Definition:Dual Category
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Definition
Let $\mathbf C$ be a metacategory.
Its dual category, denoted $\mathbf C^{\text{op} }$, is defined as follows:
Objects: | $X^{\text{op} }$, for all $X \in \operatorname{ob}\mathbf C$ | |
Morphisms: | $f^{\text{op} }: D^{\text{op} } \to C^{\text{op} }$ for all $f: C \to D$ in $\mathbf C_1$ | |
Composition: | $\left({f^{\text{op} } \circ g^{\text{op} } }\right) := \left({g \circ f}\right)^{\text{op} }$, whenever this is defined | |
Identity morphisms: | $\operatorname{id}_{X^{\text{op} } } := \operatorname{id}_X^{\text{op} }$ |
It can be seen that this comes down to the metacategory obtained by reversing the direction of all morphisms of $\mathbf C$.
Also known as
Many authors call $\mathbf C^{\text{op}}$ the opposite category of $\mathbf C$.
Others use e.g. $f^*$ in place of $f^{\text{op} }$. As the character $*$ is used so often already in mathematics, the form $f^{\text{op} }$ is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Dual Category is Category
- Results about dual categories can be found here.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.2$