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Opposite category

## Summary

In category theory, a branch of mathematics, the opposite category or dual category Cop 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, ${\displaystyle (C^{\text{op}})^{\text{op}}=C}$.

## 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 ≤op by
xop y if and only if yx.
The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/supremum, down-set/up-set, ideal/filter etc. This order theoretic duality is in turn a special case of the construction of opposite categories as every ordered set can be understood as a category.

## Properties

Opposite preserves products:

${\displaystyle (C\times D)^{\text{op}}\cong C^{\text{op}}\times D^{\text{op}}}$  (see product category)

Opposite preserves functors:

${\displaystyle (\mathrm {Funct} (C,D))^{\text{op}}\cong \mathrm {Funct} (C^{\text{op}},D^{\text{op}})}$ [2][3] (see functor category, opposite functor)

Opposite preserves slices:

${\displaystyle (F\downarrow G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})}$  (see comma category)

## References

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 at the nLab
• Danilov, V.I. (2001) [1994], "Dual Category", Encyclopedia of Mathematics, EMS Press
• 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 978-0199237180. OCLC 740446073.