# Direct models of the computational lambda-calculus

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3 var x 1 : A 1,..., x n : A n x i : A i let Γ M : A Γ, x : A N : B Γ let x = M in N : B * Γ : I, Γ M : A Γ N : B Γ M, N : A B π i Γ M : A 1 A 2 Γ π i (M) : A i constant f : A B Γ M : A Γ f(m) : B Fig. 1. Basic terms of the computational lambda-calculus λ Γ, x : A M : B Γ λx : A.M : A B app Γ M : A B Γ N : A Γ MN : B Fig. 2. Higher-order terms of the computational lambda-calculus 3

5 Rule Syntax Semantics var x 1 : A 1,..., x n : A n x i : A i = A 1 A n π i Ai λ Γ, x : A M : B = Γ A f TB Γ (λx : A.M) : A TB = Γ λf (TB) A ev Γ M : (TB) A = Γ f (TB) A Γ N : A = Γ g A Γ MN : TB = Γ f,g (TB) A A ev TB Γ : 1 = Γ! 1, Γ M : A = Γ f A Γ N : B = Γ g B Γ M, N : A B = Γ f,g A B π i Γ M : A 1 A 2 = Γ f A1 A 2 Γ π i (M) : A i = Γ f A1 A 2 π i Ai f : A B Γ M : A = Γ g A Γ f(m) : B = Γ g A f B Fig. 4. The Meta-language and its categorical semantics: The lambda-calculus part ing the monadic-style transform into the Meta-language and then using the semantics of the latter. The monadic-style transform sends a type A of the computational lambda-calculus to a type A of the Meta-language (Fig. 6), and it sends a sequent (x 1 : A 1,...,x ( n : A n M : A) of the computational lambda-calculus to a sequent x 1 : A 1,...,x n : A n M : T ( A )) of 5

6 Rule Syntax Semantics [ ] Γ M : A = Γ f A Γ [M] : A = Γ f A η TA let Γ M : TA = Γ f TA Γ, x : A N : TB = Γ A g TB Γ let x M in N : TB = Γ id,f Γ TA t T(Γ A) Tg TTB µ TB Fig. 5. The Meta-language and its categorical semantics: The monadic part (A B) = A B I = 1 (A B) = ( T ( B )) (A ) (LA) = T ( A ) Fig. 6. Monadic-style transform from the computational lambda-calculus into Moggi s Meta-language: Types the Meta-language (Fig. 7). Because the monadic-style transform is a compiler as explained in Section 1.1, we can conclude that Moggi s semantics of the computational lambda-calculus essentially performs a compilation. By a direct semantics, I mean a semantics that doesn t compile. So, is there a direct-style categorical semantics of the computational lambda-calculus? If so, what kind of categories do we need? And what is the semantic counterpart of the monadic-style transform? In other words, is there a solution for X and the two dashed arrows in figure 8? In this article, we ll find X, and we ll show that X is the only reasonable solution. We ll call the new categories direct λ C -models. They are abstract Kleisli-categories with extra structure. We get abstract Kleisli-categories by identifying the available algebraic structure on Kleisli categories. As we shall see, every monad induces an abstract Kleisli-category. What s more interest- 6

7 x = [x] (let x = M in N) = ( let x M in N ) = [ ] M, N = ( let x M in let y N in [ x, y ] ) (π i (M)) = ( let x M in [π i (x)] ) (f(m)) = ( let x M in f(x) ) [M] = [ M ] (µ(m)) = ( let x M in x ) (λx : A.M) = [ λx : A. ( M )] (MN) = ( let f M in let y N in (fy) ) Fig. 7. Monadic-style transform from the computational lambda-calculus into Moggi s Meta-language: Terms Syntax Semantics direct-style Computational lambda-calculus X monadic-style MS transform Meta-language λ C -models Fig. 8. The missing direct models, marked by X ing is that from every abstract Kleisli-category K, we can recover a monad that induces K, up to isomorphism. Direct λ C -models are to λ C -models what abstract Kleisli-categories are to monads. Because Moggi s semantics of the computational lambda-calculus in a λ C -model C uses only morphisms of the 7

8 form A TB, we can translate it into a direct semantics that uses direct λ C -models. Because every λ C -model induces a direct λ C -model, and every direct λ C -model arises from a λ C -model, we neither loose nor gain generality by moving from λ C -models to direct λ C -models. We introduce direct λ C -models in Section 2. In Section 3, we show that the computational lambda-calculus forms an internal theory for each direct λ C -model. In Section 5, we analyse exhaustively the mathematical relation between λ C -models and direct λ C - models. 1.4 New insights from direct models As it turns out, direct λ C -models draw our attention towards previously inconspicuous classes of well-behaved morphisms: So-called thunkable morphisms, central morphisms, copyable morphisms, and discardable morphisms. As we shall see in Section 4, these classes are closed under composition (i.e.they form categories) and more, which means that good program properties propagate along the term structure. This seems to be important for justifying direct-style code optimisations. We ll also see in Section 4 that, for each computational effect, there is the fascinating mathematical challenge to find out how these classes are related. Abstract Kleisli-categories are interesting from a purely mathematical point of view too. That every monad induces an abstract Kleisli-category, and every abstract Kleisli-category arises from a monad, follows from a stronger result: An evident category of abstract Kleisli-categories is reflective in an evident category of λ C -models (Theorem 5.3). The analogous result holds for direct λ C -models and λ C -models (Theorem 5.18). The full subcategory of the category of monads that is equivalent under the reflection to the category of abstract Kleisli-categories is formed by the monads that fulfil Moggi s equalizing requirement (Theorem 5.23). 1.5 Related work I found direct λ C -models by analysing Hayo Thielecke s -categories [17,18], which are direct models for call-by-value languages with higher-order types and continuations. (Roughly speaking, continuations bring the power of jumps into functional programming.) As shown in [4], -categories are the direct λ C -models that correspond to continuations monads. Another approach to semantics of call-by-value languages is given by Freyd categories (called so by John Power see [14,13]). A Freyd category comprises a given category C for modelling values, and a category K for modelling arbitrary terms. The main difference between Freyd categories and abstract Kleisli-categories is that the latter don t have a given category of values, but one that can be equationally defined. One more approach is given by the partially closed focal premonoidal categories of Ralf Schweimeier and Alan Jeffrey [15]. They have three given 8

9 categories: One for modelling values, one for modelling central expressions, and one for modelling arbitrary expressions. Jeffrey and Schweimeier use a nice graphical presentation of direct-style programs (see [8]). All approaches build on symmetric premonoidal categories, which were introduced by John Power and Stuart Anderson [1]. A gentle mathematical introduction to premonoidal categories, and an explanation of their relation to strong monads, is part of see [12]. However, I shall explain all necessary aspects of symmetric premonoidal categories in this article. For direct semantics for call-by-value with recursion, we have traced monoidal categories [6,5]. They, however, rule out important premonoidal models, and therefore exclude many computational effects other than divergence, as Hasegawa pointed out in the conclusion of his thesis [6]. This limitation is broken by the partially traced partially closed focal premonoidal categories of Schweimeier and Jeffrey [15]. Anna Bucalo, Alex Simpson, and myself, in our article on equational lifting monads [3], use abstract Kleisli-categories as a vital tool to prove representation theorems for dominion 2 Direct models In this section, we define the direct λ C -models by giving structure in three layers: Direct λ C -models Precartesian abstract Kleisli-categories Abstract Kleisli-categories The hierarchy of direct models corresponds to the following hierarchy of monadic models: λ C -models Computational cartesian models Monads In Section 2.1, we define abstract Kleisli-categories, and find the direct semantics for the monadic term formation rules of the computational lambdacalculus. In Section 2.2, we add structure to get precartesian abstract Kleislicategories, and find the direct semantics for the basic term formation rules. In Section 2.3, we add structure to get direct λ C -models, and find the direct semantics for the higher-order term formation rules. 9

10 2.1 Abstract Kleisli-categories Suppose that T a monad with multiplication µ and unit η on a category C, and let C T be the Kleisli category of T. As shown in [9], there is an adjunction C F T GT C T such that (1) G T F T = T, (2) the adjunction s unit is η, (3) letting ε be the counit, the G T εf T = µ, and (4) F T is the identity on objects. The adjunction induces some structure on the Kleisli category: Let L = def F T G T ϑ = def F T η We also have the counit ε, which is a natural isomorphism L Id. For each object A, we have ϑ A : A LA. However, ϑ is not generally a natural transformation Id L, as we shall see. Of course, L, ϑ, and ε fulfil certain equations. This leads us to the following definition: Definition 2.1 An abstract Kleisli-category is A category K A functor L : K K A transformation 2 ϑ : Id L A natural transformation ε : L Id such that ϑl is a natural transformation L L 2, and Id ϑ L Id ϑ L L ϑl L 2 ϑ ϑl L2 Lϑ id ε id Lε L Id L Let s pronounce ϑ thunk and ε force, agreeing with Hayo Thielecke s terminology for -categories, which turned out to be special abstract Kleislicategories (see [4]). Example 2.2 The well-known category Rel of sets and relations, which is isomorphic to the Kleisli category of the covariant powerset monad (see e.g. [7]). For a set A, let LA be the powerset of A. For a relation R : A B, and 2 By a transformation from a functor F : C D to a functor G : C D, I mean a map that sends each object A of C to an arrow FA GA 10

11 sets X A and Y B, let (where R[X] is the image of X under R) X(LR)Y Y = R[X] xϑx X = {x} Xε x x X Note that, for every abstract Kleisli-category, L forms a comonad on K with comultiplication ϑl and counit ε. The next definition is crucial. Its importance will emerge gradually in this article. Definition 2.3 A morphism f : A is called thunkable if A B of an abstract Kleisli-category K ϑ LA f B Lf ϑ LB The category G ϑ K is defined as the subcategory of K determined by all objects and the thunkable morphisms. Example 2.4 In Rel, the thunkable morphisms are the total functions. Note that, if K is the abstract Kleisli-category of a monad T, then a morphisms f of K is thunkable if and only if f; η T = f; Tη Now suppose that K is an arbitrary abstract Kleisli-category. For each f K(A, B), let [f] = def ϑ; Lf Letting incl be the inclusion functor G ϑ K [ ] : K(incl(A), B) = (G ϑ K)(A, LB) K, there is an adjunction with unit ϑ and counit ε. Note that if K is the abstract Kleisli-category of a monad T, then [f] = F T f Now suppose that C is a λ C -model whose monad is T = (T, µ, η), and let K be the arising abstract Kleisli-category. Figures 9 and 10 show that we can use K to rewrite Moggi s monadic semantics of the monadic term formation rules and the existence judgement of the computational lambda-calculus. In [10], Moggi defines (Γ M A) to hold if the denotation f factors through η. In the presence of the equalizing requirement, which says that η A is an equalizer 11

12 Rule Syntax Monadic semantics Direct semantics [ ] Γ M : A = Γ f TA = Γ f A Γ [M] : LA = Γ f TA η TTA = Γ [f] LA µ Γ M : LA = Γ f TTA = Γ f LA Γ µ(m) : A = Γ f TTA µ TA = Γ f LA ε A Fig. 9. Monadic semantics and direct semantics of the computational lambda-calculus monadic term formation rules Rule Syntax Monadic semantics Direct semantics ex Γ M : A = Γ f TA = Γ f A Γ M A f; η T = f; Tη f is thunkable Fig. 10. Monadic semantics and direct semantics of the computational lambda-calculus: The existence judgement of η TA and Tη A, f factors through η if and only if f; η T = f; Tη. It turns out that this equation is the suitable interpretation of (Γ M A) in the absence of the equalizing requirement. This is so because in the computational lambda-calculus (Γ M A) is interderivable with the equation (Γ let x = M in [x] [M] : LA) (see [10], bottom of p. 15), whose semantics is f; η T = f; Tη. 2.2 Precartesian abstract Kleisli-categories A direct semantics of the computational lambda-calculus basic term formation rules needs some kind of tensor for modelling product types, projections for modelling variables, and diagonals for modelling pairing. In this section, we re aiming for direct models that arise as Kleisli categories of Moggi s computational cartesian models. A computational cartesian model is a category C with given finite products and a strong monad T. Now let t : A TB T(A B) be the strength, and let t : (TA) B T(A B) be the symmetric dual of t. The tensor and tensor unit are defined as in Figure 11. The diagonals, 12

13 A B = def A B I = def 1 C f = def (C A C f C TB t ) T(C B) f C = def (A C f C B TC t ) T(B C) Fig. 11. The premonoidal tensor and tensor unit on the Kleisli category of a cartesian computational model projections, and structural isomorphisms of K are the images under F T of the corresponding maps of C. This leads to the definition of precartesian abstract Kleisli-categories. I call them precartesian because they are (symmetric) premonoidal categories together with projections and diagonals that don t quite form finite products. Roughly speaking, symmetric premonoidal categories are generalised symmetric monoidal categories in that the tensor doesn t have to be a bifunctor, but only a functor in either argument. Because not everybody knows symmetric premonoidal categories, I ll introduce them here (for more, see [12]). First, a couple of auxiliary definitions: Definition 2.5 A binoidal category is A category C For each object A, a functor A ( ) : C C For each object B, a functor ( ) B : C C such that for all objects A and B (A ( ))(B) = (( ) B)(A) For the joint value, we write A B, or short AB. Definition 2.6 A morphism f : A central if for each g : B B A of a binoidal category K is called AB fb A B BA Bf BA Ag A g AB fb A B ga ga B B f A B A The centre ZK of K is defined as the subcategory determined by all objects and the central morphisms. Definition 2.7 A symmetric premonoidal category is A binoidal category C 13

14 An object I of C Four natural isomorphisms A(BC) = (AB)C, IA = A, AI = A, and AB = BA, with central components that fulfil the coherence conditions known from symmetric monoidal categories. The symmetric monoidal categories are the symmetric premonoidal categories that have only central morphisms. As you can easily check, the natural associativity implies that A ( ) and ( ) A preserve central morphisms. Therefore Proposition 2.8 The centre of a symmetric premonoidal category is a symmetric monoidal category. Now for the definition of a precartesian abstract Kleisli-category. It is true, but not obvious, that every computational cartesian model induces a precartesian abstract Kleisli-category by the definitions in Figure 11. We ll show this later in this section. Definition 2.9 A precartesian abstract Kleisli-category K is An abstract Kleisli-category K A symmetric premonoidal structure on K Finite products on G ϑ K that agree with the symmetric premonoidal structure such that G ϑ K is a subcategory of the centre. Example 2.10 We continue our example Rel. As we ve seen in Example 2.2, Rel forms an abstract Kleisli-category whose thunkable morphisms are the total functions. The tensor on objects is the cartesian product of sets. On morphisms, the tensor is given by (x, y)(r S)(x, y ) xrx ysy As is well-known and obvious, the tensor forms a symmetric monoidal structure that agrees with the finite products on the subcategory of total functions. This example is a bit weak in that all morphisms are central, but we use it anyway in this section because it is short and elegant. Now suppose that C is a computational cartesian model, and let K be the arising precartesian abstract Kleisli-category. Figure 12 shows how to use K to rewrite Moggi s monadic semantics of the computational lambda-calculus basic term formation rules. In the rest of this section, we ll show that every computational cartesian model induces a precartesian abstract Kleisli-category by the definitions in Figure 11. First we ll prove a proposition that helps checking whether a structure is a precartesian abstract Kleisli-category. Before stating the proposition, we define two important classes of morphisms: Discardable morphisms 14

15 and copyable morphisms. For an object A of a precartesian abstract Kleislicategory K, let! A : A I be the unique element of (G ϑ K)(A, I), and let δ A : A AA be the obvious diagonal. Definition 2.11 Suppose that K is a precartesian abstract Kleisli-category. A morphism f K(A, B) is called discardable if A! I f! B id I The category G! K is defined as the subcategory of K whose objects are those of K, and whose morphisms are the discardable morphisms of K. Example 2.12 In Rel, a morphism R : A B is discardable if it relates every x A with at least one element of B. So G! Rel is the same as the category Rel tot in [7]. Next we define the notion of copyable morphism. In a precartesian abstract Kleisli-category, consider the two equations A δ AA A δ AA Af fa f AB f BA B fb δ BB B Bf δ BB Both have the same solutions f, because appending the twist map, which is absorbed by δ, transforms either equation into the other. But beware the conclusion that the two upper right paths are equal they differ for continuations (proving this is outside the scope of this article). Definition 2.13 A morphism f : A B of a precartesian abstract Kleislicategory K is called copyable if the two above equations hold. G δ K is defined as the class of copyable morphisms of K. In contrast to the discardable morphisms, the copyable ones don t always form a subcategory, as we ll see for global state (Example 4.7). Example 2.14 In Rel, the copyable morphisms are the partial functions. So G δ Rel is the same as the category Rel sval in [7]. 15

16 Proposition 2.15 Suppose that K is an abstract Kleisli-category together with a binoidal tensor, an object I, and transformations δ A : A A A, π i : A 1 A 2 A i, and! A : A I. Then K determines a precartesian abstract Kleisli-category if and only if (i) All morphisms of the form [f] are central, copyable, and discardable. (ii) All components of δ, π i, and!, as well as all morphisms of the form A [f] and [f] A, are thunkable. (iii) Letting f, g = def δ;id g; f id, the transformations π 1 : A I A π 2 : I A A π 2, π 1 : A B B A π 1 ; π 1, π 1 ; π 2, π 2 : (A B) C A (B C) are natural in each argument. (iv) We have AB δ (AB)(AB) A δ A A id π 1 π 2 id π i AB A A 1 A 2 π 1 A 1 id! A 1 I π 1 A 1 A 2 π 2 A 2! id I A 2 π 2 Proof. As for the only if, all conditions hold because all morphisms of the form [f] are thunkable and by definition the category of thunkable morphisms is a subcategory of the centre and has finite products. Now for the if. Because morphisms of the form A [f] and [f] A are thunkable, the thunkable morphisms are closed under. To see this, let f (G ϑ K)(A, B). Then (recall that ϑ = [id]) A f; ϑ = A f; ϑ; L(A ϑ); L(A ε) = A f; A ϑ; ϑ; L(A ε) = A [f]; ϑ; L(A ε) = ϑ; L(A [f]); L(A ε) = ϑ; L(A f) 16

17 Because all morphisms of the form [f] are central, so are all thunkable morphisms. To see this, let f (G ϑ K)(A, B). Then for every g K(A, B ) we have A g; f B = A g; [f] B ; ε B = [f] A ; LB g; ε B = f A ; ϑ A ; LB g; ε B = f A ; B g; ϑ B ; ε B = f A ; B g Because all morphisms of the form [f] are copyable, so are all thunkable morphisms. To see this, let f (G ϑ K)(A, B). Then f; δ = f; δ; ϑ ϑ;id ε; ε id = f; ϑ; δ;id ε; ε id = [f]; δ;id ε; ε id = δ; [f] [f];id ε; ε id = δ; [f] f; ε id = δ;id f; f id Because all morphisms of the form [f] are discardable, so are all thunkable morphisms. To see this, let f (G ϑ K)(A, B). Then f;! = f; ϑ;! = [f];! =! Now we can summarise that the thunkable morphisms determine a sub binoidal category of the centre that has only morphisms which are copyable and discardable, and contains δ, π i, and!. Therefore the remaining conditions are enough to imply that, I, δ, π i, and! determine finite products on the category of thunkable morphisms. Every category with finite products determines a symmetric monoidal category (see [9], p. 159). Because the symmetric monoidal tensor of G ϑ K agrees with the binoidal product on K, the symmetric monoidal structure on G ϑ K extends to a symmetric premonoidal structure on K. Now we turn to prove a result which is slightly stronger than saying that the Kleisli category of a cartesian computational model forms a precartesian abstract Kleisli-category. This is a good time to recall two definitions from [12] and [14], respectively: Definition 2.16 Suppose that C and D are symmetric premonoidal categories. Then a functor from C to D is strict symmetric premonoidal if it sends central morphisms to such and strictly preserves the tensor, the tensor unit, and the structural isomorphisms. 17

18 Definition 2.17 A Freyd category consists of a category C with finite products, a symmetric premonoidal category K with the same objects as C, and an identity-on-objects strict symmetric premonoidal functor F : C K. Proposition 2.18 Suppose that C is a computational cartesian model. Then the functor F T : C C T together with the tensor and tensor unit defined in Figure 11 determines a Freyd category. Proof. In this proof, let s write semicolon for the composition of C T, and colon for the composition of C. The structural premonoidal isomorphisms of C T have to be the images of the evident corresponding maps of C. First we prove that, if C T forms a symmetric premonoidal category, then F T is strict symmetric premonoidal. The equation F T (A f) = A F T f follows directly from the equation id η : t = η in the definition of a computational cartesian model. Therefore, F T strictly preserves the tensor. To see that all morphisms in the image of F T are central, suppose that f is a morphism of C, and g is a morphism of C T. Then (F T f) id;id g = F T (f id);id g = f id : id g = f g : t id g; (F T f) id = id g; F T (f id) = id g : G T F T (f id) = id g : t : T(f id) = id g : f id : t By definition, premonoidal structural isomorphisms are preserved by F T. Next we check that C T forms a symmetric premonoidal category. That A ( ) preserves the composition of K follows from a routine calculation that uses the naturality of t and the equation in the definition of a computational cartesian model that involves t and µ. That A ( ) preserves the identity of K corresponds to the equation id η : t = η. By a symmetric argument, ( ) A too is a functor, so we have a binoidal category. The premonoidal structural isomorphisms are central because they are in the image of F T, and their coherence follows from sending through F T the coherence diagrams of the corresponding isos of C. So it remains to prove that the structural premonoidal isos of C T are natural in each of their arguments. The naturality of the evident iso I A A follows ( from the naturality of the corresponding map π 2 of C and the equation 1 TA t ) T(1 A) Tπ 2 TA = π 2 in the definition of a computational cartesian model. The naturality of the symmetry map follows from the naturality of the corresponding map of C. The naturality of the associativity map follows from the naturality of the corresponding map of C, and the equation in the definition of a computational cartesian model that involves the strength and the associativity map of C. Proposition 2.19 Suppose that F : C K is a Freyd category, and F has a right adjoint G with unit η and counit ε. Then K together with the L = def GF, ϑ = def Fη, and ε forms a precartesian abstract Kleisli-category, where δ, π i, and! are the images under F of the corresponding maps of C. Moreover, every morphism in the image of F is thunkable. 18

19 Proof. All morphisms in the image of F are thunkable, because the equation Fg; ϑ = ϑ; LFg is the image under F of the naturality square for η. No we use Proposition By Proposition 2.18, is in particular a binoidal tensor. For Condition (i) of Proposition 2.15, note that [f] = F(f ), where is the adjunction iso K(FA, B) = C(A, GB). Therefore, [f] is central. Because F preserves, δ, and!, all morphisms in the image of F are copyable and discardable, and therefore so is [f]. For Condition (ii), note that A [f] = A F(f ) = F(A f ). Condition (iv) obviously holds, and Condition (iii) holds because the maps that must be natural are the given structural premonoidal isos of K. The proposition that we were aiming for follows directly from Propositions 2.18 and 2.19: Proposition 2.20 Suppose that C is a computational cartesian model. Then the Kleisli category C T together with the tensor and tensor unit defined in Figure 11 forms a precartesian abstract Kleisli-category. 2.3 Direct λ C -models In this section, finally, we shall define the direct models that arise as Kleisli categories of Moggi s λ C -models. Suppose that C is a λ C -model, and let K be the arising precartesian abstract Kleisli-category. We define higher-order structure on K as in Figure 13. This leads us to the definition of direct λ C -models. Definition 2.21 A direct λ C -model is a precartesian abstract Kleisli-category K together with, for each object A, a functor A ( ) : K G ϑ K and an adjunction Λ : K(incl(B) A, C) = (G ϑ K)(B, A C) Before we prove that the definitions in Figure 13 result in a direct λ C - model, let s consider an example, and complete the direct semantics. Example 2.22 We continue our example Rel. For sets A and B, let A B be the set of relations R A B, and let (R, x)apply y def xry x(λr)s def S = {(y, z) : (x, y)rz} Now suppose that C is a λ C -model, and let K be the arising direct λ C - model. Figure 14 shows how to use K to rewrite Moggi s monadic semantics of the computational lambda-calculus higher-order term formation rules. In the rest of this section, we prove that the definitions in Figure 13 result in a direct λ C -model. Like in the preceding section, we proceed in two steps. Now is a good time to recall a definition from [14]. 19

20 Definition 2.23 A Freyd category F : C K is closed if for every objects A, the functor (F ) A has a right adjoint. Proposition 2.24 If C is a λ C -model, then the Freyd category F T : C C T is closed. Proof. In this proof, let s write semicolon for the composition of C T, and colon for the composition of C. For each morphism f C T (C A, B) we need a unique solution g C(C, A B) for the equation (A B) A apply B (F T g) A f C A The required g is λf, where λ the Curry map C(A B, TC) = C(A, (TC) B ), because (F T g) A;apply = F T (g A);apply = g A : ev. Proposition 2.25 If F : C a direct λ C -model. K is a closed Freyd-category, then K forms Proof. In this proof, let s write semicolon for the composition of C T, and colon for the composition of C. Let A be the right adjoint of (F ) A, and apply the counit. For each morphism f C(C A, B) we need a unique thunkable morphism h : C (A B) such that (A B) A apply B h A f C A Let λ be the adjunction isomorphism K((FA) B) = C(A, B C). Because F λf is a solution for h, it remains to prove that every such solution is equal to Fλf. To see this, recall that saying h is thunkable is saying that h : η T = h : Tη, let be the adjunction iso K(FA, B) = C(A, GB), and consider Fλf = Fλ(h A;apply) = Fλ([h] A; ε A;apply) = Fλ(F(h ) A; ε A;apply) = F(h : λ(ε A;apply)) = F(h ); Fλ(ε A;apply) = [h]; Fλ(ε A;apply) = h; ϑ; Fλ(ε A;apply) = h; Fη; Fλ(ε A;apply) = h; F(η : λ(ε A;apply)) = h; F(λ(Fη A; ε A;apply)) = h; F(λ(ϑ A; ε A;apply)) = h; F(λapply) = h; Fid = h 20

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