Kan-injectivity and KZ-monads Lurdes Sousa IPV / CMUC July 10, 2018 Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 1 / 52
[A. Kock, Monads for which structures are adjoints to units, 1995]: ���� -monads (lax idempotent monads) in 2-cats KZ Kock-Z¨ oberlein Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52
[A. Kock, Monads for which structures are adjoints to units, 1995]: ���� -monads (lax idempotent monads) in 2-cats KZ Kock-Z¨ oberlein [M. Escard´ o, Properly injective spaces and function spaces, 1998]: Often, in order-enriched categories, injective objects = Eilenberg-Moore algebras of a KZ-monad = Kan-injective objects Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52
[A. Kock, Monads for which structures are adjoints to units, 1995]: ���� -monads (lax idempotent monads) in 2-cats KZ Kock-Z¨ oberlein [M. Escard´ o, Properly injective spaces and function spaces, 1998]: Often, in order-enriched categories, injective objects = Eilenberg-Moore algebras of a KZ-monad = Kan-injective objects [M. Carvalho, L.S., 2011] : Kan-injectivity/KZ-monads enjoys many features resembling Orthogonality/Idempotent monads Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 2 / 52
• M. Carvalho, L. S., Order-preserving reflectors and injectivity, TA , 2011 • J. Ad´ amek, L. S., J. Velebil, Kan injectivity in order-enr. cats., MSCS , 2015 • M. Carvalho, L. S., On Kan-injectivity of locales and spaces, ACS , 2017 • L. S., A calculus of lax fractions, JPAA , 2017 • J. Ad´ amek, L. S., KZ-monadic categories and their logic, TAC , 2017 • D. Hofmann, L. S., Aspects of algebraic algebras, LMCS , 2017 Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 3 / 52
• M. Carvalho, L. S., Order-preserving reflectors and injectivity, TA , 2011 • J. Ad´ amek, L. S., J. Velebil, Kan injectivity in order-enr. cats., MSCS , 2015 • M. Carvalho, L. S., On Kan-injectivity of locales and spaces, ACS , 2017 • L. S., A calculus of lax fractions, JPAA , 2017 • J. Ad´ amek, L. S., KZ-monadic categories and their logic, TAC , 2017 • D. Hofmann, L. S., Aspects of algebraic algebras, LMCS , 2017 • M. M. Clementino, F. Lucatelli, J. Picado: joint work in progress Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 3 / 52
1. Kan-injectivity and KZ-monads 2. In locales and topological spaces 3. Lax fractions 4. Kan-injective subcategory problem Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 4 / 52
� Most of the time, the setting is order-enriched categories g A � B ≤ f Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 5 / 52
1. Kan-injectivity and KZ-monads Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52
1. Kan-injectivity and KZ-monads Monad T = ( T , η, µ ) of Kock-Z¨ oberlein type: T η ≤ η T ( ⇐ ⇒ every T -algebra ( X , α ) has α ⊢ η X ) Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52
1. Kan-injectivity and KZ-monads Monad T = ( T , η, µ ) of Kock-Z¨ oberlein type: T η ≤ η T ( ⇐ ⇒ every T -algebra ( X , α ) has α ⊢ η X ) KZ-monadic subcategory of X = Eilenberg-Moore category of a KZ-monad over X Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52
1. Kan-injectivity and KZ-monads Monad T = ( T , η, µ ) of Kock-Z¨ oberlein type: T η ≤ η T ( ⇐ ⇒ every T -algebra ( X , α ) has α ⊢ η X ) KZ-monadic subcategory of X = Eilenberg-Moore category of a KZ-monad over X Full reflective subcategory of X = Eilenberg-Moore category of an idempotent monad over X ( T η = η T ) Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 6 / 52
1. Kan-injectivity and KZ-monads T = ( T , η, µ ) idempotent: A ∈ X T iff it is orthogonal to all η X , i.e., X ( η X , A ) � X ( X , A ) X ( TX , A ) is an isomorphism. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 7 / 52
1. Kan-injectivity and KZ-monads T = ( T , η, µ ) idempotent: A ∈ X T iff it is orthogonal to all η X , i.e., X ( η X , A ) � X ( X , A ) X ( TX , A ) is an isomorphism. T = ( T , η, µ ) KZ-monad: A ∈ X T iff X ( η X , A ) � X ( X , A ) X ( TX , A ) is a right adjoint retraction. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 7 / 52
1. Kan-injectivity and KZ-monads T = ( T , η, µ ) idempotent: A ∈ X T iff it is orthogonal to all η X , i.e., X ( η X , A ) � X ( X , A ) X ( TX , A ) is an isomorphism. T = ( T , η, µ ) KZ-monad: A ∈ X T iff X ( η X , A ) � X ( X , A ) X ( TX , A ) is a right adjoint retraction. g is a right adjoint retraction if there is an adjunction ( id , β ) : f ⊣ g In order enriched categories: gf = id and fg ≤ id Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 7 / 52
1. Kan-injectivity and KZ-monads A is (left) Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52
� � 1. Kan-injectivity and KZ-monads A is (left) Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction. Equivalently: for all f : X → A , there exists a left Kan extension of f along h of the form Lan h ( f ) = ( f / h , id ). h � Y X = f f / h =( X ( h , A )) ∗ ( f ) A Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52
� � 1. Kan-injectivity and KZ-monads A is (left) Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction. Equivalently: for all f : X → A , there exists a left Kan extension of f along h of the form Lan h ( f ) = ( f / h , id ). h � Y X = f f / h =( X ( h , A )) ∗ ( f ) A k : A → B is (left) Kan-injective wrt h : X → Y , if A and B are so, and k preserves the left Kan extension of every f : X → A along k . Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52
� � � � � � � 1. Kan-injectivity and KZ-monads A is (left) Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction. Equivalently: for all f : X → A , there exists a left Kan extension of f along h of the form Lan h ( f ) = ( f / h , id ). h � Y X = f f / h =( X ( h , A )) ∗ ( f ) A k : A → B is (left) Kan-injective wrt h : X → Y , if A and B are so, and k preserves the left Kan extension of every f : X → A along k . Equivalently: ( X ( h , A )) ∗ X ( Y , A ) X ( X , A ) A X ( Y , k ) X ( X , k ) k X ( Y , B ) X ( X , B ) B ( X ( h , B )) ∗ Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 8 / 52
1. Kan-injectivity and KZ-monads For H ⊆ Mor( X ), KInj( H ) � �� � := (locally full) subcategory of objects and morphisms Kan-injective wrt all h ∈ H (Left) Kan-injective subcategory Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 9 / 52
1. Kan-injectivity and KZ-monads For H ⊆ Mor( X ), KInj( H ) � �� � := (locally full) subcategory of objects and morphisms Kan-injective wrt all h ∈ H (Left) Kan-injective subcategory For T = ( T , η, µ ) a KZ-monad over X order-enriched, X T = KInj( { η X | X ∈ X} ). Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 9 / 52
� � � 1. Kan-injectivity and KZ-monads A a (locally full) subcategory of X A is closed under left adjoint retractions, if, for every commutative diagram f A B q q ′ � Y X g with q and q ′ left adjoint retractions, whenever f ∈ A , then g ∈ A . Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 10 / 52
� � � � 1. Kan-injectivity and KZ-monads A a (locally full) subcategory of X A is an inserter-ideal, provided that, for every inserter diagram A g i g � B I ⇑ B � A I i = ins ( f , g ) f i � f A f ∈ A = ⇒ i ∈ A . Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 11 / 52
1. Kan-injectivity and KZ-monads Theorem ([CS, 2011], [ASV, 2015]) Given H ⊆ Mor ( X ) , KInj( H ) is: 1 Closed under weighted limits, i.e., the inclusion functor KInj( H ) ֒ → X creates weighted limits; 2 An inserter-ideal; 3 Closed under left adjoint retractions. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 12 / 52
1. Kan-injectivity and KZ-monads Theorem ([CS, 2011], [ASV, 2015]) Given H ⊆ Mor ( X ) , KInj( H ) is: 1 Closed under weighted limits, i.e., the inclusion functor KInj( H ) ֒ → X creates weighted limits; 2 An inserter-ideal; 3 Closed under left adjoint retractions. Corollary Every KZ-monadic subcategory enjoys properties 1, 2 and 3 above. Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 12 / 52
1. Kan-injectivity and KZ-monads Theorem ([ASV, 2015]) Let X have inserters. A reflection of X in a subcategory A is of Kock-Z¨ oberlein type (i.e. it induces a KZ-monad), iff A is an inserter-ideal of X . Category Theory 2018, Azores, 8-14 July Kan-injectivity and KZ-monads 13 / 52
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