Extranatural transformation

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

Definition

Let $F:A\times B^{\mathrm{o}\mathrm{p}}\times B\to D$ and $G:A\times C^{\mathrm{o}\mathrm{p}}\times C\to D$ be two functors of categories. A family $\eta (a,b,c):F(a,b,b)\to G(a,c,c)$ 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) $\mathrm{\forall }(g:b\to b^{\prime })\in \mathrm{M}\mathrm{o}\mathrm{r}B$, $\mathrm{\forall }a\in A$, $\mathrm{\forall }c\in C$ the following diagram commutes

$\begin{array}{ccc}F(a,b^{\prime },b)& \stackrel{F(1,1,g)}{\to }& F(a,b^{\prime },b^{\prime })\\ {}_{F(1,g,1)}\downarrow & & {}_{\eta (a,b^{\prime },c)}\downarrow \\ F(a,b,b)& \stackrel{\eta (a,b,c)}{\to }& G(a,c,c)\end{array}$

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

$\begin{array}{ccc}F(a,b,b)& \stackrel{\eta (a,b,c^{\prime })}{\to }& G(a,c^{\prime },c^{\prime })\\ {}_{\eta (a,b,c)}\downarrow & & {}_{G(1,h,1)}\downarrow \\ G(a,c,c)& \stackrel{G(1,1,h)}{\to }& G(a,c,c^{\prime })\end{array}$

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

• Template:Nlab