nLab cocontinuous functor

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 $F\colon C \to D$ is cocontinuous if and only if the functor $F^{op}: C^{op} \to D^{op}$ 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 $F \colon Set_* \rightarrow Set$ from the category of pointed sets to the category Set of sets.

Related concepts

• continuous functor

• cocomplete category

• adjoint functor theorem

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

See the references at continuous functor.