Injectivity and sections

We investigate injectivity in a comma-category C/BC/B using the notion of the “object of sections” S(f)S(f) of a given morphism f:X→Bf:X→B in C. We first obtain that f:X→Bf:X→B is injective in C/BC/B if and only if the morphism 〈1X,f〉:X→X×B〈1X,f〉:X→X×B is a section in C/BC/B and the object S(f)S(f) of sections of ff is injective in C. Using this approach, we study injective objects ff with respect to the class of embeddings in the categories ContL/BContL/B (AlgL/BAlgL/B) of continuous (algebraic) lattices over BB. As a result, we obtain both topological (every fiber of ff has maximum and minimum elements and ff is open and closed) and algebraic (ff is a complete lattice homomorphism) characterizations.