extranatural transformation

Let $F:A\backslash times\; B^\backslash mathrm\{op\}\backslash times\; B\backslash rightarrow\; D$ and $G:A\backslash times\; C^\backslash mathrm\{op\}\backslash times\; C\backslash rightarrow\; D$ be two functors of categories.

A family $\backslash eta\; (a,b,c):F(a,b,b)\backslash rightarrow\; G(a,c,c)$ is said to be natural in a and extranatural in b and c if the following holds:

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

:: $\backslash begin\{matrix\}$

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

:: $\backslash begin\{matrix\}$

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}

Extranatural transformations can be used to define wedges and thereby endsFosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [https://arxiv.org/abs/1501.02503] (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.

• Dinatural transformation

• {{nlab|id=extranatural+transformation}}

Category:Higher category theory