Duality (Category Theory)

Idea

There is a natural symmetry in the definition of a category which allows for a construction of a dual or opposite category ${\mathcal{C}}^{\mathrm{op}}$ obtained by reversing all arrows of $\mathcal{C}$. This then extends to other category theoretic notions, such as allowing for a natural definition of contravariant functors.

Category

Let $\mathcal{C}$ be a category. We define its opposite category ${\mathcal{C}}^{\mathrm{op}}$:

• The objects $|{\mathcal{C}}^{\mathrm{op}}|:=|\mathcal{C}|$ are the same as the objects of $\mathcal{C}$
• The hom-set ${\mathcal{C}}^{\mathrm{op}}[a,b]:=\mathcal{C}[b,a]$ write $f^{\mathrm{op}}$ for inhabitants of ${\mathcal{C}}^{\mathrm{op}}[b,a]$ instead of $f$.
• Identity morphisms are simply the identity morphisms in ${\mathrm{id}}_a^{\mathrm{op}}:={\mathrm{id}}_a$.
• Composition is given as $g^{\mathrm{op}}\circ f^{\mathrm{op}}:=(f\circ g{)}^{\mathrm{op}}$ Then it can easily be checked that associative and identity laws hold for ${\mathcal{C}}^{\mathrm{op}}$.

Dual notions

Under this definition of duality, the following concepts are dual:

Concept	Dual Concept
initial object	terminal object
product	coproduct
limit	colimit
monomorphism	epimorphism
equalizer	coequalizer
pullback	pushout
kernel	cokernel
fibration	cofibrations
left adjoint	right adjoint
monad	comonad
cone	cocone
section	retraction

Extension to higher categories

In 2-categories, 1-cells and 2-cells may have distinct notions of duality. The notation used in this case is ${\mathcal{C}}^{\mathrm{op}}$ for inverting 1-cells and ${\mathcal{C}}^{\mathrm{co}}$ for inverting 2-cells. They can be combined with the notation ${\mathcal{C}}^{\mathrm{coop}}$ to invert both 1- and 2-cells.