Localization of Universal Problems. Local Colimits
- Cite this article as:
- Ehresmann, A.C. Applied Categorical Structures (2002) 10: 157. doi:10.1023/A:1014342114336
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The notion of the root of the category, which is a minimal (in a precise sense) weakly coreflective subcategory, is introduced in view of defining ‘local’ solutions of universal problems: If U is a functor from C′ to C and c an object of C, the root of the comma-category c|U is called a U-universal root generated by c; when it exists, it is unique (up to isomorphism) and determines a particular form of the locally free diagrams defined by Guitart and Lair. In this case, the analogue of an adjoint functor is an adjoint-root functor of U, taking its values in the category of pro-objects of C′. Local colimits are obtained if U is the insertion from a category into its category of ind-objects; they generalize Diers' multicolimits. Applications to posets and Galois theory are given.