Categories of lax fractions Lurdes Sousa IPV / CMUC June 18, 2015 Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 1 / 19
Idempotent monads Replete full reflective subcategories Orthogonality Categories of fractions Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 2 / 19
Ordinary categories Order-enriched categories Idempotent monads Lax-idempotent monads (KZ-monads) Replete full reflective subcategories KZ-monadic subcategories Orthogonality Kan-injectivity [Carvalho, S. , 2011] [Ad´ amek, Velebil, S. , 2015] Categories of fractions Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 2 / 19
Ordinary categories Order-enriched categories Idempotent monads Lax-idempotent monads (KZ-monads) Replete full reflective subcategories KZ-monadic subcategories Orthogonality Kan-injectivity [Carvalho, S. , 2011] [Ad´ amek, Velebil, S. , 2015] Categories of fractions Categories of lax fractions Calculus of fractions Calculus of lax fractions Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 2 / 19
Setting: order-enriched categories and functors. X order-enriched category Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 3 / 19
� Setting: order-enriched categories and functors. X order-enriched category A is Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction (in Pos). h � Y X � � � a � � � a / h =( X ( h , A )) ∗ ( a ) � � � A Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 3 / 19
� Setting: order-enriched categories and functors. X order-enriched category A is Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction (in Pos). h � Y X � � � a � � � a / h =( X ( h , A )) ∗ ( a ) � � � A [M. Escard´ o, 1998]: Kan-injective objects as E.-M. algebras of KZ-monads. Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 3 / 19
� � � � � � Setting: order-enriched categories and functors. X order-enriched category A is Kan-injective wrt h : X → Y if X ( h , A ) � X ( X , A ) X ( Y , A ) is a right adjoint retraction (in Pos). h � Y X � � � a � � � a / h =( X ( h , A )) ∗ ( a ) � � � A [M. Escard´ o, 1998]: Kan-injective objects as E.-M. algebras of KZ-monads. k : A → B is Kan-injective wrt h : X → Y , if A and B are so, and (hom( h , A )) ∗ hom( Y , A ) hom( X , A ) A hom( Y , k ) hom( X , k ) k hom( Y , B ) hom( X , B ) B (hom( h , B )) ∗ i.e., ( ka ) / h = k ( a / h ), for all a : X → A . [CS, 2011] Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 3 / 19
For H ⊆ Mor( X ), KInj( H ) � �� � := subcategory of objs. and mors. Kan-injective wrt all h ∈ H Kan-injective subcategory Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 4 / 19
For H ⊆ Mor( X ), KInj( H ) � �� � := subcategory of objs. and mors. Kan-injective wrt all h ∈ H Kan-injective subcategory For a subcategory A of X , A KInj := class of all morphisms wrt to which A is Kan-injective Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 4 / 19
KZ-monadic subcategory := Eilenberg-Moore category of a KZ-monad Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 5 / 19
� � KZ-monadic subcategory := Eilenberg-Moore category of a KZ-monad 1 A subcategory A of X is KZ-monadic, iff it is reflective � � � X A ⊢ R with R η ≤ η R , and A is closed under left adjoint retractions f ∈A � B with e and e ′ l. a. r. then g ∈ A ) . (i.e., if A e � e ′ g � Y X [CS, 2011] Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 5 / 19
� � KZ-monadic subcategory := Eilenberg-Moore category of a KZ-monad 1 A subcategory A of X is KZ-monadic, iff it is reflective � � � X A ⊢ R with R η ≤ η R , and A is closed under left adjoint retractions f ∈A � B with e and e ′ l. a. r. then g ∈ A ) . (i.e., if A e � e ′ g � Y X 2 If A is a KZ-monadic subcategory of X , then: • A KInj = { f ∈ X | Rf left adj. section in A} = { f ∈ X | Rf left adj. section in X} = R -embeddings • A = KInj { η X | X ∈ X} = KInj( A KInj ) [CS, 2011] Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 5 / 19
Examples A KInj X Objs. of a KZ-monadic Injectivity of the subcategory A objects of A Top 0 continuous lattices embeddings [Scott, 1972] Top 0 continuous Scott dense embeddings [Scott, 1980] domains Loc stably locally compact locales flat embeddings [Johnstone, 1981] (=retracts of coherent locales) In all three examples, A KInj may be replaced with A a finite subcategory. [Carvalho, S., preprint] Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 6 / 19
������� A full reflective subcat. ⇒ A is a category of fractions for A Orth = { f | Rf is an iso } , up to equiv. Σ ⊆ Mor( X ) F : X → X [Σ − 1 ] Category of fractions: Category of “lax fractions”: F : X → X [Σ ∗ ] ( Fs ) ∗ · Fs = id and Fs · ( Fs ) ∗ ≤ id for all s ∈ Σ Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 7 / 19
������� A any subcat., X cocomplete ⇒ A Orth is closed under colimits in X → ⇒ A Orth admits a left calculus of fractions A Orth � � � X → full Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 8 / 19
������� A any subcat., X cocomplete ⇒ A Orth is closed under colimits in X → ⇒ A Orth admits a left calculus of fractions A Orth � � � X → full A KInj � � � X → Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 8 / 19
������� A any subcat., X cocomplete ⇒ A Orth is closed under colimits in X → ⇒ A Orth admits a left calculus of fractions A Orth � � � X → full A KInj � � � X → For A KInj with objects A KInj and convenient morphisms: Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 8 / 19
������� A any subcat., X cocomplete ⇒ A Orth is closed under colimits in X → ⇒ A Orth admits a left calculus of fractions A Orth � � � X → full A KInj � � � X → For A KInj with objects A KInj and convenient morphisms: Theorem If X has weighted colimits, then A KInj is closed under weighted colimits in X → . Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 8 / 19
� � � � � � � � � X → A KInj � � The morphisms of A KInj : ( f , g ) : h → j with ( af ) / h = ( a / j ) g h for every X Y g f Z W j � � � a � � ( af ) / h � a / j � � � A ∋ A Equivalently: (hom( h , A )) ∗ � hom( Y , A ) hom( X , A ) commutes. hom( f , A ) hom( g , A ) � hom( W , A ) hom( Z , A ) (hom( j , A )) ∗ Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 9 / 19
Definition A category of lax fractions for Σ consists of a category X [Σ ∗ ] and a functor F : X → X [Σ ∗ ] such that: 1 The functor F satisfies the conditions: (a) F ( s ) is a left adjoint section, for all s ∈ Σ; Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 10 / 19
Definition A category of lax fractions for Σ, where Σ is a subcategory of X → , consists of a category X [Σ ∗ ] and a functor F : X → X [Σ ∗ ] such that: 1 The functor F satisfies the conditions: (a) F ( s ) is a left adjoint section, for all s ∈ Σ; Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 10 / 19
� � � � � � � Definition A category of lax fractions for Σ, where Σ is a subcategory of X → , consists of a category X [Σ ∗ ] and a functor F : X → X [Σ ∗ ] such that: 1 The functor F satisfies the conditions: (a) F ( s ) is a left adjoint section, for all s ∈ Σ; r (b) For every square • • with ( f , g ) : r → s a morphism of g f � • • s Σ, the following diagram is commutative: ( Fr ) ∗ • • Fg Ff • • ( Fs ) ∗ Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 10 / 19
� � � � � � � Definition A category of lax fractions for Σ, where Σ is a subcategory of X → , consists of a category X [Σ ∗ ] and a functor F : X → X [Σ ∗ ] such that: 1 The functor F satisfies the conditions: (a) F ( s ) is a left adjoint section, for all s ∈ Σ; r (b) For every square • • with ( f , g ) : r → s a morphism of g f � • • s Σ, the following diagram is commutative: ( Fr ) ∗ • • Fg Ff • • ( Fs ) ∗ 2 If G : X → C is another functor under the above conditions, then there is a unique fuctor H : X [Σ ∗ ] → C such that HF = G . Category Theory 2015, Aveiro, 14-19 June Categories of lax fractions 10 / 19
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