kan-extensions

Schema migration via Kan extensions uses left Kan extension (Lan_K F) for forward migrations and right Kan extension (Ran_K F) for backward migrations. You provide a mapping K from old to new schema and a functor F from the old schema to Set, then apply the extension to evolve schemas and their instances across versions while maintaining compatibility.

Forward migration (Lan_K F) evolves data to a new schema by taking colimits; backward migration (Ran_K F) maintains compatibility by taking limits. Forward handles schema evolution; backward ensures old data remains accessible under the new structure through universal properties of Kan extensions.

Yes, Kan extensions integrate directly with ACSets for schema migrations. The approach handles schema transport patterns and preserves universal properties specific to ACSets structures, enabling both forward and backward compatibility in attribute-colored set migrations.

While Kan extensions are grounded in category theory, the migration workflow abstracts the mathematical complexity. You define the schema mapping K and functor F, then apply Lan_K or Ran_K; the Julia/Catlab framework handles the categorical computation underneath.

You need a mapping K from your old schema to the new schema and a functor F from the old schema to Set. These define how data transforms; Kan extensions then compute the migrated schema and instances automatically within Julia/Catlab workflows.