A Linear Category of Polynomial Functors (extensional part)

We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is reminiscent of Day's convolution on presheaves. We then make this category into a model for intuitionistic linear logic by defining an additive and exponential structure.


Introduction
Polynomial functors are (generalizations of) functors X → k C k X E k in the category of sets and functions.Both the "coefficients" C k and the "exponents" E k are sets; and sums, products and exponentiations are to be interpreted as disjoint unions, cartesian products and function spaces.All the natural parametrized algebraic datatypes arising in programming can be expressed in this way.For example, the following datatypes are polynomial: • X → List(X) for lists of elements of X, whose polynomial is List(X) = n∈N X [n]  where [n] = {0, . . ., n − 1}; • X → LBin(X) for "left-leaning" binary trees with nodes in X, whose polynomial can be written as LBin(X) = t∈T X N (t) , where T is the set of unlabeled left-leaning trees and N (t) is the set of nodes of t; • X → Term S (X) for well-formed terms built from a first-order multi-sorted signature S with variables of sort τ taken in X τ .In the last example X is a family of sets indexed by sorts rather than a single set, and expressing it as a polynomial requires "indexed" or "multi-variables" polynomial functors.
Because of this, those functors have recently received a lot of attention from a computer science point of view.In this context, they are often called containers [AAG05,MA09] and coefficients and exponents are called shapes and positions.An early use of them (with yet another terminology) goes back to Petersson and Synek [PS89]: tree-sets are a generalization of so called W-types from dependent type theory.They are inductively generated and are related to the free monads of arbitrary polynomial functors.In the presence of extensional equality, they can be encoded using usual W-types [GK09].
Polynomial functors form the objects of a category with a very rich structure.The objects (i.e., polynomial functors), the morphisms (called simulations) and many operations (coproduct, tensor, composition, etc.) can be interpreted using a "games" intuition (refer to [HH06,Hyv14] for more details): • a polynomial is a two-players game, where moves of the first player are given by its "coefficients" and counter-moves of the opponent are given by its "exponents", • a simulation between two such games is a witness for a kind of back-and-forth property between them.This category has enough structure to model intuitionistic linear logic [Hyv14].The simplest way to define this structure is to use representations of polynomial functors, called polynomial diagrams as the objects.The fact that such representations give rise to functors isn't relevant!This paper gives a functorial counterpart: a model for intuitionistic linear logic where formulas are interpreted by polynomial functors, without reference to their representations.
The difference between the two approaches is subtle.Differentiation of plain polynomials over the real numbers provides a useful analogy.It can be defined in two radically different ways: (1) on representations of polynomials with the formula: (2) on polynomials as functions with the formula: Both definitions are useful and neither is obviously reducible to the other.The first one is easier to work with for concrete polynomials but the second one applies to a larger class of functions.The model of intuitionistic linear logic previously defined [Hyv14] was "intensional": just like point (1), it used representations of polynomial functors.The model described here is "extensional": just like point (2), it applies to arbitrary functors, even though it needs not be defined for non polynomial ones.More precisely, the category defined in [Hyv14] is equivalent (Proposition 2.6) to a subcategory of a category of arbitrary functors with simulations, where the tensor and its adjoint are not always defined.
We first (Section 1) recall some notions about polynomial functors.We then define the category of polynomial functors with simulations and show that it is symmetric monoidal closed (Section 2).We relate this structure to a generalized version of Day's convolution product on presheaves in Section 2.5.We finish by showing how the additive and exponential structure from [Hyv14] can be recovered.
Related Works.The starting point of this work is the characterization of strong natural transformations between polynomial functors due to N. Gambino and J. Kock [GK09].However, our notion of morphism between polynomial functors is more general than what appears in [Koc09,GK09] or [AAG05,MA09] (where polynomial functors are called indexed containers).In particular, there can be morphisms between polynomial functors that do not share their domains and codomains.
Another inspiration is the model of "predicate transformers" from [Hyv04].This is a model for classical linear logic where formulas are interpreted by monotonic operators on subsets and proofs are interpreted by "simulations".We show here that "monotonic operator on P(I)" can be replaced by "functor on Set/ I " with the following restrictions: • we give up classical logic; • we only consider polynomial functors.

Preliminaries: Polynomial Functors
Basic knowledge about locally cartesian closed categories and their internal language (extensional dependent type theory) is assumed throughout the paper.Refer to Appendix A and B for the relevant definitions and notation.The first half of [GK09] is of crucial importance and we start by recalling some results, referring to the original article for details.From now on, C will always denote a locally cartesian closed category.
1.1.Polynomials and Polynomial Functors.A polynomial is, in the usual sense, a function of several variables that can be written as a sum of monomials, where each monomial is a product of a (constant) coefficient and several variables.A polynomial functor is similar, with the following differences • it acts on sets rather than numbers; • sums and products are the corresponding set-theoretic operations; • it may have arbitrarily many arguments: instead of a tuple (X 1 , . . ., X n ) of variables, it acts on families (X i ) i∈I , for a given set I; • the sum of monomial isn't necessarily finite and can be indexed by any set; • because C × Y ∼ = c∈C Y , we don't need constant coefficients in monomials; • each monomial is an indexed product of variables X i .A single polynomial functor is thus made up from the following data: • a set I indexing the variables, • a set A indexing the sum of monomials, • an A-indexed family of sets (D v ) v∈A where for each v ∈ A, i.e., for each monomial appearing in the sum, the set D v indexes the variables composing the monomial, • for each v ∈ A, a function D v → I giving the indices of the variables of the monomial.
For example, if the monomial uses a single variable, this function is constant...The corresponding functor is given by Categorically speaking, a K-indexed family of sets can be represented by a function with codomain K.If γ : S → K, the associated family Γ will be given by Γ k = γ −1 (k).A polynomial functor can thus be represented with • a set I indexing the variables; • a set A indexing the sum; • a function d : D → A giving an A-indexed family of sets indexing the products of variables; • a function n : D → I giving, for each variable in each product, its index.Written in full, the corresponding functor is In order to compose such functors, we need to consider functors acting on families of sets, and giving families of sets as a result.We can thus have polynomial functors taking Iindexed families and giving J-indexed families of sets.Such a functor is simply a J-indexed family of functors in the above sense and is thus of the form: (1.1) The only difference is that instead of having a set A indexing the sum, we have a J-indexed family of sets, each A j indexing the sum of the j component of the functor.Categorically speaking, it amounts to replacing the set A by a function α : A → J representing a J-indexed family of sets.The following definition now makes sense in any category: Definition 1.1.If I and J are objects of C, a polynomial diagram from I to J is a diagram P in C of the shape .
We write PolyDiag C [I, J] for the collection of polynomial diagrams from I to J. for the collection of such functors.
In the locally cartesian closed category of sets and functions, the operations ∆ n , Π d and Σ a are given by: v∈a −1 (j) Z v j∈J and the extension of a polynomial functor corresponds exactly to the formula (1.1).We have Proposition 1.3.The composition of two polynomial functors is a polynomial functor.

Sketch of proof. The composition
uses no less than four pullbacks: where square (i) is a distributivity square (diagram (A.1) in Appendix A).One can then show that ] by a sequence of Beck-Chevalley and distributivity isomorphisms (see Appendix A).
The identity functor from C/ I to itself is trivially the extension of the polynomial .
We obtain a bicategory PolyDiag C where objects are objects of C and morphisms are polynomial diagrams; and a category PolyFun C where objects are slice categories and morphisms are polynomial functors.
1.2.Strong Natural Transformations.Each polynomial functor P from C/ I to C/ J is equipped with a strength: ).A natural transformation between two strong functors is itself strong when it is compatible with their strengths.This gives the category PolyFun C a 2-category structure: objects are slice categories, morphisms are polynomial functors and 2-cells are strong natural transformations.Note that when the base category C is Set, all endofunctors are strong, and so are all natural transformations.
The following proposition is crucial as it allows to represent strong natural transformations between polynomial functors by diagrams inside the category C [GK09].
Proposition 1.4.Every strong natural transformation ρ : P 1 ⇒ P 2 between polynomial functors can be uniquely represented (up-to a choice of pullbacks) by a diagram The strong natural transformation associated to such a diagram is defined as A corollary to Proposition 1.4 is ] in PolyFun C , then P 1 and P 2 are related by . This is particularly important as it means that instead of working on polynomial functors up-to strong natural isomorphism, we can work on their representing polynomials.

Spans and Polynomial
Functors.There are two ways to lift a span to a polynomial: .

Any functor of the form [[ R ]
] is called a linear polynomial functor.
The terminology "linear" comes from the fact that [[ R ]] commutes with arbitrary colimits.More precisely, we have Lemma 1.7.If one writes R ∼ for the span R with its "legs" reversed, we have an adjunction Composition of spans via pullbacks and composition of polynomials are compatible: Lemma 1.8.The operations _ and [_] from Span C to PolyDiag C are functorial, in a "bicategorical" sense.

Symmetric Monoidal Closed Structure
2.1.SMCC Structure for Polynomial Diagrams.We start by recalling the main definition and result from [Hyv14].
Definition 2.1.The category PDSim C has: • "endo" polynomial diagrams I ← D → A → I as objects • equivalence classes of "simulation diagrams" as morphisms, where a simulation diagram from given by a diagram like The equivalence relation between such diagrams is detailed in [Hyv14] and corresponds to the equivalence between spans that form the sides of simulations.

We have ([Hyv14]):
Proposition 2.2.The operation ⊗ that acts on objects in a pointwise manner: natural in P 1 and P 3 .
Seen from the angle of "games semantics" hinted at in the introduction, this operation is a kind of synchronous, "lockstep" parallel composition: a move in the tensor of two games must be a move in each of the games, and a counter-move / response from the opponent must be a response for each move, in each of the two games.
2.2.Simulations.We start by characterizing simulations between polynomials as special 2-cells involving their extensions.
Proposition 2.3.If P 1 and P 2 are polynomial diagrams and R is a span, any 2-cell (where ρ is a strong natural transformation) is uniquely represented (up-to a choice of pullbacks) by a diagram of the shape where (i) is a distributivity square.The plain arrows represent the polynomials R P Going from diagram (2.2) to diagram (2.1) is easy: just follow the arrows and use the pullback lemma to show that square (U, R•A 1 , A 2 , D 2 ) is a pullback and that U is indeed isomorphic to R•D 2 .The morphisms α, β and γ can be read on the diagram.
Going from diagram (2.1) to diagram (2.2) is slightly messier.We choose U to be R•D 2 and look at the following: The morphisms α, β and γ come from diagram (2.1).
To define f and h, write ϕ for the natural isomorphism By naturality of ψ −1 , the following commutes: is the action of the functor ∆ d 2 on morphisms.Starting from the identity, we get This shows that the triangle (ii) commutes.
• We can then construct h from β by using the fact that X is a pullback.The only remaining thing to check is that the upper left rectangle commutes.It follows from r 1 γ = n 1 β in diagram (2.1) and the construction.Proposition 2.3 makes it natural to define simulations for arbitrary endofunctors: Definition 2.4.If F 1 and F 2 are two endofunctors over C/ I 1 and C/ I 2 , a simulation from F 1 to F 2 is given by a span I 1 ← R → I 2 and a 2-cell of the form The category FSim C is defined with: • endofunctors over slices of C as objects, • equivalence classes of simulations as morphisms.
It is routine to check that this gives a category.(Recall that the composition of two linear functors is again linear by Lemma 1.8).Note that the notion of equivalence of simulations is inherited from Span C : by Corollary 1.5 an isomorphism between [[ R ]] and [[ R ′ ]] amounts to a span isomorphism between R and R ′ .As a particular subcategory, we have Definition 2.5.The category PFSim C is the subcategory of FSim C with • polynomial endofunctors over slices of C as objects, • equivalence classes of strong simulations as morphisms where a simulation (R, ρ) is strong if and only if the natural transformation ρ is strong.
There is a functor from PDSim C to PFSim C that sends a polynomial diagram to its corresponding polynomial functor and a simulation diagram to its simulation cell.This functor is • surjective on objects by the definition of polynomial functor, • full and faithful by Proposition 2.3.This implies that Proposition 2.6.The category PDSim C and the category PFSim C are equivalent.

Tensor Product.
Recall that the tensor of two polynomials is the "pointwise cartesian product": This gives rise to an operation on polynomial functors: However, this definition is intensional because it acts on polynomial diagrams, i.e., on representations of polynomial functors.In particular, it doesn't even make sense for functors that are not polynomial.We will now show that it is possible to characterize [[P 1 ⊗ P 2 ]] by a universal property relying only on [[P 1 ]] and [[P 2 ]], thus giving an extensional definition of the tensor of polynomial functors.
To avoid confusion, we will write : we have: Proposition 2.7.Let P 1 and P 2 be polynomial functors, the polynomial functor P 1 ⊗ P 2 is a left Kan-extension along : it is universal s.t.
Corollary 2.8.If C has copowers, denoted by ⊙, we can express left Kan-extensions using coends.We then have This definition is reminiscent of the tensor of predicate transformers (Definition 7 in [Hyv04]): if P 1 : P(S 1 ) → P(S 1 ) and P 2 : P(S 2 ) → P(S 2 ) are monotonic operators on subsets, then The proof of proposition 2.7 makes heavy use of the internal language of locally cartesian closed categories.First note that a polynomial I ← D → A → I can be described by the following judgments: (1) "⊢ I type" for the object I (slice over 1), (2) "i : I ⊢ A(i) type" for the slice a : A → I in C/ I , (3) "i : I, a : A(i) ⊢ D(i, a) type" for the slice d : is a left-adjoint, it commutes with all colimits, including left Kan-extensions.We thus have To save some parenthesis, we will write n 1 • d 1 instead of n 1 (i 1 , a 1 , d 1 ) and similarly for n 2 • d 2 .
We write F 1 and To reduce verbosity, we will ignore the dependency on A 1 and A 2 , i.e., we'll "pretend" both are equal to 1.To correct that, one simply needs to add "a 1 :A 1 (i 1 ), a 2 :A 2 (i 2 )" to all contexts and make the constructions depend on a 1 and a 2 .Recall that if X and V are families indexed by U and V , X Y is the family "λ u, v .X(u) × Y (v)" indexed by U × V .We use the same notation for functions: f g stands for λ u, v .f (u), g(v) .We will (1) construct a natural transformation ε : In the internal language, if F and G are two functors from C/ U to C/ V , a natural transformation α from F to G, takes the form of ) is defined as follows: for families X and Y over I 1 and I 2 and h 1 , h 2 : To check universality, let ρ : for some functor F .We define the natural transformation Θ : Π d 1 ×d 2 ∆ n 1 ×n 2 (_) =⇒ F (_) in several steps: (1) define the type E 1 indexed by I 1 as "i :

This is well typed because
We have Θε = ρ because: where the first equality is the definition of Θ and the second follows from naturality of ρ: It works because the action of F 1 on morphisms is composition: With that in mind, we find that We now need to show that Θ is unique with this property.It follows from the fact that Θ is determined by its values on "rectangles" X Y : The second equality comes from Θε = ρ and the first one follows from the naturality square where like above, we have ( This concludes the proof that F 1 ⊗ F 2 = Lan F 1 (_) F 2 (_) and thus the proof that P 1 ⊗ P 2 = Lan P 1 (_) P 2 (_) .This operation is a tensor product.This follows for example from the fact that it is functorial in PDSim C and that PFSim C is equivalent to it (Proposition 2.6), but a direct proof is also possible.
2.4.SMCC Structure.From Propositions 2.2, 2.6 and 2.7, we can deduce that Proposition 2.9.The category PFSim C with ⊗ is symmetric monoidal closed, i.e., there is a functor natural in P 1 and P 3 .
The concrete intensional definition of P 2 ⊸ P 3 , either in its type theory version or its diagramatic version is rather verbose (Definition 3.7 or Lemma 3.8 in [Hyv14]) and won't be needed here.However just as with the tensor, it is possible to define P 2 ⊸ P 3 without referring to the representing polynomial diagrams.Not surprisingly, it takes the form a right Kan-extension.To simplify the proof, we only state the result for the case C = Set: Proposition 2.10.Given two polynomial endofunctors P 2 and P 3 respectively on Set/ I 2 and Set/ I 3 , the polynomial endofunctor P 2 ⊸ P 3 on Set/ I 2 ×I 3 is a right Kan-extension along ⊲: it is universal such that More precisely, we have P 2 ⊸ P 3 = Ran ⊲ P 2 (_) ⊲ P 3 (_) in the category of endofunctors with natural transformations.
Note that in the internal language, Y ⊲ Z is "i 2 : I 2 , i 3 : I 3 ⊢ Y (i 2 ) → Z(i 3 )".Before proving Proposition 2.10, we show: Lemma 2.12.There is a natural isomorphism Proof.In the internal language, those homsets correspond to the types Proof of Proposition 2.10.We will show, using the formulas for Kan extensions and the calculus of ends and coends [Lan98], that the adjoint to P 1 ⊗ _ (as given by proposition 2.7) is necessarily the above right Kan-extension.
Suppose P 1 , P 2 and P 3 are polynomial functors with domains I 1 , I 2 and I 3 .Let R be a span between I 1 × I 2 an I 3 .We write R ′ for the corresponding span between I 1 and I 2 × I 3 .Besides the previous lemmas and propositions, we will use: and G (note the inversion of left and right); , and similarly for ⋔; • R ′ (x) (y) ∼ = R (x y): in the internal language, they are respectively which are naturally isomorphic.
We have: Because in Set, all natural transformations are strong, these calculations show that there is a natural isomorphism between PFSim Set [P 1 ⊗ P 2 , P 3 ] and PFSim Set P 1 , Ran ⊲ P 2 (_) ⊲ P 3 (_) .Note that because adjoints are unique up-to isomorphisms, the functor we just defined is necessarily isomorphic to the one defined on polynomial diagrams in [Hyv14].This implies that P 2 ⊸ P 2 is indeed well defined and that the Kan extension exists.
The previous proof relied on the fact that C is Set in two ways: • strong natural transformations and natural transformations are the same thing, so that strong natural transformations can be expressed as an end, • Set has powers and copowers, so that we can use the end / coend formulas for _ ⊸ _ and _ ⊗ _.Proposition 2.10 holds for arbitrary C, but the sequence of computations needs to be rewritten to use only the universal properties of left and right Kan extensions, and we need to check that all the natural isomorphisms respect the strength.
2.5.Special Case: Polynomial Presheaves and Day's Convolution.When M is a small monoidal category, presheaves over M have a monoidal structure using Day's convolution product: whenever F, G : M op → Set.Moreover, this tensor has a right adjoint making presheaves a symmetric monoidal closed category.The category M = Set op is monoidal but not small.For F, G : Set → Set, the formula for Day's convolution becomes This coend needs not exist as it is indexed by a large category.However, when F and G are polynomial, this is just a special case of Corollary 2.8.Definition 2.13.Call a presheaf P : for some set A and family D : A → Set.
Corollary 2.8 implies that polynomial presheaves are closed under Day's convolution and we have the explicit formula: Moreover, the right-adjoint is also polynomial: Proposition 2.14.The category of polynomial endofunctors on Set with Day's convolution product is symmetric monoidal closed.
Proof.The category of polynomial endofunctors on Set with natural transformations between them is a (non full) subcategory of PFSim Set .It is thus closed under _ ⊗ _ and _ ⊸ _.
To show that ⊗ and ⊸ are still adjoint in this category, we can rewrite the same proof as for Proposition 2.10 and replace R everywhere by the trivial span 1, 1 .The proof carries through.
There is an explicit formula for the right-adjoint _ ⊸ _, but it is much less elegant than the formula for the tensor: where (See [Hyv14].)

Additive and Exponential Structure
3.1.Additive Structure.In [Hyv14], it was shown that the category of polynomial diagrams with simulations also has a cartesian / cocartesian structure whenever C has a well behaved coproduct.The coproduct of two diagrams, which is also their product is defined as: to f 1 + f 2 is an equivalence of category.It implies in particular the following: Lemma 3.1.If C is locally cartesian closed and extensive, we have whenever the expressions make sense.
With that in mind, we have directly that ]](y) whenever x ∈ C/ I 1 and y ∈ C/ I 2 .We can thus express the additive structure on polynomial functors without referring to the underlying polynomials.We have Lemma 3.2.If C is extensive with an initial object 0, then: (1) the unique functor from C/ 0 to itself is a zero object in FSim C , (2) if we define then _ ⊕ _ is a product as well as a coproduct in the category FSim C .(3) 0 and _ ⊕ _ are a zero object and a product/coproduct in PFSim C as well.
Proof.The first point is direct.The second point boils down to the following: because C is extensive, any span where the left leg is r 1 + r 2 with r k : I k → R k and the right leg is [s 1 , s 2 ], with s k : R k → J. Let's write R k for the obvious span I k ← R k → J: its legs are r k and s k .Extensivity of C implies that: We have: This shows that ⊕ is indeed the coproduct in FSim C .Note that this proof doesn't rely on the functors F 1 and F 2 being polynomial.The proof that it is also a product is similar.
To get the last point, i.e., that _ ⊕ _ is also a coproduct in PFSim C , one needs to show that the natural isomorphisms given preserves the strengths of natural transformations.This is left as an exercise... Another way to prove the last point is simply to use Lemma 3.1 and the fact that ⊕ is the product and coproduct in PDSim C [Hyv14].
3.2.Exponential Structure.As hinted in [Hyv14], the category of PDSim Set has free commutative ⊗-comonoids.For a set I, we write M f (I) for the collection of finite multisets of elements of I.The free commutative ⊗-comonoid for I ← D → A → I is given by where • _ * : Set → Set is the "list functor" sending a set X to the collection of finite sequences of elements in X, • c I : I * → M f (I) sends a sequence to its equivalence class under permutations (multiset).
Conjecture 3.3.In PFSim Set , the free commutative ⊗-comonoid over F is given by This conjecture is a strengthening of the following lemma: Lemma 3.4.If we write PFSim Set∼ for the category of polynomial functors and simulations, where two simulations (R, ρ) and (R ′ , ρ ′ ) are identified when R ∼ = R ′ , then: • PFSim Set∼ with ⊗ and ⊸ is symmetric monoidal closed, • PFSim Set∼ with 0 and ⊕ is cartesian and cocartesian, • PFSim Set∼ has free commutative ⊗-comonoids given by Proof.The first two points follow from earlier results in the paper, and the third one is a consequence of the fact that (3.1) is the free commutative ⊗-comonoid in PDSim Set∼ [Hyv14].It gives the extensional definition of the lemma since it implies that [ It doesn't look too difficult to extend the proof that (3.1) gives the free commutative ⊗comonoid in PDSim Set∼ (Proposition 3.2 in [Hyv14]) to the whole of PDSim Set .A complete and concise (or at least readable) proof of this fact would be most welcome and would prove the conjecture... 3.3.Failure of Classical (Linear) Logic.It is natural to ask if the resulting model for intuitionistic linear logic can be extended to a model for classical linear logic, i.e., if the monoidal closed structure of the category PFSim Set can be extended to a * -autonomous structure.The answer is, perhaps unsurprisingly, no.
The full proof isn't very enlightening but let's look at what happens with the "natural" choice of ⊥ def = 1 ← 1 → 1 → 1 as a potential dualizing object.Take arbitrary objects A and B in C and define the polynomial functor P A,B (X) = A × X B .As in the case of presheaves on sets (page 17), there is a simple explicit formula for P ⊥ A,B = P A,B ⊸ ⊥: we have Asking that the canonical simulation from P A,B to P ⊥⊥ A,B is an isomorphism would imply (by a variant of Corollary 1.5) that the canonical map from A to A B A in C is an isomorphism.This is not possible in general: Lemma 3.5.Any cartesian closed category C in which the canonical natural transformation from A to A B A is an isomorphism is posetal.
If C is also cocartesian, then it becomes a (possibly large) "pre boolean algebra".
Proof.Because C is cartesian closed, we may use simply typed λ-calculus as its internal language.We'll write A B as B ⇒ A as is natural in type theory.Taking both A and B to be C × C above, we have the following isomorphism: Because C is cartesian closed, it is thus a (possibly large) Heyting semi-lattice; and if C is also cocartesian, it becomes a Heyting algebra.Now, the existence of a transformation from (A ⇒ B) ⇒ A to A amounts to saying the law of Peirce is satisfied.This makes the Heyting algebra boolean...Note however that the model of predicate transformers from [Hyv04] can be seen as a "proof irrelevant" variant of PFSim Set : • we collapse the categories Set/ I into preorders, making each Set/ I equivalent to the algebra of subsets of I, • we collapse the categories of spans into preorders, making each Span[I, J] equivalent to the algebra of relations between I and J, • we identify simulations (R, α) and (R, β).This gives a non-trivial * -autonomous category: • polynomial functors on Set/ I become monotonic transformations on P(I), all of which are in fact "polynomial", • simulations become relations satisfying a closure property, • duality gives P ⊥ (x) = P (x) whenever P : P(I) → P(I), where y is the complement of y with respect to I. • the dualizing object is 1, the unit of the tensor, i.e., 1 = ⊥.• However, the category is not compact closed as _ ⊗ _ is different from its dual.where each G(a) is a subgroup of the automorphisms of D(a), acting in an obvious way on X D(a) .Formula 2.3 from Section 2.5 has a natural generalization to this context.Does it work as an intensional formula for Day's convolution?What about the analogous to formula 2.4? .This type for equality is "extensional" because having an inhabitant of Id X (u, v) implies that u and v are definitionally (extensionally) equal.This is reflected in its interpretation: where eq(f 1 , f 2 ) is the equalizer of f 1 and f 2 .(This is indeed a slice over the interpretation of Γ.) 1) Proof.As compositions of polynomial functors, both [[ R ]][[P 1 ]] and [[P 2 ]][[ R ]] are polynomial (Proposition 1.3).We can use Proposition 1.4 to represent the strong natural transformation ρ by the following diagram: 1 and P 2 R , whose extensions are [[ R ]][[P 1 ]] and [[P 2 ]][[ R ]]; and the dashed arrows represent the strong natural transformation between them, as in diagram (1.3).
c 2 and ϕ tw = c 2 , c 1 , where tw in C[C × C, C × C] exchanges the left and right components of a pair.However, Because of the isomorphism above, ϕ must be of the form λf. x, y for some x y in C[1, C].This implies that ϕ id = ϕ tw, and thus, that c 1 = c 2 .Because C is cartesian closed, any f 1 , f 2 in C[A, B] correspond precisely to constants f 1 and f 2 in C[1, A ⇒ B] and are thus equal.The category C is thus posetal.