nLab cocontinuous functor

Contents

Context

Category theory

Limits and colimits

Contents

Idea

A functor is cocontinuous if it preserves small colimits.

Typically one only considers cocontinuous functors whose domain and codomain are cocomplete categories (have all small colimits).

Note that F:CDF\colon C \to D is cocontinuous if and only if the functor F op:C opD opF^{op}: C^{op} \to D^{op} between opposite categories is a continuous functor.

Examples

Example

Every left adjoint functor is cocontinuous, since left adjoints preserve colimits.

Not every functor is cocontinuous:

Example

A counterexample of a dis-cocontinuous functor is the forgetful functor F:Set *SetF \colon Set_* \rightarrow Set from the category of pointed sets to the category Set of sets.

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References

See the references at continuous functor.

Last revised on April 17, 2024 at 15:46:57. See the history of this page for a list of all contributions to it.