Injectivity in a comma-category C/B is investigated using the notion of the “object of sections” S(f) of a given morphism f:X→B in C. We first obtain that f:X→B is injective in C/B if and only if the morphism 1X,f:X→X×B is a section in C/B and the object S(f) of sections of f is injective in C. Using this approach, we study injective objects f with respect to the class of embeddings in the categories ContL/B (AlgL/B) of continuous (algebraic) lattices over B. As a result, we obtain both topological (every fiber of f has maximum and minimum elements and f is open and closed) and algebraic (f is a complete lattice homomorphism) characterizations.
F. Cagliari, S.Mantovani (2006). Injectivity and sections. JOURNAL OF PURE AND APPLIED ALGEBRA, 204, 79-89 [10.1016/j.jpaa.2005.03.003].
Injectivity and sections
CAGLIARI, FRANCESCA;
2006
Abstract
Injectivity in a comma-category C/B is investigated using the notion of the “object of sections” S(f) of a given morphism f:X→B in C. We first obtain that f:X→B is injective in C/B if and only if the morphism 1X,f:X→X×B is a section in C/B and the object S(f) of sections of f is injective in C. Using this approach, we study injective objects f with respect to the class of embeddings in the categories ContL/B (AlgL/B) of continuous (algebraic) lattices over B. As a result, we obtain both topological (every fiber of f has maximum and minimum elements and f is open and closed) and algebraic (f is a complete lattice homomorphism) characterizations.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
