Kan extension

In mathematical category theory, functors , which are the universal approximation to the solution of the equation, are called Kan extensions . The construction is named after Daniel M. Kan , who constructed such extensions as Limites and Kolimites in 1960 . ${\displaystyle ?\circ F=X}$

definition

There are two dual definitions: one extension is called left-sided because it is defined by a universal property in which the Kan extension occurs as a source, while the other extension is called right-sided because it is the target of a universal transformation.

Kan extension on the left

Let , and categories, L , X , F and M be functors and and natural transformations . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \sigma }$ ${\displaystyle \alpha }$

The left-hand Kan expansion of a functor along a functor is a pair that satisfies the following universal property: ${\displaystyle X\colon {\mathcal {A}}\to {\mathcal {C}}}$ ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}$ ${\displaystyle (L\colon {\mathcal {B}}\to {\mathcal {C}},\varepsilon \colon X\to L\circ F)}$

For each and every one there is exactly one with , with . ${\displaystyle M\colon {\mathcal {B}}\to {\mathcal {C}}}$ ${\displaystyle \alpha \colon X\to M\circ F}$ ${\displaystyle \sigma \colon L\to M}$ ${\displaystyle \sigma _{F}\circ \varepsilon =\alpha }$ ${\displaystyle \sigma _{F}(A)=\sigma \left(F(A)\right)}$

Kan extension on the right

Let , and categories, R , X , F and M be functors and and natural transformations. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \delta }$ ${\displaystyle \mu }$

The right-hand Kan expansion of a functor along a functor is a pair that satisfies the following universal property: ${\displaystyle X\colon {\mathcal {A}}\to {\mathcal {C}}}$ ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}$ ${\displaystyle (R\colon {\mathcal {B}}\to {\mathcal {C}},\eta \colon R\circ F\to X)}$

For each and every one there is exactly one with , with . ${\displaystyle M\colon {\mathcal {B}}\to {\mathcal {C}}}$ ${\displaystyle \mu \colon M\circ F\to X}$ ${\displaystyle \delta \colon M\to R}$ ${\displaystyle \eta \circ \sigma _{F}=\mu }$ ${\displaystyle \delta _{F}(A)=\delta \left(F(A)\right)}$

literature

• Saunders Mac Lane : Categories for the Working Mathematician (= Graduate Texts in Mathematics . 5). 2nd edition. Springer, New York et al. 1998, ISBN 0-387-98403-8 .