Search results for "Cartesian closed category"

showing 4 items of 4 documents

Wellfounded Trees and Dependent Polynomial Functors

2004

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by in- vestigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed cat- egories.

Class (set theory)Pure mathematicsCartesian closed categoryFunctorType theoryMathematics::Category TheoryComputer Science::Logic in Computer ScienceWellfounded trees locally cartesian closed categories categorical logicTree (set theory)PrewellorderingCategory theoryForgetful functorMathematics
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The cartesian closed bicategory of generalised species of structures

2007

AbstractThe concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature — including of course Joyal's original notion — together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our secon…

FunctorGeneral MathematicsSubstitution (logic)species of structures analytic functorPresheafComposition (combinatorics)BicategoryMathematics::Algebraic TopologyAlgebraCartesian closed categoryCombinatorial speciesMathematics::Category Theorybicategory cartesian closed categoriesMathematicsJournal of the London Mathematical Society
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On operads, bimodules and analytic functors

2017

We develop further the theory of operads and analytic functors. In particular, we introduce a bicategory that has operads as 0-cells, operad bimodules as 1-cells and operad bimodule maps as 2-cells, and prove that this bicategory is cartesian closed. In order to obtain this result, we extend the theory of distributors and the formal theory of monads.

General Mathematics0102 computer and information sciences01 natural sciencesMathematics::Algebraic TopologyQuantitative Biology::Cell BehaviorMathematics::K-Theory and HomologyMathematics::Quantum AlgebraMathematics::Category Theory18D50 55P48 18D05 18C15FOS: MathematicsAlgebraic Topology (math.AT)Category Theory (math.CT)Mathematics - Algebraic Topology0101 mathematicsMathematicsFunctorOperad bimodule analytic functor bicategoryTheoryMathematics::Operator AlgebrasApplied Mathematics010102 general mathematicsOrder (ring theory)Mathematics - Category Theory16. Peace & justiceBicategoryAlgebraCartesian closed category010201 computation theory & mathematicsBimodule
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Polynomial functors and polynomial monads

2009

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.

Pure mathematicsPolynomialFunctorGeneral MathematicsMathematics - Category Theory18C15 18D05 18D50 03G30517 - AnàlisiMonad (functional programming)BicategoryMathematics::Algebraic TopologyCartesian closed categoryMathematics::K-Theory and HomologyMathematics::Category TheoryPolynomial functor polynomial monad locally cartesian closed categories W-types operadsFOS: MathematicsPolinomisCategory Theory (math.CT)Mathematics
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