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In mathematics, specifically in category theory, an extranatural transformation[1] is a generalization of the notion of natural transformation.
Definition
Let
and
two functors of categories.
A family
is said to be natural in a and extranatural in b and c if the following holds:
is a natural transformation (in the usual sense).
- (extranaturality in b)
,
,
the following diagram commutes
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57913e1604fe81b3c332a597a4fb686ec65a3ff3)
- (extranaturality in c)
,
,
the following diagram commutes
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f9d7666efd8ea4f550c07c559278ec26c273bc)
Properties
Extranatural transformations can be used to define wedges and thereby ends[2] (dually co-wedges and co-ends), by setting
(dually
) constant.
Extranatural transformations can be defined in terms of Dinatural transformations.[2]
See also
External links
References
- ^ Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966)
- ^ a b Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1]