A type of higher-order two-dimensional sketch is defined which has models in suitable 2-categories. It has as special cases the ordinary sketches of Ehresmann and certain previously defined generalizations of one-dimensional sketches. These sketches allow the specification of constructions in 2-categories such as weighted limits, as well as higher-order constructions such as exponential objects and subobject classifiers, that cannot be sketched by limits and colimits. These sketches are designed to be the basis of a category-based methodology for the description of functional programming languages, complete with rewrite rules giving the operational semantics, that is independent of the usual specification methods based on formal languages and symbolic logic. A definition of ‘path grammar’, generalizing the usual notion of grammar, is given as a step towards this goal.

Based on a categorical semantics for impredicative calculi of dependent types we prove several dependence and independence results. Especially we prove that there exists a model where all usual syntactical concepts can be interpreted with only one exception: in the model the strong sum of a family of propositions indexed over a proposition need not be isomorphic to a proposition again.

The method of proof consists of restricting the set of propositions in the well known PERw model due to E. Moggi. The subsets of PERw considered in this paper are inspired by the subset ExpO of PERw introduced by Freyd et al.

Finally we show that a weak and a strong notion of sub-locally-cartesian-closed-category coincide under rather mild completeness conditions.

A subtyping relation ≤ between types is often accompanied by a typing rule, called subsumption: if a term a has type T and T≤U, then a has type U. In presence of subsumption, a well-typed term does not codify its proof of well typing. Since a semantic interpretation is most naturally defined by induction on the structure of typing proofs, a problem of coherence arises: different typing proofs of the same term must have related meanings. We propose a proof-theoretical, rewriting approach to this problem. We focus on F≤, a second-order lambda calculus with bounded quantification, which is rich enough to make the problem interesting. We define a normalizing rewriting system on proofs, which transforms different proofs of the same typing judgement into a unique normal proof, with the further property that all the normal proofs assigning different types to a given term in a given environment differ only by a final application of the subsumption rule. This rewriting system is not defined on the proofs themselves but on the terms of an auxiliary type system, in which the terms carry complete information about their typing proof. This technique gives us three different results:

— Any semantic interpretation is coherent if and only if our rewriting rules are satisfied as equations.

— We obtain a proof of the existence of a minimum type for each term in a given environment.

— From an analysis of the shape of normal form proofs, we obtain a deterministic typechecking algorithm, which is sound and complete by construction.

The partially ordered models of linear logic, a logical system developed by J. Y. Girard, turn out to be a class of quantales called Girard quantales. The notion of a quantaloid is a natural categorical generalization of a quantale and is one possible way of keeping track of types. In this paper, Girard quantales are generalized to Girard quantaloids. The general theory is developed and several key examples are studied. Turning our attention to the theory of categories enriched in a bicategory, it is then shown that if G is a Girard quantaloid, then the quantaloids Bim(G), Matr(G), and Mon(G), consisting of G-bimodules, G-matrices and G-monads respectively, are also Girard quantaloids.