Applied Categorical Structures 6: 87–103, 1998. 87 c  1998 Kluwer Academic Publishers. Printed in the Netherlands. Categorical Aspects in Projective Geometry CLAUDE-ALAIN FAURE and ALFRED FR ¨ OLICHER Section de Math´ ematiques, Universit´ e de Gen` eve, Case postale 240, CH-1211 Gen` eve 24, Switzerland (Received: 8 December 1995; accepted: 5 June 1996) Abstract. After introducing morphisms between projective geometries, some categorical questions are examined. It is shown that there are three kinds of embeddings and two kinds of quotients. Furthermore the morphisms decompose in a canonical way into four factors. Mathematics Subject Classifications (1991). 18B99, 51A10. Key words: morphisms and homomorphisms of projective geometries, subgeometries and sub- spaces, quotient geometries, initial and final morphisms, sections and retractions, decomposition of a morphism. Introduction Categorical questions in projective geometry are new, since morphisms between projective geometries have been introduced only recently; cf. [3, 4] and [5]. Many questions about the category P roj of projective geometries ought to be studied; in this paper we examine three of these topics, namely embeddings, quotients and factorizations of morphisms. Since a morphism between projective geometries is a map between the under- lying point sets which in general is not defined for all points of the source, the natural forgetful functor has its values in the so-called partial-map category P ar. The objects of P ar are the sets, and the morphisms from X to Y are the partial maps, i.e. the maps f : X \ N → Y where N ⊆ X. We call N the kernel of f and we write f : X−→ Y and N = Ker f . P ar is known to be equivalent to the category Set ⋆ of pointed sets; cf. [1]. By an embedding we understand a monomorphism which is initial with respect to the forgetful functor P roj → P ar. The embeddings turn out to be, up to isomorphy, the inclusions of the usual (projective) subgeometries. But subgeome- tries can have undesired features. For instance the dimension of a subgeometry can be bigger than the one of the ambiant geometry. It will be shown as an example that the usual real projective plane contains a subgeometry of dimen- sion d for any d  ℵ 0 . Several equivalent geometric conditions which prevent such a pathological behavior will be given. The respective embeddings will be called proper embeddings. Still smaller is the class of the embeddings which VTEX(ZJ) PIPS No.: 115592 MATHKAP APCS232.tex; 5/02/1998; 13:54; v.7; p.1