nLab
local colimit

Local colimits

Context

2-Category theory

Limits and colimits

Local colimits

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 BV\mathbf{B} V of a monoidal category VV, 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 BB has local coproducts, then we can construct its bicategory Mat(B)Mat(B), whose objects are families of objects of BB and whose morphisms are matrices, which also has local coproducts.
  • If BB has local coequalizers, thenh we can construct its bicategory Mod(B)Mod(B), whose objects are monads in BB and whose morphisms are bimodules, which also has local coequalizers.
  • Thus, if BB is locally cocomplete, we can construct its bicategory Prof(B)=Mod(Mat(B))Prof(B) = Mod(Mat(B)), whose objects are categories enriched in BB and whose morphisms are profunctors, which is also locally cocomplete. In particular, Prof = Prof(BSet)Prof(\mathbf{B} Set) is locally cocomplete.

Last revised on October 20, 2017 at 06:46:57. See the history of this page for a list of all contributions to it.