词条 | Dual (category theory) |
释义 |
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about Cop. Also, if a statement is false about C, then its dual has to be false about Cop. Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and Cop are equivalent as categories. In the case when C and its opposite Cop are equivalent, such a category is self-dual.[1] Formal definitionWe define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms. Let σ be any statement in this language. We form the dual σop as follows:
Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. Duality is the observation that σ is true for some category C if and only if σop is true for Cop.{{sfn|Mac Lane|1978|p=33}}{{sfn|Awodey|2010|p=53-55}} Examples
Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category Cop is an epimorphism.
x ≤new y if and only if y ≤ x. This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices.
See also
References1. ^{{cite book|author1=Jiří Adámek|author2=J. Rosicky|title=Locally Presentable and Accessible Categories|url=https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62|year=1994|publisher=Cambridge University Press|isbn=978-0-521-42261-1|page=62}}
2 : Category theory|Duality theories |
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