We investigate injectivity in a comma-category C/B 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 <1_X, f>:X--> XxB 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.

Injectivity and sections / F. Cagliari, S. Mantovani. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 204:1(2006), pp. 79-89.

Injectivity and sections

S. Mantovani
Ultimo
2006

Abstract

We investigate injectivity in a comma-category C/B 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 <1_X, f>:X--> XxB 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.
dcpo, Scott topology, exponentiable morphism, partial product.
Settore MAT/03 - Geometria
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/14745
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