Extranatural transformation

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

Definition

Let $F:A\times B^\mathrm{op}\times B\rightarrow D$ and $G:A\times C^\mathrm{op}\times C\rightarrow D$ 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

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

• (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

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

Properties

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

Extranatural transformations can be defined in terms of Dinatural transformations.^[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