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
  • (extranaturality in c) , , the following diagram commutes

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

  1. Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966)
  2. 1 2 Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint
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