A Linear Category of Polynomial Functors
نویسندگان
چکیده
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 7→ k AkXk in the category of sets and functions. Both the “coefficients” Ak and the “exponents” Dk 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 7→ List(X) for lists of elements of X, whose polynomial is List(X) = n∈NX [n] where [n] = {0, . . . , n− 1}; • X 7→ 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 7→ TermS(X) for well-formed terms built from a first-order multi-sorted signature S with variables of sort τ taken in Xτ . 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 Received by the editors April 2, 2014. 1998 ACM Subject Classification: F.3.2, F.4.1.
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A Linear Category of Polynomial Functors (extensional part)
Abstract. 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 an...
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