Deriving Category Theory from Type Theory

10 years 10 months ago
Deriving Category Theory from Type Theory
This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories. It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics. We begin by showing how the concept of a category can be derived from some simple and primitive mechanisms of monadic type theory. We then show how the notion of a category with finite products can model the most fundamental syntactical constructions of (algebraic) type theory. The idea of naturality is shown to capture, in a syntax free manner, the notion of substitution, and therefore provides a syntax free coding of a multiplicity of type theoretical constructs. Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed λ-calculus with binary product...
Roy L. Crole
Added 09 Aug 2010
Updated 09 Aug 2010
Type Conference
Year 1993
Authors Roy L. Crole
Comments (0)