Examples
- An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤new by
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- x ≤new y if and only if y ≤ x.
- For example, there are opposite pairs child/parent, or descendant/ancestor.
- The category of Boolean algebras and Boolean homomorphisms is equivalent to the opposite of the category of Stone spaces and continuous functions.
- The category of affine schemes is equivalent to the opposite of the category of commutative rings.
- The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups.
- By the Gelfand-Neumark theorem, the category of localizable measurable spaces (with measurable maps) is equivalent to the category of commutative Von Neumann algebras (with normal unital homomorphisms of *-algebras).
Read more about this topic: Opposite Category
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)