nLab local colimit

Definition

A local colimit in a bicategory is a colimit in a hom-category that is preserved by the composition functor. A bicategory has local colimits of some shape if its hom-categories have colimits of those shapes that are preserved in each variable by the composition functors. A bicategory with all (small) local colimits is called locally cocomplete.

Examples

• In the delooping $\mathbf{B} V$ of a monoidal category $V$, any colimit preserved in each variable by the tensor product gives a local colimit. In particular, in the delooping of a closed monoidal category all colimits are local colimits.
• More generally, in a closed bicategory all colimits in hom-categories are local colimits.
• If $B$ has local coproducts, then we can construct its bicategory $Mat(B)$, whose objects are families of objects of $B$ and whose morphisms are matrices, which also has local coproducts.
• If $B$ has local coequalizers, then we can construct its bicategory $Mod(B)$, whose objects are monads in $B$ and whose morphisms are bimodules, which also has local coequalizers.
• Thus, if $B$ is locally cocomplete, we can construct its bicategory $Prof(B) = Mod(Mat(B))$, whose objects are categories enriched in $B$ and whose morphisms are profunctors, which is also locally cocomplete. In particular, Prof = $Prof(\mathbf{B} Set)$ is locally cocomplete.