element is an equivalence class p r � K � L ] K [ J • • Q ∗ : J � K Q ∗ ( J , K ) = J ( J , QK ) Examples: • is • � J • Q ∗ : K is Q ∗ ( K , J ) = J ( QK , J ) • � J Id J ( J , J ′ ) = J ( J , J ′ ) • Id J : J is • ✤ Q ∗ • Q ∗

Prosections � K such that Q ∗ ⊗ S ∼ A prosection for Q is a profunctor S : J = Id J • � Q ∗ The isomorphism corresponds to a morphism σ : S Definition � K and a morphism A prosection for Q is a profunctor S : J • � Q ∗ such that σ : S Q ∗ ⊗ σ � Q ∗ ⊗ Q ∗ ǫ � Id J Q ∗ ⊗ S is an isomorphism � S ′ such that � ( S ′ , σ ′ ) is t : S A morphism of prosections ( S , σ ) σ ′ t = σ The category of prosections is denoted Ps ( Q )

� � Analysis of Prosections In general an element of ( Q ∗ ⊗ S )( J , J ′ ) is an equivalence class of pairs j s � K , QK � J ′ ] K [ J • k j j � J ′ lifts to K � j ∗ K and If Q is an opfibration, QK j k j s � K , QK � J ′ ] = [ J s � K � j ∗ K , Q ( j ∗ K ) J ′ ] [ J • • Proposition � K a profunctor, ( Q ∗ ⊗ S )( J , J ′ ) For Q an opfibration and S : J • s � K ′ ] QK ′ = J ′ where the equivalence consists of equivalence classes [ J • relation is generated by s ∼ ¯ s if there exists k such that s � K ′ K ′ J J • k ¯ ¯ K ′ K ′ J J • s ¯ with Qk = 1 J ′

Analysis of Prosections � Q ∗ For a prosection ( S , σ ), σ : S σ ( s ) � QK ) s σ � ( J � K ) ✤ ( J • � Id J Induces Q ∗ ⊗ S σ ( s ) � J ′ ) s � K ′ ] QK ′ = J ′ �− [ J → ( J • Proposition � K J ( S , σ ) is a prosection if and only if for every J there exists s J : J • such that (1) σ ( s J ) = 1 J (so in particular QK J = J ) , � K we have (2) for every s : J • s QK � K QK ] K QK = [ J σ s � QK s � K ] K QK [ J • •

Distributivity II Theorem � J an opfibration and Γ : K � Set we have For Q : K Γ K ∼ lim lim = lim lim Γ K ← − − → − → ← − J ∈ J K ∈ K J ( S ,σ ) ∈ Ps ( Q ) s ∈ S ( J , K ) • We can write lim − s ∈ S ( J , K ) Γ K as an iterated limit to get an equivalent ← form of the isomorphism Γ K ∼ lim lim = lim lim lim Γ K ← − − → − → ← − ← − QK = J J ( S ,σ ) ∈ Ps ( Q ) J s ∈ S ( J , K ) • If ( S , σ ) is representable S = Φ ∗ , for Φ an actual section, then − s ∈ S ( J , K ) ∼ lim = ΓΦ J ←

� � � � � � � Distributivity II (continued) Theorem lim − → Q Set K Set K � Set J Set J Set J Set J lim ← − J V ∗ Set Set lim Set Ps ∗ ( Q ) op Set Ps ∗ ( Q ) op Set Ps ( Q ) op Set Ps ( Q ) op Set Ps ( Q ) op Set Ps ( Q ) op → Ps ( Q ) − lim ← − U Ps ∗ ( Q ) the category of pointed prosections � K , σ : S � Q ∗ , s : J � K , ( S , σ ) a – Objects ( S , σ, s ) , S : J • • prosection � S ′ such that � ( S ′ , σ ′ , s ′ ) t : S – Morphisms ( t , j , k ) : ( S , σ, s ) σ ′ t = σ and j � J J ′ J ′ J s ′ • • ts K ′ K ′ K K k � Ps ( Q ) � K forgetful functors V : Ps ∗ ( Q ) op U : Ps ∗ ( Q ) and

Properties of Prosections Proposition (1) If Q has a prosection, then Q is pseudo epi ( FQ ∼ = GG ⇒ F ∼ = G ) (2) If ( S , σ ) is a prosection, then S is total (lim → K S ( J , K ) = 1 for every J) − (3) Ps ( Q ) is closed under connected colimits in Set J op × K Proof. (1) FQ ∼ = GQ ⇒ F ∗ ⊗ Q ∗ ∼ = G ∗ ⊗ Q ∗ ⇒ F ∗ ⊗ Q ∗ ⊗ S ∼ = G ∗ ⊗ Q ∗ ⊗ S ⇒ F ∗ ∼ = G ∗ ⇒ F ∼ = G (2) S is total ⇔ T ∗ ⊗ S ∼ � ✶ ) = T ∗ ( T : ? Q ∗ ⊗ S ∼ = Id J ⇒ T ∗ ⊗ Q ∗ ⊗ S ∼ = T ∗ ⊗ Id J ⇒ T ∗ ⊗ S ∼ = T ∗ → α S α ) ∼ → α ( Q ∗ ⊗ S α ) ∼ → α Id J ∼ (3) Q ∗ ⊗ (lim = lim = lim = Id J − − −