nLab cocontinuous functor

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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.

basic properties of…

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.