R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....popular in various areas of category theory in the 60 s. In particular, much of universal algebra can be formulated in terms of monads [16] It soon became clear that monads are in fact a 2 categorical concept, not restricted to the 2 category of categories, functors, and natural transformations [26]. Here we are interested in monads in an enriched context, so called strong monads over a symmetric monoidal closed category X . Such an X is equipped with ffl a functor X Theta X Omega Gamma X (called a tensor product) that is associative, i.e. Omega Z and X Omega (Y Omega Z) are ....

Ross Street. The formal theory of monads. J. Pure Appl. Algebra, 2:149--168, 1972.

....construction for their comonad, one obtains a full subcategory Adjem of Adj . Similarly, one obtains a full subcategory Adj k of Adj by considering those adjunctions that coincide with the co Kleisli construction. As category of comonads we consider a strict version of the one de ned by Street in [26]: De nition 26 (Comonads) We denote by Comon the category whose objects are comonads (T ; on an arbitrary category B ; and whose morphisms from (T ; on B to (T 0 ; 0 ; 0 ) on B 0 are functors K : B B 0 satisfying KT = T 0 K, K = 0 K and K = 0 K. With our ....

Ross Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972. 25

....with respect to the linear exponential comonads. 3 De nition 4 Let C, D be models for linear logic. The functor F : C D is linearly distributive 1 if and only if F is monoidal (with structure m I ; m C;C 0 ) and is equipped with a distributive law in the sense of Beck [9] see also [48]) F F respecting the comonoid structure, in the sense that I e F (C) F (C) d F (C) F (C) F (C) F (I) m I F (e C ) F ( C) C F (d C ) F ( C C) m C; C ( C C ) commutes. 2.4 Examples Examples derived from basic set theory Sup lattices. Let ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

....sense: M , being a monoid in BimodT (K) X; X) acts on itself by composition, while N has a similar (N; N) bimod structure, which is transferred by change of base along the morphisms (f; f h ) g; g h ) M N . 4.5. Remark. Our de nition of morphism of monads above is a special case of that of [Str72], where we have restricted the 1 cells to be representable bimodules. This is in accordance with the similar restriction we imposed on morphisms of lax algebras. Our de nition of 2 cells mimics that of transformation between morphisms of multicategories [Her99b, Def. 6.6] 4.6. Remark. ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

....partly because the discussion of arrows and transformations would distract the reader from the simple ideas presented here and partly because the relationship is at the mere object level very tight. Still, a word or two on the matter for the reader interested in such things and familiar with [ST1] is warranted. In [ST1] Street defined for any 2 category C a 2category Mnd(C) whose objects are the monads in C. He showed that the objects of Mnd(Mnd(C) are distributive laws and thus a definition of arrow between distributive laws and of transformation between those has already been ....

....discussion of arrows and transformations would distract the reader from the simple ideas presented here and partly because the relationship is at the mere object level very tight. Still, a word or two on the matter for the reader interested in such things and familiar with [ST1] is warranted. In [ST1] Street defined for any 2 category C a 2category Mnd(C) whose objects are the monads in C. He showed that the objects of Mnd(Mnd(C) are distributive laws and thus a definition of arrow between distributive laws and of transformation between those has already been provided. The adaptation of ....

[Article contains additional citation context not shown here]

R. Street. The formal theory of monads. Preprint, 2000.

....with the simplest case, which nevertheless contains all the essential details: oplax colimits for oplax 21 functors G : 1 Fib, where 1 is the terminal (one object) category. In this case G amounts to a comonad ( G; G) p p in Fib, and its oplax colimit to its Kleisli object p ( G;G) [Str72a]. To structure the proof, we state some auxiliary lemmas rst, in line with the construction of colimits outlined above. We omit the laborious calculations involved in the veri cation that the constructions provided satisfy the relevant properties axioms; details can be found in [Her93] In any ....

....veri cation that the constructions provided satisfy the relevant properties axioms; details can be found in [Her93] In any 2 category K, given a comonad hG : A A; i, we write (U; for an oplax cocone for it, i.e. U : A C and : U)UG satisfying U = 1 U and G = U . Recall, e.g. from [Str72a], that when (U; is a colimiting oplax cocone, U has a left adjoint F which gives a resolution for the comonad G, i.e. the adjunction F a U generates the comonad hG : A A; i. 5.1. Lemma. Given a comonad h( G; G) E #p B E #p B ; i for E #p B , a bred ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

....Let T be a V monad on V . Then, one can speak of the Kleisli V category Kl(T ) for T , and there is a V functor j : V Gamma Kl(T ) that is the identity on objects, has a right adjoint, and satisfies the usual universal property of Kleisli constructions, as explained in Street s article [15]. Since j has a right adjoint, it preserves tensors. So if we restrict j to the finitely presentable objects of V , we have a bijective on objects finite tensor preserving V functor from V f to the full sub V category Kl(T ) f of Kl(T ) determined by the objects of V f . Dualizing, we have a ....

Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149168.

....rst constructing the (generalized) autonomous category V chu = hV; T ichu , where T is terminal in V , and then recovering the categories hV; Xichu , X 2 V ob , as the categories of endo modules for certain monads in V chu . Monads and modules in bicategories are well understood, cf. e.g. [19], 5] or more recently [14] Since Barr s rst construction step di c 1999 Kluwer Academic Publishers. Printed in the Netherlands. kchu.tex; 4 06 1999; 13:24; p.1 2 rectly carries over to any closed bicategory X with local nite limits and coequalizers, we are led to generalize the notion ....

Street, R. The formal theory of monads. J. Pure Appl. Algebra 2 (1972), 149-168.

....are able to simplify the study of distributive laws between them. This we do in Section 4. We express our results for such monads on an object in an ord cat category, where ord denotes the 2 category of antisymmetric ordered sets. 1.3. A brief word on the level of generality may be helpful. In [STR] Street defined and studied monads on objects in an arbitrary 2 category. His results are easily extended to monads on objects in bicategories either directly or by using the coherence theorem which states that each bicategory is biequivalent to a 2 category. It has become clear that ....

....M. Barr s diagram macros. 1991 Mathematics Subject Classification: 06D10. Key words and phrases: KZ doctrine, adjunction, completely distributive. 1 2 Our modifications (3 cells) are mere inequalities and we assume that any instance of is antisymmetric. 1.4. Section 3 is in the spirit of [STR] and the results are valid in any bicategory. We add a new formulation of distributive laws that is useful when the ambient bicategory does not necessarily admit the construction of algebras in the terminology of [STR] or, said otherwise, Eilenberg Moore objects are not known to exist) This ....

[Article contains additional citation context not shown here]

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

....structural recursion ( 22] Theorem 5.1) ii) Since the category B coalg is isomorphic to D Coalg, the category of Eilenberg Moore coalgebras for the comonad (D; ffi; ffl) cofreely generated by B, the monad (T; j) lifts to the category D Coalg. iii) By using an adjoint lifting theorem (see [8,19,17,18]) a lifting of the monad T to D Coalg is equivalent to a distributive law : TD ) DT of T over D. The first step above is intricate because of the use of a structural recursion theorem. So we shall give a different proof. 5.2 Feedback from our framework Now we use our framework to analyse the ....

R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149-- 168

....conditions [3] Moreover, 3 our de nitions and analysis naturally live at the level of 2 categories, so that level of generality makes the choices clearest and the proofs simplest. Mathematically, this puts our analysis exactly at the level of generality of the study of monads by Street in [15], but see also Johnstone s [6] for an analysis of adjoint lifting that extends to this setting. The 2 categorical treatment clari es the conditions needed for adjoint lifting. The topic of our study, distributivity for monads and comonads, agrees with that of MacDonald and Stone [9, 10] when ....

....monads and comonads, agrees with that of MacDonald and Stone [9, 10] when restricted to Cat. Mulry [12] has also done some investigation into liftings to Kleisli categories. Much of the abstract work of the rst four technical sections of this paper is already in print, primarily in Street s paper [15]. But that is an old paper that was directed towards a mathematical readership; it contains no computational examples or analysis; and the material relevant to us is interspersed with other work that is not relevant. We happily acknowledge Street s contribution, but thought it worthwhile to repeat ....

[Article contains additional citation context not shown here]

R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149-168

....of our present work. Having identi ed categories with monads, we could expect the rest of the structure (namely functors and natural transformations) to follow from this identi cation. This is not quite as straightforward: Street s original formulation of the 2 category of monads in a 2 category [Str72] was designed to account for the category of algebras construction, and his de nition does not apply to obtain the 2 category of internal categories in a direct way. Street has recently elaborated an alternative version to sort out this di culty [Str99] see also Remark 6.4) We will address this ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

....using structural recursion ( 22] Theorem 5.1) ii) Since the category B coalg is isomorphic to D Coalg, the category of Eilenberg Moore coalgebras for the comonad (D; cofreely generated by B, the monad (T; lifts to the category D Coalg. iii) By using an adjoint lifting theorem (see [8,19,17,18]) a lifting of the monad T to D Coalg is equivalent to a distributive law : TD ) DT of T over D. The rst step above is intricate because of the use of a structural recursion theorem. So we shall give a di erent proof. 5.2 Feedback from our framework Now we use our framework to analyse the ....

R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149{ 168

....M , being a monoid in BimodT (K) X; X) acts on itself by composition, while N has a similar (N; N) bimod structure, which is transferred by change of base along the morphisms (f; f h ) g; g h ) M N . 4.5. Remarks. Our de nition of morphism of monads above is a special case of that of [Str72], where we have restricted the 1 cells to be representable bimodules. This is in accordance with the similar restriction we imposed on morphisms of lax algebras. Our de nition of 2 cells mimics that of transformation between morphisms of multicategories [Her99b, Def. 6.6] Consistently with our ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

.... monad is an obvious variation on that of a monoidal monad , and is strictly dual to that of monoidal comonad used e.g. by Boardman [Bo] Following an earlier version of this paper, McCrudden [McC] has explained how Hopf monads t into the general framework of monads on objects in a 2 category [S], and has proved general versions of Proposition 2.2 and Theorem 7.1 in this context. Acknowledgements. This paper is a faithful write up of my talk at the Category Meeting in Coimbra (July 1999) and I would like to thank the organizers of this meeting for inviting me. I am grateful to J. ....

R. Street, The formal theory of monads, J. Pure Appl. Alg. 2 (1972), 149-168. 20

....c 0 ) to C 2 (F 1 c; F 2 c 0 ) are equal. Adjunctions are the basic tool to de ne data types, while monads are used to model computations (see [Mog89b] Their de nition can be rephrased in the language of 2 categories and most of their properties can be proved in such a formal setting (see [Str72]) so these standard tools can be applied in a di erent 2 category, e.g. that of indexed categories (see Section 2) De nition 1.6 Let c and c 0 be objects of a 2 category C. A monad over c is a triple (T; s.t. c c c T c id c ) T T ; T ) T c c (T ; T ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2, 1972.

....corresponding to id TF via the following sequence of steps C(F; T op ; TF ) D( TF; TF ) C(F; op ; TF ) C(F; S op ; TF ) D( SF; TF ) Moreover, Y: is a 2 natural transformation. Since monads are a 2 categorical concept (see [Str72]) the 2 functor maps monads in Cat to monads in CAT. Then, the statement of Theorem 4.9 about lifting of monads follows immediately from Proposition 4.11. It remains to de ne the lifting t of a tensorial strength t for a monad (T ; over a small category C. Proposition 4.12 If C is a ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2, 1972. 29

....of our present work. Having identified categories with monads, we could expect the rest of the structure (namely functors and natural transformations) to follow from this identification. This is not quite straightforward: Street s original formulation of the 2 category of monads in a 2 category [Str72] was designed to account for the category of algebras construction, and his definition does not apply to obtain the 2 category of internal categories in a direct way. Street has recently elaborated an alternative version to sort out this difficulty [Str99] see also Remark 6.4) We will address ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

....Let T be a V monad on V . Then, one can speak of the Kleisli V category Kl(T ) for T , and there is a V functor j : V Gamma Kl(T ) that is the identity on objects, has a right adjoint, and satisfies the usual universal property of Kleisli constructions, as explained in Street s article [15]. Since j has a right adjoint, it preserves tensors. So if we restrict j to the finitely presentable objects of V , we have an identity on objects (strictly) finite tensor preserving V functor from V f to the full sub V category Kl(T ) f of Kl(T ) determined by the objects of V f . Dualising, we ....

Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972) 149-- 168.

.... = N(D C ) 4) as (ND) ND) colim On = lim (ND) On = lim N(D Sigma n ) N(lim D Sigma n ) N(D colim Sigma n ) N(D C ) In fact, from (4) we obtain a natural isomorphism : N ) N( C ) making the pair (N; into a morphism W C W of monads (see [40] for this notion) We thus get a functor C WC e W sending a W C algebra D C W D to the W algebra (ND) N(D C ) N( W ) ND. Moreover, the diagram (N ) N( C ) N ) N ) succ # = N( C ) # N( succC ) commutes, and so the functor C WC ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

....conditions [3] Moreover, our definitions and analysis naturally live at the level of 2 categories, so that level of generality makes the choices clearest and the proofs simplest. Mathematically, this puts our analysis exactly at the level of generality of the study of monads by Street in [11], but see also Johnstone s [6] for an analysis of adjoint lifting that extends to this setting. Formally, we recall the definition of 2 category in Section 2, define the notion of a monad in a 2 category, and characterise the Eilenberg Moore construction in those terms. We also explain a dual, ....

....to this setting. Formally, we recall the definition of 2 category in Section 2, define the notion of a monad in a 2 category, and characterise the Eilenberg Moore construction in those terms. We also explain a dual, yielding the Kleisli construction. This is all essentially in Street s paper [11]. In Section 3, we give another dual, yielding accounts of the Eilenberg Moore and Kleisli constructions for comonads. Then, in Section 4 lies the heart of the paper, in which we consider the eight possible combinations of monads and comonads, characterise two of them, explain how they arise in ....

[Article contains additional citation context not shown here]

R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149--168

....(R S SD ) T D ) R STD ) SH) The action on natural transformations : H ) H 0 is de ned similarly. The counit and comultiplication are de ned as = R S and = T SS S S . It is worth noticing that the result follows also axiomatically by dualising in the spirit of [29], the characterisation of distibutive laws in terms of monad liftings given in [1] Taking S = and R = C , we see lifting as a comonad on the Kleisli category Kl(T ) when it behaves like a very weak exponential; it is weak in the sense that we cannot expect its coKleisli category to be ....

R. Street. Formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

....the B coalgebras if UB e T = TUB , ie the diagram C C T UB UB e T B Coalg B Coalg commutes. Cf [15] When both T and e T are monads, e T lifts the monad T to the B coalgebras if the forgetful functor UB : B Coalg C (together with the identity natural transformation) is a monad morphism [27] from e T to T . 2 Remark 5.1 A monad e T lifts a monad T = hT ; j; i to the B coalgebras if and only if UB e T = TUB and, for every B coalgebra k : X BX , the diagram TX X T 2 X BTX BT 2 X BX e T 2 (k) e T (k) k jX X B X BjX commutes. 2 Consider now T to be the monad freely ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

....0 o : The functor GA is given on objects and arrows by projecting on I and ff. In fact, GA is a fibration [5] We can now proceed to describe the relational completion monad. We recall that a monad in a 2 category is a 1 cell with a monoid structure given by unit and multiplication 2 cells [17]. Thus in cat 2 a monad is an arrow of cat 2 together with unit and multiplication 2 cells. We note that a monad on A : A 0 Gamma A 1 in cat 2 is specified by a monad M 0 in cat on A 0 , a monad M 1 in cat on A 1 and a 2 cell M : M 1 A = AM 0 . These must satisfy equations involving ....

....Now the definition of Gamma is straightforward. For any object (oe; R) of grD(A) the natural isomorphism required has components the identity on C 1 (oe) The reader will have no trouble verifying that C is a monad on GD(A) in cat 2 . The category grD(A) C 0 of Eilenberg Moore algebras [17] for the monad C 0 has objects pairs (oe; R) where oe is the database signature of a relationally complete database with relations R. The Eilenberg Moore algebras set=M (I) C 1 for C 1 has objects database signatures with interpretations for the operations of relational algebra. There is a ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

....it is these that we shall call T morphisms, the detailed definition being given below. We shall use the name strict T morphism for those preserving the structure on the nose; they retain a certain theoretical importance, as being the morphisms of the Eilenberg Moore object . Various authors [11,2,26] have also pointed out the importance of lax morphisms, and these too will play a prominent role in our analysis; we define a lax T morphism from a T algebra (A; a) to a T algebra (B; b) to be a pair (f; f) where f : A B is a morphism in K, and f is a 2 cell, not necessarily ....

R.H. Street, The formal theory of monads, J. Pure and Appl. Algebra, 2:149--168, 1972.

....stability is guaranteed whenever the object A is functionally complete, i.e. when j I has a right adjoint. We spell this out in more detail for categories and fibrations in the following subsections. Further details on indeterminates and functional completeness can be found in [HJ93] We refer to [Str72] for the relevant definitions of comonads and their associated morphisms, as well as Kleisli objects for them in a 2 category. Anyway, these concepts are not essential to understand what follows. 5.1. Stability of initial algebras in a distributive bicategory The material in this subsection is ....

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

No context found.

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149-168, 1972.

No context found.

R. Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2:149--168, 1972.

No context found.

R. Street, The Formal Theory of Monads, J. Pure Appl. Algebra, Vol. 2, pp. 149-168, 1972.

No context found.

R. Street. The formal theory of monads. Journal for Pure and Applied Algebra, 2:149--168, 1972.

No context found.

R. Street. The formal theory of monads. J. Pure Appl. Algebra 2 (1972) 149--168.

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