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
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