This is a complete version of the work which was published as a summary [in: Proc. Natl. Acad. Sci. USA 52, 1506–1511 (1964; Zbl 0141.00603)]. We are very happy to see it with C. McLarthy’s preface and the author’s commentary after four decades. The author is well-known as the founder of topos theory, and it is undoubtedly exciting to see the origins of topos theory in such a form. The theory in the paper is more fully developed in the joint textbook with R. Rosebrugh [“Sets for mathematics” (Cambridge University Press, Cambridge) (2003; Zbl 1031.18001)], which was preceded by the joint textbook with S. H. Schanuel [“Conceptual mathematics. A first introduction to categories” (Buffalo Workshop Press, Buffalo, NY) (1991; Zbl 0889.18002)].