To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Opposite category

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, ${\displaystyle (C^{\text{op}})^{\text{op}}=C}$.

• 1/3
Views:
2 641
28 280
30 418
• Category Theory Part 1 of 3: Categories
• Category Theory 7.2: Monoidal Categories, Functoriality of ADTs, Profunctors
• Category Theory 5.1: Coproducts, sum types

## 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)