Dinatural transformation

Short description: Generalization of natural transformations

In category theory, a branch of mathematics, a dinatural transformation $\alpha$ between two functors

$S,T:C^{\mathrm{o}\mathrm{p}}\times C\to D,$

written

$\alpha :S\ddot{\to }T,$

is a function that to every object $c$ of $C$ associates an arrow

${\alpha }_c:S(c,c)\to T(c,c)$ of $D$

and satisfies the following coherence property: for every morphism $f:c\to c^{\prime }$ of $C$ the diagram center commutes. Note the direction of $S(f,g)$ is opposite along $f$ in the first component since it is contravariant.

The composition of two dinatural transformations need not be dinatural.

See also

• Extranatural transformation
• Natural transformation

References

• Fosco, Loregian (22 July 2021), (Co)end Calculus, Cambridge University Press, doi:10.1017/9781108778657, ISBN 9781108746120, https://books.google.com/books?id=cfIuEAAAQBAJ&dq&pg=PA23 
• Dubuc, Eduardo; Street, Ross (1970). "Dinatural transformations". Reports of the Midwest Category Seminar IV. Lecture Notes in Mathematics. 137. pp. 126–137. doi:10.1007/BFb0060443. ISBN 978-3-540-04926-5. https://books.google.com/books?id=vTF7CwAAQBAJ&pg=PA126. 

External links

• dinatural transformation in nLab