Extranatural transformation

In mathematics, specifically in category theory, an extranatural transformationEilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966) is a generalization of the notion of natural transformation.

Definition

Let $F:A\times B^\mathrm\times B\rightarrow D$ and $G:A\times C^\mathrm\times C\rightarrow D$ 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'') $\forall (g:b\rightarrow b^\prime)\in \mathrm\, B$, $\forall a\in A$, $\forall c\in C$ the following diagram commutes

:: $\begin F(a,b',b) & \xrightarrow & F(a,b',b') \\ _\downarrow\qquad & & _\downarrow\qquad \\ F(a,b,b) & \xrightarrow & G(a,c,c) \end$

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

:: $\begin F(a,b,b) & \xrightarrow & G(a,c',c') \\ _\downarrow\qquad & & _\downarrow\qquad \\ G(a,c,c) & \xrightarrow & G(a,c,c') \end$

Properties

Extranatural transformations can be used to define wedges and thereby endsFosco Loregian, ''This is the (co)end, my only (co)friend'', arXiv preprin

/ref> (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

* {{nlab, id=extranatural+transformation

References

Higher category theory