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 ${\displaystyle G:A\times C^{\mathrm {op} }\times C\rightarrow D}$ 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

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

• (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

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

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

External links

• extranatural+transformation in nLab

References

1. Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966)
2. Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint