Opposite category

In category theory, a branch of mathematics, the opposite category or dual category C^op of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, $(C^{\text{op}})^{\text{op}}=C$ .

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

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).^[1]

Properties

Opposite preserves products:

(C\times D)^{\text{op}}\cong C^{\text{op}}\times D^{\text{op}}

(see product category)

Opposite preserves functors:

(\mathrm {Funct} (C,D))^{\text{op}}\cong \mathrm {Funct} (C^{\text{op}},D^{\text{op}})

^[2]^[3] (see functor category, opposite functor)

Opposite preserves slices:

(F\downarrow G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})

(see comma category)

References

↑ "Is there an introduction to probability theory from a structuralist/categorical perspective?". MathOverflow. Retrieved 25 October 2010.
↑ H. Herrlich, G. E. Strecker, Category Theory, 3rd Edition, Heldermann Verlag, ISBN 978-3-88538-001-6, p. 99.
↑ O. Wyler, Lecture Notes on Topoi and Quasitopoi, World Scientific, 1991, p. 8.

Opposite category in nLab
Danilov, V.I. (2001) [1994], "Dual Category", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. p. 33. ISBN 1441931236. OCLC 851741862.
Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 53–55. ISBN 0199237182. OCLC 740446073.

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[MO1-1] "Is there an introduction to probability theory from a structuralist/categorical perspective?". MathOverflow. Retrieved 25 October 2010.

[2] H. Herrlich, G. E. Strecker, Category Theory, 3rd Edition, Heldermann Verlag, ISBN 978-3-88538-001-6, p. 99.

[3] O. Wyler, Lecture Notes on Topoi and Quasitopoi, World Scientific, 1991, p. 8.

Opposite category

Examples

Properties

See also

References