Extranatural transformation

In mathematics, specifically in category theory, an extranatural transformation^[1] is a generalization of the notion of natural transformation.

Definition

Let ${\displaystyle F:A\times B^{\mathrm {op} }\times B\rightarrow D}$ and ${\displaystyle G:A\times C^{\mathrm {op} }\times C\rightarrow D}$ be two functors of categories. A family ${\displaystyle \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c)}$ is said to be natural in a and extranatural in b and c if the following holds:

• ${\displaystyle \eta (-,b,c)}$ is a natural transformation (in the usual sense).
• (extranaturality in b) ${\displaystyle \forall (g:b\rightarrow b^{\prime })\in \mathrm {Mor} \,B}$, ${\displaystyle \forall a\in A}$, ${\displaystyle \forall c\in C}$ the following diagram commutes

${\displaystyle {\begin{matrix}F(a,b',b)&\xrightarrow {F(1,1,g)} &F(a,b',b')\\_{F(1,g,1)}\downarrow \qquad &&_{\eta (a,b',c)}\downarrow \qquad \\F(a,b,b)&\xrightarrow {\eta (a,b,c)} &G(a,c,c)\end{matrix}}}$

• (extranaturality in c) ${\displaystyle \forall (h:c\rightarrow c^{\prime })\in \mathrm {Mor} \,C}$, ${\displaystyle \forall a\in A}$, ${\displaystyle \forall b\in B}$ the following diagram commutes

${\displaystyle {\begin{matrix}F(a,b,b)&\xrightarrow {\eta (a,b,c')} &G(a,c',c')\\_{\eta (a,b,c)}\downarrow \qquad &&_{G(1,h,1)}\downarrow \qquad \\G(a,c,c)&\xrightarrow {G(1,1,h)} &G(a,c,c')\end{matrix}}}$

Properties

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

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

See also

• Dinatural transformation