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, of which they are a special case.[2]
See also
Dinatural transformation
External links
extranatural+transformation in nLab
References
Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966)
Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1]
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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