Dual category

to a category $\mathcal{C}$

The category $\mathcal{C}^o$ with the same objects as $\mathcal{C}$ and with morphism sets $\mathrm{Hom}_{\mathcal{C}^o}(A,B) = \mathrm{Hom}_{\mathcal{C}}(B,A)$ ( "reversal of arrows" ). Composition of two morphisms $u$ and $v$ in the category $\mathcal{C}^o$ is defined as composition of $v$ with $u$ in $\mathcal{C}$. The concepts and the statements in the category $\mathcal{C}$ are replaced by dual concepts and statements in $\mathcal{C}^o$. Thus, the concept of an epimorphism is dual to the concept of a monomorphism; the concept of a projective object is dual to that of an injective object, that of the direct product to that of the direct sum, etc. A contravariant functor on $\mathcal{C}$ becomes covariant on $\mathcal{C}^o$.

A dual category may sometimes have a direct realization; thus, the category of discrete Abelian groups is equivalent to the category dual to the category of compact Abelian groups (Pontryagin duality), while the category of affine schemes is equivalent to the category dual to the category of commutative rings with a unit element.

Comments

The dual category to a category $\mathcal{C}$ is also called the opposite category, and one also uses the notation $\mathcal{C}^{\mathrm{op}}$ (see Category).