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Monads from Comonads Comonads from Monads Monads from Comonads—Prologue WG 2.8 Marble Falls Monads from Comonads Comonads from Monads Ralf Hinze Computing Laboratory, University of Oxford Wolfson Building, Parks Road, Oxford, OX1 3QD, England [email protected] http://www.comlab.ox.ac.uk/ralf.hinze/ March 2011 University of Oxford—Ralf Hinze 1-39 Monads from Comonads—Some context WG 2.8 Marble Falls Buy one get one free! A common form of sales promotion (BOGOF). University of Oxford—Ralf Hinze 2-39 Monads from Comonads—Some context WG 2.8 Marble Falls 1 Monads Monads, a success story. University of Oxford—Ralf Hinze 3-39 Monads from Comonads—Some context WG 2.8 Marble Falls A ! M B University of Oxford—Ralf Hinze 4-39 Monads from Comonads—Some context WG 2.8 Marble Falls A monad consists of a functor M and natural transformations r : I ! M j : MM ! M The two operations have to go together: j · rM = id M j · Mr = id M j · Mj = j · j M MM Mj j MMM MM rM id M M j M j M g g r j MM M MM j University of Oxford—Ralf Hinze 5-39 Monads from Comonads—Some context WG 2.8 Marble Falls 1 Comonads Comonads, not exactly a success story. University of Oxford—Ralf Hinze 6-39 Monads from Comonads—Some context WG 2.8 Marble Falls N A ! B University of Oxford—Ralf Hinze 7-39 Monads from Comonads—Some context WG 2.8 Marble Falls A comonad consists of a functor N and natural transformations e : N ! I d : N ! NN The two operations have to go together: Ne · d = id N eN · d = id N Nd · d = dN · d d NN e N NN d N id N N d Nd d e g g N NN NNN NN dN University of Oxford—Ralf Hinze 8-39 Monads from Comonads—Some context WG 2.8 Marble Falls The simplest of all: the product comonad. N = − × X e = outl d = id M outr University of Oxford—Ralf Hinze 9-39 Monads from Comonads—Some context WG 2.8 Marble Falls Why has the product comonad not taken off? University of Oxford—Ralf Hinze 10-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls 2 Adjunctions • One of the most beautiful constructions in mathematics. • They allow us to transfer a problem to another domain. • They provide a framework for program transformations. University of Oxford—Ralf Hinze 11-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls Let C and D be categories. The functors L : C D and R : C ! D are adjoint, L a R, L ≺ C ? D R if and only if there is a bijection between the hom-sets C (L A; B) =∼ D(A; R B) that is natural both in A and B. The witness of the isomorphism is called the left adjunct. It allows us to trade L in the source for R in the target of an arrow. Its inverse is the right adjunct. University of Oxford—Ralf Hinze 12-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls Perhaps the best-known example of an adjunction is currying. − × X ≺ C ? C Λ : C (A × X ; B) =∼ C (A; B X ) (−)X The left adjunct Λ is also called curry and the right adjunct Λ◦ is also called uncurry. University of Oxford—Ralf Hinze 13-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls − × X ≺ C ? C Λ : C (A × X ; B) =∼ C (A; B X ) (−)X The images of the identity are function application and the return of the state monad. app = Λ◦ id : C (B X × X ; B) r = Λ id : C (A; (A × X )X ) University of Oxford—Ralf Hinze 14-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls The images of the identity are the counit and the unit of the adjunction. : LR ! I η : I ! RL An alternative definition of adjunctions builds solely on these units, which have to satisfy L · Lη = id L R · ηR = id R LRL RLR R η L R L η id id L L L R R R University of Oxford—Ralf Hinze 15-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls 2 Comonads and monads Every adjunction L a R induces a comonad and a monad. N = LR M = RL e = r = η d = LηR j = RL University of Oxford—Ralf Hinze 16-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls The curry adjunction induces the state monad. M A = (A × X )X r = Λ id j = Λ (app · app) The monad supports stateful computations, where the state X is threaded through a program. University of Oxford—Ralf Hinze 17-39 Monads from Comonads—Adjunctions WG 2.8 Marble Falls The curry adjunction induces the costate comonad. N A = AX × X e = app d = (Λ id) × X The context can be seen as a store AX together with a memory location X , a focus of interest. University of Oxford—Ralf Hinze 18-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls 3 Transforming natural transformations • Adjunctions provide a framework for program transformations. • All the operations we have encountered so far are natural transformations. • To deal effectively with those we develop a little theory of ‘natural transformation transformers’. University of Oxford—Ralf Hinze 19-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls 3 Post-composition Every adjunction L a R gives rise to an adjunction L− a R− between functor categories. L L− ≺ ≺ C ? D then C E ? DE R R− C E (LF; G) =∼ DE (F; RG) University of Oxford—Ralf Hinze 20-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls We write b−c for the lifted left adjunct and b−c◦ for its inverse. α : LF ! G β : F ! RG bαc : F ! RG bβc◦ : LF ! G The lifted adjuncts can be defined in terms of the units of the underlying adjunction: bαc = Rα · ηF bβc◦ = G · Lβ University of Oxford—Ralf Hinze 21-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls 3 Pre-composition Post-composition dualizes to pre-composition. Consequently, every adjunction L a R also induces an adjunction −R a −L. L −R ≺ ≺ C ? D then E D ? E C R −L E D (FR; G) =∼ E C (F; GL) University of Oxford—Ralf Hinze 22-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls We write d−e◦ for the lifted left adjunct and d−e for its inverse. β : FR ! G α : F ! GL dβe◦ : F ! GL dαe : FR ! G Again, the lifted adjuncts can be defined in terms of the units of the underlying adjunction: dβe◦ = βL · Fη dαe = G · αR An aide-mémoire: b−c turns an L in the source to an R in the target, while d−e turns an L in the target to an R in the source. University of Oxford—Ralf Hinze 23-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls 3 Transformation transformers If we combine b−c and d−e, we can send natural transformations of type LF ! GL to transformations of type FR ! RG. The order in which we apply the adjuncts does not matter. bdαec = dbαce dbβc◦e◦ = bdβe◦c◦ University of Oxford—Ralf Hinze 24-39 Monads from Comonads—Program transformations WG 2.8 Marble Falls If we assume that L and R are endofunctors, then we can nest b−c and d−e arbitrarily deep. α : Lm F ! GLn dbαcm en : Rn F ! GRm An aide-mémoire: the number of bs corresponds to the number of Ls in the source, and the number of ds corresponds to the number of Ls in the target. University of Oxford—Ralf Hinze 25-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls 4 Monads from comonads • Assume that a left adjoint is at the same time a comonad. • Then its right adjoint is a monad! • Dually, the left adjoint of a monad is a comonad. University of Oxford—Ralf Hinze 26-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls The ‘transformation transformers’ allow us to systematically turn the comonadic operations into monadic ones and vice versa. r = bec : I ! R e = brc◦ : L ! I j = bdddeec : RR ! R d = ddbj c◦e◦e◦ : L ! LL University of Oxford—Ralf Hinze 27-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls The curry adjunction, provides an example, where the left adjoint L = − × X is also a comonad. e = outl d = id M outr Consequently, L’s right adjoint R = (−)X is a monad with operations r = boutlc = Λ outl j = bddid M outreec = Λ (app · (app M outr)) The resulting structure is known as the reader monad. University of Oxford—Ralf Hinze 28-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls A × X ! B =∼ A ! B X University of Oxford—Ralf Hinze 29-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls Every (co)monad comes equipped with additional operations. The product comonad might provide a getter and an update operation: get = outr : L ! ∆X update (f : X ! X ) = id × f : L ! L where ∆X is the constant functor. The transforms of get and update correspond to operations called ask and local in the Haskell monad transformer library. ask = boutrc = Λ outr : I ! R ∆X local (f : X ! X ) = bdid × f ec = Λ (app · (id × f )) : R ! R University of Oxford—Ralf Hinze 30-39 Monads from Comonads—Comonads from monads WG 2.8 Marble Falls 4 Proof We have to show that the comonadic laws imply the monadic laws and vice versa.
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