Definition:Continuous Functor

Definition

Let $\mathbf C$, $\mathbf D$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a functor.

Then $F$ is continuous if and only if for all diagrams $D: \mathbf J \to \mathbf C$ with limit ${\varprojlim \,}_j \, D_j$:

$\map F {{\varprojlim \,}_j \, D_j} \cong {\varprojlim \,}_j \, F D_j$

where $F D: \mathbf J \to \mathbf D$ is the diagram obtained by composition of $F$ with $D$, and $\mathbf J$ is an arbitrary metacategory.

Also see

• Functor Preserving Limits
• Cocontinuous Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.5$: Definition $5.25$