Incategory theory, a branch ofmathematics, theopposite category ordual category${\displaystyle C^{\text{op}}}$ of a givencategory${\displaystyle C}$ is formed by reversing themorphisms, 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,${\displaystyle (C^{\text{op}}{)}^{\text{op}}=C}$.

• An example comes from reversing the direction of inequalities in apartial order. So ifX is aset and ≤ a partial order relation, we can define a new partial order relation ≤^op by

x ≤^opy if and only ify ≤x.

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 beunderstood as a category.

• Given asemigroup (S, ·), one usually defines the opposite semigroup as (S, ·)^op = (S, *) wherex*y ≔y·x for allx,y inS. So also for semigroups there is a strong duality principle. Clearly, the same construction works for groups, as well, and is known inring theory, too, where it is applied to the multiplicative semigroup of the ring to give the opposite ring. Again this process can be described by completing a semigroup to a monoid, taking thecorresponding opposite category, and then possibly removing the unit from that monoid.
• The category ofBoolean algebras and Booleanhomomorphisms isequivalent to the opposite of the category ofStone spaces andcontinuous functions.
• The category ofaffine schemes isequivalent to the opposite of the category ofcommutative rings.
• ThePontryagin duality restricts to an equivalence between the category ofcompactHausdorffabeliantopological groups and the opposite of the category of (discrete) abelian groups.
• By the Gelfand–Naimark theorem, the category of localizablemeasurable spaces (withmeasurable maps) is equivalent to the category of commutativeVon Neumann algebras (withnormalunital homomorphisms of*-algebras).^[1]

Opposite preserves products:

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

Opposite preservesfunctors:

${\displaystyle (\mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}(C,D){)}^{\text{op}}\cong \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}(C^{\text{op}},D^{\text{op}})}$^[2][3] (seefunctor category,opposite functor)

Opposite preserves slices:

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

• Dual object
• Dual (category theory)
• Duality (mathematics)
• Adjoint functor
• Contravariant functor
• Opposite functor

• Opposite category at thenLab
• 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.
• Herrlich, Horst; Strecker, George E. (1979).Category Theory. SSPM (Sigma Series in Pure Mathematics) 01. Heldermann.ISBN 978-3-88538-001-6.

• Category theory

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