Kan extensions and descent theory Fernando Lucatelli Nunes 1 1 PhD - PowerPoint PPT Presentation

Pseudo-Kan extension Facets of Descent Kan extensions and descent theory Fernando Lucatelli Nunes 1 1 PhD program at University of Coimbra, under supervision of Maria Manuel Clementino Category Theory - CT 2015, University of Aveiro

Pseudo-Kan extension Facets of Descent Setting A : C op → CAT p ∈ Mor ( C ) , Desc A ( p ) and (assuming that A ( p )! ⊣ A ( p ) ) the category of (Eilenberg Moore) algebras of A ( p )( A ( p )!) [Janelidze and Tholen 1997] Facets of descent II

Pseudo-Kan extension Facets of Descent Setting A : C op → CAT p ∈ Mor ( C ) , Desc A ( p ) and (assuming that A ( p )! ⊣ A ( p ) ) the category of (Eilenberg Moore) algebras of A ( p )( A ( p )!) [Ross Street 1976] Limits indexed by category-valued 2-functors [Ross Street 1980] Correction to: “Fibrations in bicategories”

Pseudo-Kan extension Facets of Descent Aim Work Investigate whether formal methods and commuting properties of bilimits are useful in proving theorems of descent theory in the context of Facets of Descent II Talk Give a proof of Bénabou-Roubaud Theorem using this approach

Pseudo-Kan extension Facets of Descent Outline Pseudo-Kan extension 1 Definition Factorization Eilenberg Moore Algebras Descent Object Commuting property Facets of Descent 2 (Usual context of) Facets of Descent Bénabou-Roubaud Theorem

� Pseudo-Kan extension Facets of Descent Definition Right pseudo-Kan extension : S → ˙ Fully Faithful v S Obj ( ˙ S ) = { e } ∪ Obj ( S ) 2-category H � � ˙ S , H PS [ v , H ] PS [ S , H ] PS

� � Pseudo-Kan extension Facets of Descent Definition Right pseudo-Kan extension : S → ˙ Fully Faithful v S Obj ( ˙ S ) = { e } ∪ Obj ( S ) 2-category H ε : [ v , H ] PS ◦ Ps- R an v → Id � � ˙ S , H : Id → Ps- R an v ◦ [ v , H ] PS η PS Id L ∼ s : = ( ε L ) ◦ ( L η ) [ v , H ] PS ⊣ Ps- R an v ( U ε ) ◦ ( η U ) ∼ t : = Id U plus coherence [ S , H ] PS

Pseudo-Kan extension Facets of Descent Definition Pointwise pseudo-Kan extension Theorem (Pointwise pseudo-Kan extension) Given a pseudofunctor D : S → H , � � ˙ Ps- R an v D ( a ) = S ( a , v − ) , D bi , provided that these weighted bilimits exist in H .

Pseudo-Kan extension Facets of Descent Definition Pointwise pseudo-Kan extension Theorem (Pointwise pseudo-Kan extension) Given a pseudofunctor D : S → H , � � ˙ Ps- R an v D ( a ) = S ( a , v − ) , D bi , provided that these weighted bilimits exist in H . We get actually extensions (up to pseudonatural equivalences): the counit of the biadjunction is a pseudonatural equivalence (since v is fully faithful) We always assume that the pointwise right pseudo-Kan extensions exist.

� � � Pseudo-Kan extension Facets of Descent Factorization Factorization and Comparison D : ˙ S → H S , Obj ( ˙ ˙ Fully faithful v : S → S ) = { e } ∪ Obj ( S ) v -comparison: η D e : D ( e ) → Ps- R an v ( D ◦ v ) ( e ) v -“factorizations”: For each morphism f : e → a of ˙ S , η D e D ( e ) Ps - R an v ( D ◦ v ) ( e ) ∼ = D ( f ) D ( a )

� � Pseudo-Kan extension Facets of Descent Factorization Factorization and Comparison D : ˙ S → H S , Obj ( ˙ ˙ Fully faithful v : S → S ) = { e } ∪ Obj ( S ) v -comparison: η D e : D ( e ) → Ps- R an v ( D ◦ v ) ( e ) v -“factorizations”: For each morphism f : a → e of ˙ S , η D e � Ps - R an v ( D ◦ v ) ( e ) D ( e ) ∼ = D ( f ) D ( a )

Pseudo-Kan extension Facets of Descent Factorization v -effective S , Obj ( ˙ ˙ v : S → S ) = { e } ∪ Obj ( S ) D : ˙ S → H Comparison: η D e : D ( e ) → Ps- R an v ( D ◦ v ) ( e ) Definition Herein, if η D e : D ( e ) → Ps- R an v ( D ◦ v ) ( e ) is an equivalence, we say that D is v -effective. This means that D is v -effective if it is “in the image” of the global right pseudo-Kan extension.

� Pseudo-Kan extension Facets of Descent Eilenberg Moore Algebras The 2 -category A dj Free Adjunction (Street and Schanuel) We denote by A dj the 2-category generated by the diagram u � a e f with 2-cells : Id a → uf η : fu → Id e ε satisfying the triangular identities. We define the full inclusion m : Mnd → A dj , with Obj ( Mnd ) = { a }

� � � Pseudo-Kan extension Facets of Descent Eilenberg Moore Algebras Eilenberg Moore Factorization Each adjunction in a 2-category H corresponds to a diagram D : A dj → H . The m -factorization gives the Eilenberg Moore factorization (if H is bicategorically complete) and the Eilenberg Moore comparison 1-cell. D ( e ) Ps - R an v ( D ◦ v ) ( e ) ∼ = D ( u ) D ( a ) Thereby D is m -effective if and only if the right adjoint D ( u ) is monadic. [S. Schanuel and R. Street 1986] The Free Adjunction

� � � � Pseudo-Kan extension Facets of Descent Descent Object The category ∆ Definition We denote by ˙ ∆ the category generated by the diagram d 0 ∂ 0 d � 1 0 � 2 � 3 s 0 ∂ 1 d 1 ∂ 2 with the usual cosimplicial identities ∂ k d i ∂ i d k − 1 , if i < k = s 0 d 0 = Id 1 s 0 d 1 Id 1 = d 1 d d 0 d = We define the full inclusion g : ∆ → ˙ ∆ , with Obj ( ˙ ∆) = Obj (∆) ∪ { 0 }

Pseudo-Kan extension Facets of Descent Descent Object Descent Object We define the full inclusion g : ∆ → ˙ ∆ , with Obj ( ˙ ∆) = Obj (∆) ∪ { 0 } Theorem (Descent Object) If D : ∆ → H has a (pointwise) right pseudo-Kan extension along g , then Ps- R an g D ( 0 ) is indeed the descent object of D .

Pseudo-Kan extension Facets of Descent Descent Object Descent Object We define the full inclusion g : ∆ → ˙ ∆ , with Obj ( ˙ ∆) = Obj (∆) ∪ { 0 } Theorem (Descent Object) If D : ∆ → H has a (pointwise) right pseudo-Kan extension along g , then Ps- R an g D ( 0 ) is indeed the descent object of D . Main observation: This is a very easy description of the descent object!

Pseudo-Kan extension Facets of Descent Commuting property Diagram of effective diagrams We consider ˙ ˙ v : S → S j : R → R Obj ( ˙ Obj ( ˙ S ) = { e } ∪ Obj ( S ) R ) = { z } ∪ Obj ( R ) Theorem (Main commuting property) � � Given a pseudofunctor M : ˙ ˙ S → R , H PS M is v -effective � � ˙ The image of M ◦ v : S → R , H PS has only j -effective diagrams. Then M ( e ) : ˙ R → H is j -effective as well.

� � Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Factorizations C with pullbacks; 1 A : C op → CAT ; 2 p ∈ C ( E , B ) 3 ˙ ∆ → C p � B E × B E × B E ��� E × B E � E

� � Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Factorizations C with pullbacks; 1 A : C op → CAT ; 2 p ∈ C ( E , B ) 3 D esc : ˙ A ∆ → CAT (descent diagram induced by p ) p A ( p ) � A ( E ) A ( B ) � A ( E × p E ) ��� A ( E × p E × p E )

� � � Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Factorizations C with pullbacks; 1 A : C op → CAT ; 2 p ∈ C ( E , B ) 3 A ( p )! ⊣ A ( p ) 4 D esc : ˙ A ∆ → CAT (descent diagram induced by p ) p A ( p ) � A ( E ) A ( B ) � A ( E × p E ) ��� A ( E × p E × p E ) A dj A : A dj → CAT (adjunction induced by p ) p A ( p ) � A ( E ) A ( B ) A ( p )!

� � � � Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Factorizations C with pullbacks; 1 A : C op → CAT ; 2 p ∈ C ( E , B ) 3 A ( p )! ⊣ A ( p ) 4 D esc : ˙ A dj A ∆ → CAT A : A dj → CAT p p ( g -factorization - natural ( m -factorization - natural isomorphism) isomorphism) A ( p ) � A ( p ) � A ( B ) A ( E ) A ( B ) A ( E ) Desc A ( p ) Alg A ( p )

� � � � Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Factorizations D esc : ˙ A dj A ∆ → CAT A : A dj → CAT p p ( g -factorization - natural ( m -factorization - natural isomorphism) isomorphism) A ( p ) � A ( p ) � A ( B ) A ( E ) A ( B ) A ( E ) Desc A ( p ) Alg A ( p )

Pseudo-Kan extension Facets of Descent (Usual context of) Facets of Descent Usual definitions Definitions D esc : ˙ p is of A -effective descent if A ∆ → CAT is g -effective; p A dj p is of A -monadic descent if A : A dj → CAT is p m -effective.

Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem Assumed Lemmas A : C op → CAT Lemmas Every split epimorphism p is of A -effective descent (proved in Facets of Descent II) ; Every split epimorphism p is of A -monadic descent provided that A ( p )! ⊣ A ( p ) (easy consequence of Beck’s Theorem). They are corollaries of properties of the right pseudo-Kan extensions along m : Mnd → A dj and g : ∆ → ˙ ∆ (respectively).

� � � Pseudo-Kan extension Facets of Descent Bénabou-Roubaud Theorem Bénabou-Roubaud Theorem Bénabou-Roubaud Theorem A : C op → CAT A ( q )! ⊣ A ( q ) for all q ∈ Mor ( C ) The natural isomorphisms A ( q ) A ( B ) A ( E ) ∼ A ( d 0 ) A ( q ) = � A ( E × q E ) A ( E ) A ( d 1 ) satisfy Beck Chevalley condition Then p is A -monadic descent if and only if p is A -effective descent