Abstract
Projective geometries studied as Pasch geometries possess morphisms and homomorphisms. A homomorphic image of a projective geometry is shown to be projective. A projective geometry is shown to be Desarguesian iff it is a homomorphic image of a higher dimensional one, which in a sense is dual to the classical imbedding theorem. Semi-linear maps induce morphisms which are homomorphisms iff the associated homomorphisms of skewfields are isomorphisms. Projective geometries form categories with morphisms as well as homomorphisms and Desarguesian ones form a subcategory with Desarguesian homomorphisms.
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Bhattrai, H. Categories of Projective Geometries with Morphisms and Homomorphisms. Geometriae Dedicata 78, 111–120 (1999). https://doi.org/10.1023/A:1005222520379
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DOI: https://doi.org/10.1023/A:1005222520379