Kan Extension

Given functors and , the left Kan extension is the universal (initial) functor through which factors up to a natural transformation ; the right Kan extension is the dual (terminal) one. When is (co)complete they are pointwise, computed object-by-object as

Extending a functor defined on a subcategory to all of is a Kan extension along the inclusion.

Proposition, latching assembly on a finite poset

On a finite partially-ordered-set the pointwise left Kan extension is a colimit computed by latching objects. At each the part already determined by proper sub-objects is , so the new content is the complement of the latching map’s image. Summing the new content over recovers the global value, and the passage between the value and its increments is Möbius inversion. Dually, the right Kan extension along a refinement inclusion gives the terminal value compatible with what is already known, namely the canonical warm-start / minimum-norm completion.

References

• Daniel M. Kan, “Adjoint functors”, Trans. Amer. Math. Soc. 87 (1958)
• Saunders Mac Lane, “Categories for the Working Mathematician” (1971), Ch. X