In this paper we define a logical framework, called &;lambda;TT, that is well suited for semantic analysis. We introduce the notion of a fibration [Lscr]1 : [Fscr]1 [xrarr ] [Cscr]1 being internally definableThe definability as used in this paper should not be confused with Bénabou's ‘definability’ (Bénabou 1985). in a fibration [Lscr]2 : [Fscr]2 [xrarr ] [Cscr]2. This notion amounts to distinguishing an internal category Lin [Lscr]2 and relating [Lscr]1 to the externalization of L through a pullback. When both [Lscr]1 and [Lscr]2 are term models of typed calculi [Lscr]1 and [Lscr]2, respectively, we say that [Lscr]1 is an internal typed calculus definable in the frame language [Lscr]2. We will show by examples that if an object language is adequately represented in λTT, then it is an internal typed calculus definable in the frame language λTT. These examples also show a general phenomenon: if the term model of an object language has categorical structure S, then an adequate encoding of the language in λTT imposes an explicit internal categorical structure S in the term model of λTT and the two structures are related via internal definability. Our categorical investigation of logical frameworks indicates a sensible model theory of encodings.

We study three comonads derived from the comma construction. The induced coalgebras correspond to the three concepts displayed in the title of the paper. The comonad that yields the *-autonomous categories is, in essence, the Chu construction, which has recently awaken much interest in computer science. We describe its couniversal property. It is right adjoint to the inclusion of *-autonomous categories among autonomous categories, with lax structure-preserving morphisms. Moreover, this inclusion turns out to be comonadic: *-autonomous categories are exactly the Chu-coalgebras.

The traditional notions of strong and weak normalization refer to properties of a binary reduction relation. In this paper we explore an alternative approach to normalization, in which we bypass the reduction relation and instead focus on the normalization function, that is, the function that maps a term to its normal form. We work in an intuitionistic metalanguage, and characterize a normalization function as an algorithm that picks a canonical representative from the equivalence class of convertible terms. This means that we also get a decision algorithm for convertibility.Such a normalization function can be constructed by building an appropriate model and a function quote, which inverts the interpretation function. The normalization function is then obtained by composing the quote function with the interpretation function. We also discuss how to get a simple proof of the property that constructors are one-to-one, which is usually obtained as a corollary of Church–Rosser and normalization in the traditional sense.We illustrate this approach by showing how a glueing model (closely related to the glueing construction used in category theory) gives rise to a normalization algorithm for a combinatory formulation of Gödel System T. We then show how the method extends in a straightforward way when we add cartesian products and disjoint unions (full intuitionistic propositional logic under a Curry–Howard interpretation) and transfinite inductive types such as the Brouwer ordinals.