Generalization of natural transformations
In mathematics, specifically in category theory, an extranatural transformation[1] is a generalization of the notion of natural transformation.
Definition
Let
and
be 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)}\downarrow \qquad &&_{\eta (a,b',c)}\downarrow \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/4598644a997f63e324ed37ecb9c3a9f765cb9886)
- (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)}\downarrow \qquad &&_{G(1,h,1)}\downarrow \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/6aef3ffc41a61b418759a484a4db3a4e0f9e1370)
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, of which they are a special case.[2]
See also
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]
External links
- extranatural+transformation at the nLab