Polynomial functors and polynomial monads

Abstract

We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.

41 pages, latex, 2 ps figures generated at runtime by the texdraw package (does not compile with pdflatex). v2: removed assumptions on sums, added short discussion of generalisation, and more details on tensorial strength.

References in corpus (5)

• Polynomial functors and opetopes
• Framed bicategories and monoidal fibrations
• Local fibered right adjoints are polynomial
• Tambara functors
• Pseudo-categories

Cited by in corpus (66)

• Compact Closed Bicategories
• Homotopy theory for algebras over polynomial monads
• Polynomial functors and opetopes
• The Frobenius Condition, Right Properness, and Uniform Fibrations
• Incomplete Tambara functors
• Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees
• Homotopy Theoretic Models of Type Theory
• When is a container a comonad?
• Operads as polynomial 2-monads
• Database queries and constraints via lifting problems
• Koszul duality for operadic categories
• Internal algebra classifiers as codescent objects of crossed internal categories
• Differential bundles and fibrations for tangent categories
• $\infty$-Operads as Analytic Monads
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Faà di Bruno for operads and internal algebras
• Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses
• Whole-grain Petri nets and processes
• Learners' Languages
• Operadic categories as a natural environment for Koszul duality
• Functorial Data Migration
• Poly: An abundant categorical setting for mode-dependent dynamics
• The incidence comodule bialgebra of the Baez-Dolan construction
• Categorification of Hopf algebras of rooted trees
• Constructing Infinitary Quotient-Inductive Types
• A Linear Category of Polynomial Diagrams
• A type theory for cartesian closed bicategories
• A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory
• Universal properties of bicategories of polynomials
• Computads for weak $\omega$-categories as an inductive type
• Polynomial Functors and Shannon Entropy
• The naturality of natural deduction (II). Some remarks on atomic polymorphism
• Polynomial pseudomonads and dependent type theory
• Local fibered right adjoints are polynomial
• Polynomial functors and trees
• Bi-incomplete Tambara functors as $\mathcal{O}$-commutative monoids
• Polynomials as spans
• Dirichlet Polynomials form a Topos
• Companions, Causality and Codensity
• Kripke-Joyal forcing for type theory and uniform fibrations
• A Linear Category of Polynomial Functors (extensional part)
• Algebraic models of simple type theories: a polynomial approach
• The Universal Exponentiable Arrow
• Generic bicategories
• Polynomial and Analytic Functors and Monads, revisited
• Differential 2-rigs
• Functorial aggregation
• From dependent type theory to higher algebraic structures
• Canonicity and homotopy canonicity for cubical type theory
• Dirichlet Functors are Contravariant Polynomial Functors
• Cartesian Factorization Systems and Grothendieck Fibrations
• $W$-Types in Categories of Coalgebras
• Implicit automata in typed $\lambda$-calculi II: streaming transducers vs categorical semantics
• Types are Internal $\infty$-Groupoids
• W-types in setoids
• Familial Monads as Higher Category Theories
• Stabilized profunctors and stable species of structures
• Combinatorial Dyson-Schwinger equations and inductive data types
• Formalizing two-level type theory with cofibrant exo-nat
• Automata and coalgebras in categories of species
• Dependent Inductive and Coinductive Types are Fibrational Dialgebras
• Pattern Runs on Matter: The Free Monad Monad as a Module over the Cofree Comonad Comonad
• Comodule Representations of Second-Order Functionals
• Monoid Structures on Indexed Containers
• Polynomials in homotopy type theory as a Kleisli category
• Syntax and Consequence Relations -- A Categorical Perspective