Preservation (Category Theory)

Definition

Let $F:\mathcal{C}\to \mathcal{D}$ be a functor. We say $F$ preserves limits of shape $\mathcal{I}$ if for every diagram $D:\mathcal{I}\to \mathcal{C}$ and every limiting cone $(L,\varphi )$ over $D$, the image $(FL,F\varphi )$ is a limiting cone over the diagram $F\circ D$ in $\mathcal{D}$.

Explicitly, if ${\varphi }_i:L\to D_i$ is a natural transformation from the constant functor ${\Delta }_L$ to $D$ satisfying the universal property of the limit, then $F{\varphi }_i:FL\to F(D_i)$ must satisfy the universal property of the limit for $F\circ D$.

Preservation of Colimits

Dually, $F$ preserves colimits of shape $\mathcal{I}$ if for every diagram $D:\mathcal{I}\to \mathcal{C}$ and every colimiting cocone $(C,\psi )$ over $D$, the image $(FC,F\psi )$ is a colimiting cocone over $F\circ D$ in $\mathcal{D}$.

Properties

1. A functor is continuous if it preserves all small limits.
2. A functor is cocontinuous if it preserves all small colimits.
3. Every covariant representable functor $Hom(X,-):\mathcal{C}\to Set$ preserves all limits.
4. Every contravariant representable functor $Hom(-,X):{\mathcal{C}}^{op}\to Set$ sends colimits in $\mathcal{C}$ to limits in $Set$.

Theorems

Adjoint Functor Theorem

If $F:\mathcal{C}\to \mathcal{D}$ is a left adjoint, then $F$ preserves all colimits. If $G:\mathcal{D}\to \mathcal{C}$ is a right adjoint, then $G$ preserves all limits.

Preservation by Equivalence

Equivalences of categories preserve and reflect both limits and colimits.

References

• riehl2016-category-theory