Oeuvres complètes et commentées. Esquisses et structures monoïdales fermées. Partie IV - 2. Editée et commentée par Andrée Charles Ehresmann.

*(French)*Zbl 0561.01028
Supplément 2 au Vol. XXIII (1982) des Cahiers de Topologie et Géométrie Différentielle. Amiens: Mme A. C. Ehresmann, U. E. R. de Mathématiques. IV, 823 p. (1983).

This is the final volume of Charles Ehresmann’s collected works. Parts I- 1 and I-2 are reviewed above, Part II-1 is Zbl 0496.01009, Part II-2 is Zbl 0529.01017, Part III-1 and III-2 are Zbl 0452.01016 and Zbl 0452.01017, while Part IV-1 is Zbl 0508.01026. The contents consists of six long papers written between 1972 and 1979, together with four brief non-technical notes, as well as the usual Comments and Synopsis by A. C. Ehresmann. The first of the long papers is about sketched structures. One of the main topics of this paper is several different completions of a sketch to a suitably complete category. These constructions are of interest in current work in theoretical computer science concerning abstract data types, since constructing such a completion includes constructing the initial algebra described by the data type. The second paper is on tensor products of topological ringoids. (A ringoid is an additive category.) The four remaining long papers are a series of works on ”multiple functors”: i.e., multiple categories. These papers are relevant to current work in categorical homotopy theory. They contain many intricate ”geometrical” constructions of a sort that is becoming increasingly popular. The Synopsis section is quite brief since the papers are written in a style that is almost easily understood by other category theorists. The Comments are mainly concerned with summarizing later results in the topics covered by the papers. In particular, there are several comments on the connections between the study of multiple categories and homotopy theory. In general, the Comments are an excellent guide to the current literature in the topics covered by the papers.

Reviewer: J.W.Gray