Lax-algebraic theories and closed objects Dirk Hofmann University of Aveiro dirk@mat.ua.pt 1
� � � � � A lax-algebraic theory T is a triple T = ( T , V , ξ ) consisting of a monad T = ( T, e, m ), a quantale V = ( V , ⊗ , k ) and a map ξ : T V → V such that (M e ) 1 V ≤ ξ · e V , (M m ) ξ · Tξ ≤ ξ · m V , T ( ⊗ ) T k � (Q ⊗ ) T ( V × V ) (Q k ) T 1 T V T V ξ ! ≤ ≤ ξ � V , � V , 1 V × V k ⊗ (Q W ) ( ξ X ) X : P V → P V T is a natural transformation. 2
Examples. (a). I V = ( 1 , V , 1 V ) is a strict lax-algebraic theory. (b). Let T = ( T, e, m ) be a monad where T is taut and let V be a (ccd)-quantale. Then T V = ( T , V , ξ V ) is a lax-algebraic theory, where � ξ V : T V → V , x �→ { v ∈ V | x ∈ T ( ↑ v ) } . ⊗ V = ( L , V , ξ ⊗ ) is a strict lax-algebraic theory for each (c). L quantale V , where ξ ⊗ : L V → V . ( v 1 , . . . , v n ) �→ v 1 ⊗ . . . ⊗ v n () �→ k 3
The bicategory V - Mat : • objects: sets X , Y ,. . . • morphism: V -matrices r : X × Y → V , • composition: s · r ( x, z ) = � y ∈ Y r ( x, y ) ⊗ s ( y, z ) We extent T : Set → Set to T ξ : V - Mat → V - Mat by putting ξ r : TX × TY → V . T � ( x , y ) �→ ξ · Tr ( w ) w ∈ T ( X × Y ): T π X ( w )= x , T π Y ( w )= y Here ξ T r T ( X × Y ) − − → T V − → V . 4
� � � � � � � The following statements hold. ξ ( r ◦ ) = T ξ ( r ) ◦ . (a). For each V -matrix r : X − → Y , T ξ f and Tf ◦ ≤ T ξ f ◦ . (b). For each function f : X → Y , Tf ≤ T (c). T ξ s · T ξ r ≤ T ξ ( s · r ) provided that T satisfies (BC), and ξ ( s · r ) provided that (Q = ξ s · T ξ r ≥ T ⊗ ) holds. T (d). The natural transformations e and m become op-lax, that is, for every V -matrix r : X − → Y we have the inequalities: e Y · r ≤ T ξ r · e X , m Y · T ξ r ≤ T ξ r · m X . ξ T e X m X � T ξ X T ξ T ξ X T ξ X X � T ξ r T ξ T ξ r � T ξ r r � ≤ � ≤ � T � T ξ Y T ξ T ξ Y ξ Y Y e Y m Y 5
� � � � � � � Let T = ( T , V , ξ ) be a lax-algebraic theory. • A T -algebra ( T -category) is a pair ( X, a : TX − → X ) s. t. e X � m X � and X TX TTX TX � � � � � ≤ � � � T ξ a � � a � a � ≤ � � � � 1 X � � � � X. X TX a � k → a ( x, x ) T ξ a ( X , x ) ⊗ a ( x , x ) → a ( m X ( X ) , x ) • A map f : X → Y between T -algebras ( X, a ) and ( Y, b ) is a lax homomorphism ( T -functor) if T f � a ( x , x ) → b ( Tf ( x ) , f ( x )) . TX TY a � ≤ � b � Y X f 6
• The resulting category of T -algebras and lax homomorphisms we denote by T - Alg . Examples. (a). For each quantale V , I V - Alg = V - Cat . In particular, I 2 - Alg ∼ + - Alg ∼ = Ord and I P = Met . (b). U 2 - Alg ∼ = Top . + - Alg ∼ (c). U P = Ap . V - Alg ∼ ⊗ (d). L = V - MultiCat . 7
� � � � � Let T = ( T , V , ξ ) and T ′ = ( T ′ , V ′ , ξ ′ ) be lax-algebraic theories. • A morphism ( j, ϕ ) : T ′ → T of lax-algebraic theories is a pair ( j, ϕ ) consisting of a monad morphism j : T ′ → T and a lax homomorphism of quantales ϕ : V → V ′ such that ξ ′ · T ′ ϕ ≤ ϕ · ξ · j V . j V T ′ V T V T ′ ϕ T ′ V ′ ≤ ξ ξ ′ V ′ V ϕ 8
From now on we consider a strict lax-algebraic theory T = ( T , V , ξ ) where T satisfies (BC). Examples. (a). The identity theory I V , for each quantale V . ⊗ V = ( L , V , ξ ⊗ ). (b). For each quantale V , the theory L (c). Any lax-algebraic theory T = ( T , V , ξ ) with a (BC)-monad T , ⊗ = ∧ and ξ a Eilenberg-Moore algebra. + = ( U , P (d). The theory U P + ). + , ξ P 9
Then • V becomes a T -algebra ( V , hom ξ ) where hom ξ = hom · ξ , that is, hom ξ ( v , v ) = hom( ξ ( v ) , v ) . • the tensor product ⊗ on V can be transported to T - Alg by putting ( X, a ) ⊗ ( Y, b ) = ( X × Y, c ) where c ( w , ( x, y )) = a ( x , x ) ⊗ b ( y , y ) . 10
X ? When X ⊗ has a right adjoint Note that 1 → Y X X ⊗ 1 → Y Hence we consider { f : ˆ X → Y | f is a lax homomorphism } , where a ( x , x ) if T !( x ) = e 1 ( ⋆ ), a ( x , x ) = ˆ ⊥ else; and � d ( p , h ) = hom( a ( Tπ X ( q ) , x ) , b ( T ev( q ) , h ( x ))) . q ∈ T ( Y X × X ) ,x ∈ X q �→ p 11
Letv X = ( X, a ) be a T -algebra. • Assume that a · T ξ a = a · m X . Then d is transitive. • Assume that the structure d on V X is transitive. Then a · T ξ a = a · m X . • Each T -algebra is closed in T - Alg . • Each V -category is closed in T - Alg provided that Te · e = m ◦ · e . 12
� � The following assertions hold. • � : V I → V is a lax homomorphism. • hom( v, ) : V → V is a lax homomorphism for each v ∈ V . • v ⊗ : V → V is a lax homomorphism for each v ∈ V which satisfies T v � T 1 T V ≥ ξ ! � V . 1 v • For each T -algebra I , � : V I → V is a lax homomorphism. 13
� � � � � � � � � T - Kleisli . objects: sets X , Y , . . . morphism: V -matrices a : TX − → Y . ξ a · m ◦ composition: b ◦ a := b · T X , m ◦ X � TX TY TX TTX � � � � T ξ a � � � � � TY � a � b � � � b ◦ a � � � � � b � � Y Z Z 14
� � � Then e ◦ X : TX − → X is a lax identity for “ ◦ ”, that is a ◦ e ◦ e ◦ X = a and X ◦ a ≥ a . Moreover, c ◦ ( b ◦ a ) = ( c ◦ b ) ◦ a . X ) is a T -algebra iff e ◦ ( X, a : TX − → X ≤ a and a ◦ a ≤ a . Example: U 2 • e ◦ X is also a left unit (precisely) if we restrict ourself to those a : UX − → Y where { x ∈ UX | a ( x , y ) = true } is closed in UX . • This restriction of U 2 - Kleisli is 2-equivalent to CSet (where a morphism from X to Y is a finitely additive map c : PX → PY ). 15
� � Let X = ( X, a ) and Y = ( Y, b ) be T -algebras. • A ( T , V )- bimodule ψ : ( X, a ) − ◦ ( Y, b ) is a matrix ψ : TX − → → Y such that ψ ◦ a ≤ ψ and b ◦ ψ ≤ ψ . • For ( T , V )-categories ( X, a ) and ( Y, b ), and a V -matrix ψ : TX − → Y , the following assertions are equivalent. ◦ ( Y, b ) is a ( T , V )-bimodule. (a). ψ : ( X, a ) − → (b). Both ψ : | X | ⊗ Y → V and ψ : X op ⊗ Y → V are ( T , V )-functors. 16
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