Adjunction of semifunctors: Categorical structures in nonextensional lambda calculus

• Published in: Theoretical computer science Vol. 41; no. 1; pp. 95 - 104
• Main Author: Hayashi, Susumu
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 1985; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Miscellaneous; Theoretical computing
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/0304-3975(85)90062-3

Some connections between λ-calculus and category theory have been known. Among them, it has been shown by Lambek that cartesian closed categories (ccc for short) can be identified with extensional typed λ-calculus (cf. Lambek (1980), and Lambek and Scott (1986)). In this paper we introduce the notion of adjunction of semifunctors (for simplicity, we refer to this as ‘semiadjunction’) and, by the aid of this notion, we define the notion of semi cartesian closed category (semi-ccc for short). Some categorical or algebraic systems aimed to represent λ-calculus will turn out to be special cases of semi-ccc. Another intersting connection between ccc and λ-calculus is Scott's embedding of λ-theory into a ccc (cf. Scott (1980)). (This will be referred to as Scott embedding.) We will show that any semiadjunction is embeddable in an adjunction (of functors) and Scott embedding is a special case.