Galois Connections As Morphisms
Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets. Especially, it is possible to compose Galois connections: given Galois connections (f ∗, f ∗) between posets A and B and (g ∗, g ∗) between B and C, the composite (g ∗f ∗, f ∗g ∗) is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, this categories display auto duality, that are quite fundamental for obtaining other duality theorems. More special kinds of morphisms that induce adjoint mappings in the other direction are the morphisms usually considered for frames (or locales).
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Famous quotes containing the word connections:
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)