Extranatural transformation
In mathematics, specifically in category theory, an extranatural transformation[1] is a generalization of the notion of natural transformation.
Definition
Let [math]\displaystyle{ F:A\times B^\mathrm{op}\times B\rightarrow D }[/math] and [math]\displaystyle{ G:A\times C^\mathrm{op}\times C\rightarrow D }[/math] be two functors of categories. A family [math]\displaystyle{ \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c) }[/math] is said to be natural in a and extranatural in b and c if the following holds:
- [math]\displaystyle{ \eta(-,b,c) }[/math] is a natural transformation (in the usual sense).
- (extranaturality in b) [math]\displaystyle{ \forall (g:b\rightarrow b^\prime)\in \mathrm{Mor}\, B }[/math], [math]\displaystyle{ \forall a\in A }[/math], [math]\displaystyle{ \forall c\in C }[/math] the following diagram commutes
- [math]\displaystyle{ \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} }[/math]
- (extranaturality in c) [math]\displaystyle{ \forall (h:c\rightarrow c^\prime)\in \mathrm{Mor}\, C }[/math], [math]\displaystyle{ \forall a\in A }[/math], [math]\displaystyle{ \forall b\in B }[/math] the following diagram commutes
- [math]\displaystyle{ \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} }[/math]
Properties
Extranatural transformations can be used to define wedges and thereby ends[2] (dually co-wedges and co-ends), by setting [math]\displaystyle{ F }[/math] (dually [math]\displaystyle{ G }[/math]) constant.
Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.[2]
See also
References
External links
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