ARPE September presentation : Generalised Species of Structures - PowerPoint PPT Presentation

ARPE September presentation : Generalised Species of Structures Marcelo Fiore, University of Cambridge Younesse Kaddar Ecole Normale Supérieure Paris-Saclay Friday 27 th September, 2019

Table of contents 1. Introduction 2. Context 3. Studied article: “Generalised Species of Structures: Cartesian Closed and Difgerential Structure” 4. What’s next? 5. Conclusion 1

Introduction

Motivations Mind map of explored fjelds over my years at the ENS 2

Author & Institution Marcelo Fiore University of Cambridge – Computer Laboratory 3

Project Generalised Species of Structures [10, 11, 8, 1] Project will involve • presheaf categories (cf. M1 at Oxford [16]) • monoidal and higher categories (cf. M2 at Macquarie in Sydney [15]) • homotopy type theory (cf. L3 at Nottingham [14]) 4

Project Generalised Species of Structures [10, 11, 8, 1] Project will involve • presheaf categories (cf. M1 at Oxford [16]) • monoidal and higher categories (cf. M2 at Macquarie in Sydney [15]) • homotopy type theory (cf. L3 at Nottingham [14]) 4

Project Generalised Species of Structures [10, 11, 8, 1] Project will involve • presheaf categories (cf. M1 at Oxford [16]) • monoidal and higher categories (cf. M2 at Macquarie in Sydney [15]) • homotopy type theory (cf. L3 at Nottingham [14]) 4

Context

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs Motto: Data type/computation structures from a combinatorial perspective, and vice versa. 5

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs Motto: Data type/computation structures from a combinatorial perspective, and vice versa. 5

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs Motto: Data type/computation structures from a combinatorial perspective, and vice versa. 5

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs perspective, and vice versa. 5 Motto: Data type/computation structures from a combinatorial

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs Motto: Data type/computation structures from a combinatorial perspective, and vice versa. 5

Species of structures Generalised species of structures 2008: “The cartesian closed bicategory of generalised species of structures” by Fiore, Gambino, Hyland, and Winskel [10] Generalisation of • 1981: Joyal’s species of structures [13] • algebraic account of types of labelled combinatorial structures • structural counterparts of counting formal power series • 2003: Relational model of Ehrhard–Regnier’s differential linear logic [6, 5, 4] • enrichment of Girard’s linear logic [12] • extra rules to produce derivatives of proofs perspective, and vice versa. 5 Motto: Data type/computation structures from a combinatorial

Species of structures Generalised species of structures Also related to: • Para-toposes [9], in connection to • higher-dimensional category theory (opetopes) [7, 3] • resource calculi [18] • Homotopy Type Theory [2]: their calculus can be mimicked therein 6 • ∞ -categories via polynomial functors [17]

Species of structures Generalised species of structures Also related to: • Para-toposes [9], in connection to • higher-dimensional category theory (opetopes) [7, 3] • resource calculi [18] • Homotopy Type Theory [2]: their calculus can be mimicked therein 6 • ∞ -categories via polynomial functors [17]

Species of structures Generalised species of structures Also related to: • Para-toposes [9], in connection to • higher-dimensional category theory (opetopes) [7, 3] • resource calculi [18] • Homotopy Type Theory [2]: their calculus can be mimicked therein 6 • ∞ -categories via polynomial functors [17]

Species of structures Generalised species of structures Also related to: • Para-toposes [9], in connection to • higher-dimensional category theory (opetopes) [7, 3] • resource calculi [18] • Homotopy Type Theory [2]: their calculus can be mimicked therein 6 • ∞ -categories via polynomial functors [17]

Species of structures Generalised species of structures Also related to: • Para-toposes [9], in connection to • higher-dimensional category theory (opetopes) [7, 3] • resource calculi [18] • Homotopy Type Theory [2]: their calculus can be mimicked therein 6 • ∞ -categories via polynomial functors [17]

Joyal’s species: defjnition Combinatorial species of structures P A functor groupoid of fjnite sets Equivalently : P given by a family of symmetric group actions such that 7 P : B − → Set ↑ _ [=]: P [ n ] × S n − → P [ n ] p [ id ] = p p [ σ ][ τ ] = p [ σ · τ ] ∀ p ∈ P [ n ] , σ, τ ∈ S n

Joyal’s species: intuition • action of P = abstract rule of transport of structures (structural equivalence) Figure 1: Schematic representation of a species 8 Intuition: For a species P : B − → Set • P ( U ) = structures of type P parameterised by the set of tokens U

Equality of species: Joyal’s species: generating series Natural isomorphism (pointwise bijection + well-behaved with transport) calculus of species 9 Generating series of P : B − → Set: � | P [ n ] | x n P ( x ) = n ! n ≥ 0 Arithmetic on generating functions ( + , × , ◦ , ∂ ) ← → Combinatorial

Joyal’s species: examples Examples 10 • Endofunctions: • End ( U ) := Hom Set ( U , U ) ∀ U ∈ B • End ( σ )( f ) := σ f σ − 1 ∀ f ∈ End ( U , U ) , σ ∈ Hom B ( U , V ) • Underlying set species U : B ֒ → Set • X n := Hom B ( n , − ) • Terminal species: exp( X ) := U �− − → { U } • Initial species: 0 := U �− − → ∅

Joyal’s species: addition and multiplication • Addition: Figure 2: Multiplication of species 11 ( P + Q )( U ) := P ( U ) + Q ( U ) • Multiplication: Day’s tensor product � U 1 , U 2 ∈ B P · Q := P ( U 1 ) × Q ( U 2 ) × Hom B ( U 1 + U 2 , − ) � In [ B , Set ] : � ( P · Q )( U ) = P ( U 1 ) × Q ( U 2 ) U 1 ⊔ U 2 = U

Joyal’s species: differentiation • Difgerentiation: Figure 3: Difgerentiation of species 12 ( d / d x ) P = P ( − + x )

Joyal’s species: composition • Composition: Figure 4: Composition of species 13 � T ∈ B ( Q ◦ P )( U ) := Q ( T ) × ( P · · · · · P )( U ) � �� � | T | fois � In [ B , Set ] : � � ( Q ◦ P )( U ) = Q ( U ) × P ( u ) U∈ Part U u ∈U

Joyal’s species: composition Analytic endofunctors on Set In this respect: Composition of species = Composition of corresponding functors 14 Species ← → Coeffjcients of analytic endofunctors on Set

Studied article: “Generalised Species of Structures: Cartesian Closed and Differential Structure”