Some articles on cartesian morphisms, morphisms, morphism:
Fibred Category - Formal Definitions - Cartesian Morphisms and Functors
... of E, then the subcategory of F consisting of those objects x for which φ(x)=S and those morphisms m satisfying φ(m)=idS, is called the fibre category (or fibre) over S ... The morphisms of FS are called S-morphisms, and for x,y objects of FS, the set of S-morphisms is denoted by HomS(x,y) ... The image by φ of an object or a morphism in F is called its projection (by φ) ...
... of E, then the subcategory of F consisting of those objects x for which φ(x)=S and those morphisms m satisfying φ(m)=idS, is called the fibre category (or fibre) over S ... The morphisms of FS are called S-morphisms, and for x,y objects of FS, the set of S-morphisms is denoted by HomS(x,y) ... The image by φ of an object or a morphism in F is called its projection (by φ) ...
Fibred Category - Formal Definitions - Fibred Categories and Cleaved Categories
... most flexible and economical definition of fibred categories is based on the concept of cartesian morphisms ... one presented in Grothendieck (1959) the definition in terms of cartesian morphisms was introduced in Grothendieck (1974) in 1960–1961 ... category (or a fibred E-category, or a category fibred over E) if each morphism f of E whose codomain is in the range of projection has at least one inverse image, and moreover the composition m o n of any two ...
... most flexible and economical definition of fibred categories is based on the concept of cartesian morphisms ... one presented in Grothendieck (1959) the definition in terms of cartesian morphisms was introduced in Grothendieck (1974) in 1960–1961 ... category (or a fibred E-category, or a category fibred over E) if each morphism f of E whose codomain is in the range of projection has at least one inverse image, and moreover the composition m o n of any two ...
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