A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the previous sections. We also characterise when an equational lifting monad is dominical (Theorem 3) and discuss other miscellaneous properties of lifting monads. 2 Preliminaries In this section, we briefly review facts we require about monads [11, 1] monoidal categories [11] strong monads [10, 14] and idempotent splittings. The reader may prefer to skip this section, and refer back to it as and when necessary. First, some general remarks about our policy towards structure preserving functors between categories with additional structure. In all cases, the correct general notion of ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

....the previous sections. We also characterise when an equational lifting monad is dominical (Theorem 3) and discuss other miscellaneous properties of lifting monads. 2 Preliminaries In this section, we briefly review facts we require about monads [10, 1] monoidal categories [10] strong monads [9, 13] and idempotent splittings. The reader may prefer to skip this section, and refer back to it as and when necessary. Given a monad (T , #, on a category C, we write C T for the Kleisli category, J : C # C T for the associated left adjoint functor, and K : C T # C for its right adjoint. We ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

....takes as fundamental a commutative monoidal structure on C, which models the tensor product of linear logic (see [6, 14] If C is a monoidal closed category, in particular a ccc, then it can be enriched over itself by taking C(A; B) to be the object B A . The equations for t are taken from [5], where a one one correspondence is established between functorial and tensorial strengths 3 : the rst two equations say that t is a tensorial strength of T , so that T is a C enriched functor. the last two equations say that and are natural transformations between C enriched functors, ....

....B (obtained by rst evaluating the rst argument and then the second) There is also 3 A functorial strength for an endofunctor T is a natural transformation st A;B : B A (TB) TA which internalizes the action of T on morphisms. a dual notion of pairing, A;B = c A;B ; B;A ; T c B;A (see [5]) which amounts to rst evaluating the second argument and then the rst. The reason why a functional type A B in a programming language (like ML) cannot be interpreted by the exponential B A (as done in a ccc) is fairly obvious; in fact the application of a functional procedure to an ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

.... B id A B A B R A TB t A;B T (A B) id A B I A B A T 2 B t A;TB T (A TB) T t A;B T 2 (A B) where r and are the natural isomorphisms r A : 1 A) A ; A;B;C : A B) C A (B C) Remark 3. 3 The diagrams above are taken from [Koc72], where a characterisation of strong monads is given in terms of C enriched categories (see [Kel82] Kock xes a commutative monoidal closed category C (in particular a cartesian closed category) and in this setup he establishes a one one correspondence between strengths st A;B : B A (TB) ....

....Kock xes a commutative monoidal closed category C (in particular a cartesian closed category) and in this setup he establishes a one one correspondence between strengths st A;B : B A (TB) TA and tensorial strengths t A;B : A TB T (A B) for a endofunctor T over C (see Theorem 1. 3 in [Koc72]) Intuitively a strength st A;B internalises the action of T on morphisms from A to B, and more precisely it makes (T ; st) a C enriched endofunctor on C enriched over itself (i.e. the hom object C(A; B) is B A ) In this setting the diagrams of De nition 3.2 have the following meaning: the ....

[Article contains additional citation context not shown here]

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

....coproduct Phi. We add to this the following: Definition 13 (Strong Monads) A strong monad is a monad h Sigma( Delta) ji on C, together with a natural transformation oe : Sigma( Delta) Omega Delta Sigma( Delta Omega Delta) such that the diagrams in Table 11 commute. See [20] cf. [9, 10, 11]. 42 Table 11 Strong Monads Sigma Sigma SigmaA SigmaA Sigma A fflffl Sigma SigmaA A fflffl Sigma SigmaA A SigmaA SigmaA j SigmaA E E E E E E E E E E E E E E E E Sigma SigmaA A fflffl SigmaA Sigmaj A oo y y y y y y y y y y y y y y y y SigmaA ( SigmaA) Omega ....

Anders Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

....an elementary description that will be more widely recognized. The molding will be performed quite independently of cartesian closedness. The starting point in generality is the notion of a V tensored category Y. Here V is a category with an arbitrary tensor product, as described by Anders Kock [9], and Y is a category with a V action on it given by a tensor preserving functor from V into Endo(Y) Our interest narrows to the special case of V s tensor product being the (cartesian) product. We will hereafter designate such a category V as X. Furthermore, we demand that Y have its own ....

A. Kock, Strong functors and monoidal monads. Archiv der Mathematik, 23 (1972) 113-120.

....formulates a similar closure property of the free construction: if the free construction can be cut down to a cartesian closed category then there the associated monad and the natural transformations that come with it can be internalized. This concept was introduced by Anders Kock [Kock, 1970, Kock, 1972] It has recently found much interest under the name computational monads through the work of Eugenio Moggi [Moggi, 1991] Theorem 6.1.9. For any signature Sigma and set E of inequalities the composition U ffi F is a locally continuous functor on CONT. Proof. The action of U ffi F on ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

....formulates a similar closure property of the free construction: if the free construction can be cut down to a cartesian closed category then there the associated monad and the natural transformations that come with it can be internalized. This concept was introduced by Anders Kock [Kock, 1970, Kock, 1972] It has recently found much interest under the name computational monads through the work of Eugenio Moggi [Moggi, 1991] 92 Samson Abramsky and Achim Jung Theorem 6.1.9. For any signature 6 and set E of inequalities the composition U ffi F is a locally continuous functor on CONT. Proof. ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

....to distinguish the copyable natural transformations from the shapely ones arises because c fl : Gamma) ThetaF 1)F is copyable, as witnessed by A Theta(1 ThetaF 1) id Theta c fl A ThetaF 1 A ThetaF 1 id Thetal c fl FA c fl but is not usually cartesian. Recall that a strength ([Koc72, Mog89, CS92]) for an endo functor F : C C is a transformation A;B : FA ThetaB F (A ThetaB) which is natural in A and B, and both unitary and associative, meaning that the following diagrams commute for all objects A; B and C ( F A) ThetaB) ThetaC Thetaid F (A ThetaB) ThetaC F ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

....t L(A Theta B) as the unique map making the square below into a pullback. A Theta B id A Theta B A Theta LB id Theta j t L(A Theta B) j Then A Theta LB t L(A Theta B) is natural in A and B and provides a (unique) strength for the monad in the sense of Kock [9] (see also [11, 12] The costrength LA Theta B t 0 L(A Theta B) can be defined analogously. A symmetric map LA Theta LB L(A Theta B) is obtained as the unique morphism making the square below into a pullback. A Theta B id A Theta B LA Theta LB j Theta j L(A ....

....is obtained as the unique morphism making the square below into a pullback. A Theta B id A Theta B LA Theta LB j Theta j L(A Theta B) j It is straightforward to verify that ffi Lt 0 ffi t = ffi Lt ffi t 0 ; 3) i.e. the monad is commutative in the sense of [9]. We end this section by noting that the following (apparently arbitrary) diagram always commutes for dominical lifting monads. LA Delta LA Theta LA Lhj; idi R L(LA Theta A) t (4) This diagram will be very important in the sequel. 3 Equational lifting monads In the ....

[Article contains additional citation context not shown here]

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113 -- 120, 1972.

....x = # ffi take ffi x = ffi x follows from the assumption about x. The induced morphism into LA ThetaLA is unique since h# ffi ; i is a monomorphism by the second equation above. 2 3 Shapely Functors Shapely functors will be defined using two properties of the list functor, its strength [Koc72] and stability, which we will now review. Construct 0 as B hnil; idi LA ThetaB oe cons Thetaid A ThetaLA ThetaB H H H H H H H H H H H hnil; idi j L(A ThetaB) ThetaB 0 oe h A ThetaL(A ThetaB) ThetaB id Theta 0 where h = hcons ffi (id Thetac) 0 i. Define the strength of ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

....) X ThetaZ ) Y ThetaZ ) which distributes data of type Z over X Y . It can be defined directly as the uncurried form of [x : X: z : Z: inl Y ThetaZ Delta hx; zi; y : Y: z : Z: inr X ThetaZ Delta hy; zi] More generally, for types S and T and type variable X we can define a strength [Koc72, Mog89, CS92] tau R;S;X;T : T [R=X ] S)T [R ThetaS=X ] provided that fv(S) fv(T [R=X] OE: 1) We will construct the strength from other, simpler parametrising operations, as follows. The co strength tau R;S;X;T 1 : R)T [S=X ] T [R ThetaS=X ] is given by y : R: map S;R ThetaS;X;T Delta (z : S: ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

....fact is now easily verified: Theorem 4.3. L; j; is a monad on Ass. In fact, it is easy to check that (L; j; is a commutative strong monad in the sense of (Kock 1970) The strength is a family of morphisms X Theta LY L(X Theta Y ) obtained from (f 7 Lf) Y X LY LX as in (Kock 1972). Note that, because Ass is well pointed the strength is determined by the monad. Commutativity is an easily verified property of the strength. However, we will not explicitly use these facts in this paper. Observe that Mod is closed under application of the lift functor L. So (L; j; cuts ....

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113 -- 120, 1972.

No context found.

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23, 1972.

No context found.

A. Kock. Strong functors and monoidal monads. Archiv der Mathematik, 23:113--120, 1972.

No context found.

A. Kock, Strong functors and monoidal monads, Archiv der Mathematik 23 (1972), 113 -- 120.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC