Extranatural transformation

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

Definition

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

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

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

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

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

Properties

Extranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting $F$ (dually $G$) constant.

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

See also

• Dinatural transformation

External links

• extranatural+transformation at the nLab