A cartesian bicategory of polynomial functors in homotopy type theory Samuel Mimram June 18, 2021 This is joint work with Eric Finster, Maxime Lucas and Thomas Seiller. 1 Part I Polynomials and polynomial functors 2 We can categorify this notion: replace natural numbers by elements of a set. X P(X ) = X Eb b2B Categorifying polynomials A polynomial is a sum of monomials X P(X ) = X ni 0≤i<k (no coefficients, but repetitions allowed) 3 Categorifying polynomials A polynomial is a sum of monomials X P(X ) = X ni 0≤i<k (no coefficients, but repetitions allowed) We can categorify this notion: replace natural numbers by elements of a set. X P(X ) = X Eb b2B 3 Polynomial functors This data can be encoded as a polynomial P, which is a diagram in Set: p E B where • b 2 B is a monomial −1 • Eb = P (b) is the set of instances of X in the monomial b. X X X X ::: b 4 Polynomial functors This data can be encoded as a polynomial P, which is a diagram in Set: p E B where • b 2 B is a monomial −1 • Eb = P (b) is the set of instances of X in the monomial b. It induces a polynomial functor P : Set ! Set J K X X 7! X Eb b2B 4 Polynomial functors For instance, consider the polynomial corresponding to the function p E B • • • • • • • The associated polynomial functor is P (X ): Set ! Set J K X 7! X × X t X × X × X 5 A polynomial is finitary when each monomial is a finite product. Polynomial functors For instance, consider the polynomial corresponding to the function p N 1 . • • • • The associated polynomial functor is P (X ): Set ! Set J K X 7! X × X × X × ::: 6 Polynomial functors For instance, consider the polynomial corresponding to the function p N 1 . • • • • The associated polynomial functor is P (X ): Set ! Set J K X 7! X × X × X × ::: A polynomial is finitary when each monomial is a finite product. 6 Polynomial functors: typed variant We will more generally consider a \typed variant" of polynomials P p I s E B t J this means that • each monomial b has a \type s(b) 2 J", • each occurrence of a variable e 2 E has a type s(e) 2 I . j1 j2 jn−1jn ::: b i 7 Polynomial functors: typed variant We will more generally consider a \typed variant" of polynomials P p I s E B t J this means that • each monomial b has a \type s(b) 2 J", • each occurrence of a variable e 2 E has a type s(e) 2 I . It induces a polynomial functor P (X ): SetI ! SetJ J K 0 1 X Y (Xi )i2I 7! @ Xs(e)A −1 −1 b2t (j) e2p (b) j2J 7 Proposition The composite of two polynomial functors is again polynomial: I P J Q K Set J K Set J K Set Q ◦ P J K J K Proof. Basically the usual one: XYXY Q ◦ P (Xi ) = Xi J K J K ∼ XXYY = Xi ∼ XY = Xi The category of polynomial functors Given a set I , we have an \identity" polynomial functor: I id I id I id I 8 Proof. Basically the usual one: XYXY Q ◦ P (Xi ) = Xi J K J K ∼ XXYY = Xi ∼ XY = Xi The category of polynomial functors Given a set I , we have an \identity" polynomial functor: I id I id I id I Proposition The composite of two polynomial functors is again polynomial: I P J Q K Set J K Set J K Set Q ◦ P J K J K 8 The category of polynomial functors Given a set I , we have an \identity" polynomial functor: I id I id I id I Proposition The composite of two polynomial functors is again polynomial: I P J Q K Set J K Set J K Set Q ◦ P J K J K Proof. Basically the usual one: XYXY Q ◦ P (Xi ) = Xi J K J K ∼ XXYY = Xi ∼ XY = Xi 8 The category of polynomial functors We can thus build a category PolyFun of sets and polynomial functors: • an object is a set I , • a morphism F : I ! J is a polynomial functor P : SetI ! SetJ J K 9 Polynomial vs polynomial functors A polynomial P p I s E B t J induces a polynomial functor P : SetI ! SetJ J K We have mentioned that composition is defined for polynomials. However, on polynomials, it is not strictly associative: we can build a bicategory Poly of sets an polynomial functors. This suggests that 2-cells are an important part of the story! 10 We can build a bicategory Poly of sets, polynomials and morphisms of polynomials. Morphisms between polynomials A morphism between two polynomials is p I s E B t J " y β I E 0 B0 J s0 p0 t0 We send operations to operators, preserving typing and arities: j1 j2 jn−1jn j1 j2 jn−1jn ::: ::: b 7! β(b) i i 11 Morphisms between polynomials A morphism between two polynomials is p I s E B t J " y β I E 0 B0 J s0 p0 t0 We send operations to operators, preserving typing and arities: j1 j2 jn−1jn j1 j2 jn−1jn ::: ::: b 7! β(b) i i We can build a bicategory Poly of sets, polynomials and morphisms of polynomials. 11 Morphisms between polynomial functors A morphism between polynomial functors P ; Q : SetI ! SetJ J K J K is a \suitable" natural transformation, and we can build a 2-category PolyFun. 12 Cartesian structure The category PolyFun is cartesian. Namely, given two polynomial functors in Poly P : I ! JQ : I ! K i.e., in Cat, P : SetI ! SetJ Q : SetI ! SetK J K J K we have, in Cat, hP; Qi : SetI ! SetJ × SetK =∼ SetJtK and the constructions preserve polynomiality: in PolyFun, hP; Qi : I ! (J t K) 13 SetI × SetJ ! SetK J SetI ! (SetK )Set J SetI ! SetSet ×K which suggests defining the closure as [J; K] = SetJ × K for LL-people: this looks like !J K. ` Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK 14 J SetI ! (SetK )Set J SetI ! SetSet ×K which suggests defining the closure as [J; K] = SetJ × K for LL-people: this looks like !J K. ` Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK SetI × SetJ ! SetK 14 J SetI ! SetSet ×K which suggests defining the closure as [J; K] = SetJ × K for LL-people: this looks like !J K. ` Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK SetI × SetJ ! SetK J SetI ! (SetK )Set 14 which suggests defining the closure as [J; K] = SetJ × K for LL-people: this looks like !J K. ` Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK SetI × SetJ ! SetK J SetI ! (SetK )Set J SetI ! SetSet ×K 14 for LL-people: this looks like !J K. ` Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK SetI × SetJ ! SetK J SetI ! (SetK )Set J SetI ! SetSet ×K which suggests defining the closure as [J; K] = SetJ × K 14 Closed structure For the closed structure, we can hope for the same: given, in PolyFun, P : I t J ! K i.e., in Cat, P : SetI tJ ! SetK we have SetI tJ ! SetK SetI × SetJ ! SetK J SetI ! (SetK )Set J SetI ! SetSet ×K which suggests defining the closure as [J; K] = SetJ × K for LL-people: this looks like !J K. 14 ` Closed structure In terms of operations, the intuition behind the bijection PolyFun(I t J; K) =∼ PolyFun(I ; SetJ × K) is that we can formally transform operations as follows I J I ::: ::: ::: ::: K J K 15 Closed structure In terms of operations, the intuition behind the bijection PolyFun(I t J; K) =∼ PolyFun(I ; Set=J × K) is that we can formally transform operations as follows I J I ::: ::: ::: ::: K J K 15 One can restrict to polynomial functors which are finitary: we can then take [I ; J] = Setfin=I × J or rather [I ; J] = N=I × J Finitary polynomial functors are also known as normal functors (introduced by Girard). Closed structure There are two problems with our closure. The first one is that [I ; J] = Set=I × J is too large to be an object of our category. 16 or rather [I ; J] = N=I × J Finitary polynomial functors are also known as normal functors (introduced by Girard). Closed structure There are two problems with our closure. The first one is that [I ; J] = Set=I × J is too large to be an object of our category. One can restrict to polynomial functors which are finitary: we can then take [I ; J] = Setfin=I × J 16 Finitary polynomial functors are also known as normal functors (introduced by Girard).