Inmathematics, specifically incategory theory, anextranatural transformation^[1] is a generalization of the notion ofnatural transformation.

Let${\displaystyle F:A\times B^{\mathrm{o}\mathrm{p}}\times B\to D}$ and${\displaystyle G:A\times C^{\mathrm{o}\mathrm{p}}\times C\to D}$ be twofunctors of categories.A family${\displaystyle \eta (a,b,c):F(a,b,b)\to G(a,c,c)}$ is said to benatural inaand extranatural inbandc if the following holds:

• ${\displaystyle \eta (-,b,c)}$ is a natural transformation (in the usual sense).
• (extranaturality inb)${\displaystyle \mathrm{\forall }(g:b\to b^{\mathrm{\prime }})\in \mathrm{M}\mathrm{o}\mathrm{r}B}$,${\displaystyle \mathrm{\forall }a\in A}$,${\displaystyle \mathrm{\forall }c\in C}$ the followingdiagram commutes

• (extranaturality inc)${\displaystyle \mathrm{\forall }(h:c\to c^{\mathrm{\prime }})\in \mathrm{M}\mathrm{o}\mathrm{r}C}$,${\displaystyle \mathrm{\forall }a\in A}$,${\displaystyle \mathrm{\forall }b\in B}$ the following diagram commutes

Extranatural transformations can be used to define wedges and therebyends^[2] (dually co-wedges and co-ends), by setting${\displaystyle F}$ (dually${\displaystyle G}$) constant.

Extranatural transformations can be defined in terms ofdinatural transformations, of which they are a special case.^[2]

• Dinatural transformation

• extranatural+transformation at thenLab

• Higher category theory

• Articles with short description
• Short description matches Wikidata