Opposite Category

Let $\mathcal C$ be a category .

Definition

The opposite category $\mathcal C^{\mathrm{op}}$ is the category defined by:

• $\mathrm{Ob}(\mathcal C^{\mathrm{op}})=\mathrm{Ob}(\mathcal C)$;
• for objects $A,B$, $\mathrm{Hom}_{\mathcal C^{\mathrm{op}}}(A,B) := \mathrm{Hom}_{\mathcal C}(B,A).$ So a morphism $f:A\to B$ in $\mathcal C^{\mathrm{op}}$ is “the same arrow” as a morphism $f:B\to A$ in $\mathcal C$.

Composition and identities

• The identity morphism on an object $A$ is the same arrow $\mathrm{id}_A$ as in $\mathcal C$.
• Composition is reversed: if $f:A\to B,\quad g:B\to C \quad \text{in } \mathcal C^{\mathrm{op}},$ then in $\mathcal C$ these correspond to $f:B\to A$ and $g:C\to B$, and the composite in $\mathcal C^{\mathrm{op}}$ is defined by $g\circ_{\mathcal C^{\mathrm{op}}} f := f\circ_{\mathcal C} g.$

Basic facts

• Taking opposites is involutive: $(\mathcal C^{\mathrm{op}})^{\mathrm{op}}=\mathcal C$.
• Many constructions come in dual pairs via $\mathcal C \leftrightarrow \mathcal C^{\mathrm{op}}$.
  For instance, a morphism is a monomorphism in $\mathcal C$ iff it is an epimorphism in $\mathcal C^{\mathrm{op}}$.

Examples

1. Posets as categories: A partially ordered set $(P,\le)$ can be viewed as a category with a unique morphism $p\to q$ iff $p\le q$.
   Its opposite category corresponds to the reversed order $\ge$.

2. Contravariance: A contravariant functor $F:\mathcal C\to \mathcal D$ can be packaged as an ordinary functor $F:\mathcal C^{\mathrm{op}}\to \mathcal D$.

3. One-object categories from groups: A group $G$ defines a category with one object $*$ and $\mathrm{End}(*)=G$.
   The opposite category corresponds to reversing multiplication (the “opposite group”); inversion $g\mapsto g^{-1}$ gives an isomorphism $G\cong G^{\mathrm{op}}$.