Comprehension Categories and the Semantics of Type Dependency

Jacobs, B., Comprehension categories and the semantics of type dependency, Theoretical Computer Science 107 (1993) 169-207. A comprehension category is defined as a functor 8: E-+B’ satisfying (a) cod 0 9 is a fibration, and (b) fis Cartesian in E implies that qfis a pullback in B. This notion captures many structures which are used to describe type dependency (like display-map categories (Taylor (1986), Hyland and Pitts (1989) and Lamarche (1988)), categories with attributes (Cartmell (1978) and Moggi (1991)), Dcategories (Ehrhard (1988)) and comprehensive fibrations (Pavlovic (1990)). It also captures comprehension as occurring in topos theory and as described by Lawvere’s (1970) hyperdoctrines. This paper is meant as an introduction to these comprehension categories. A comprehension category will be called closed if it has appropriate dependent products and sums. A few examples of closed comprehension categories will be described here; more of them may be found in Jacobs (1991); applications occur in Jacobs (1991) and Jacobs et al. (1991).