does category theory subsume type theory

We say the simply typed lambda calculus (with products) (STLC) is the internal language of Cartesian closed categories, but it is really just one presentation of the internal language and not even the most 'direct' one. A primary computational topic here is E-unification which operates entirely at the level of equational logic, i.e. Meanwhile, from the perspective of equational logic, presentation is all there is. From the perspective of universal algebra, a variety (in the universal algebra sense, or a finitary algebraic category from a CT perspective) forgets its presentation. Second, many of the distinctions category theory/model theory are purposely trying to blur are of primary interest in type theory/logic.Īs a very basic example, all presentations of the axioms of a group give rise to the same class of models (namely groups). Indeed, type theory predates category theory, and the creation of category theory was not motivated by these concerns. Some of the consequences of that are, first, each can exist autonomously. I see category theory as (roughly) being to type theory what model theory is to logic. My view is more or less similar to chi's. If category theory does subsume type theory, should the dialectical differences of the two disciplines be retained, and if so, why? For the sake of historical or cultural value? To retain different but salient differences of instructional or theoretical emphasis? What might these be? However, a multiplicity of dialects probably also limits the size of a receptive audience, and, should a polyglot approach be taken, the length and complexity of exposition is compounded.

If this is so, can the language and formal/graphical descriptions used by category theorists replace those of type theorists? And should they (e.g., in pedagogy and academic publishing)?ĭifferent formalisms can inspire novel perspectives and lay bare conceptual connections that might otherwise be obscure. However, by my naive understanding of the relationship between the two historically convergent disciplines, the latter entirely subsumes the former. The subtleties of the correspondence between type theory and category theory are outside my ken.

Does category theory subsume type theory