A local colimit in a bicategory is a colimit in a hom-category that is preserved by the compositionfunctor. 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 of a monoidal category , 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 has local coproducts, then we can construct its bicategory , whose objects are families of objects of and whose morphisms are matrices, which also has local coproducts.
If has local coequalizers, then we can construct its bicategory , whose objects are monads in and whose morphisms are bimodules, which also has local coequalizers.
Thus, if is locally cocomplete, we can construct its bicategory , whose objects are categories enriched in and whose morphisms are profunctors, which is also locally cocomplete. In particular, Prof = is locally cocomplete.
Last revised on March 3, 2021 at 02:42:21.
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