Kan extension
Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.
An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors.
In Categories for the Working Mathematician Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that
Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on 'constrained optimization'.
A Kan extension proceeds from the data of three categories
and two functors
and comes in two varieties: the "left" Kan extension and the "right" Kan extension of along .
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