Kan extension

In the mathematical category theory is called functors, which are the universal approximation to the solution of the equation, as Kan extensions. The design is named after Daniel M. Kan, who designed such extensions in 1960 as Limites and colimits.

Definition

There are two dual definitions: an extension to the left side called because it is defined by a universal property in which the Kan extension occurs as the source, while the other extension is called the right side because it is the goal of a universal transformation.

Left-sided Kan extension

Be, and categories, L, X, F and M functors and natural transformations and.

The left- Kan extension of a functor along a functor is a pair that satisfies the following universal property:

For each and every there is exactly one, with.

Right-sided Kan extension

Be, and categories, R, X, F and M functors and natural transformations and.

The right- Kan extension of a functor along a functor is a pair that satisfies the following universal property:

For each and every there is exactly one, with.