Corestriction

In mathematics, a corestriction[1] of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.

Given any subset we can consider the corresponding inclusion of sets as a function. Then for any function , the restriction of a function onto can be defined as the composition .

Analogously, for an inclusion the corestriction of onto is the unique function such that there is a decomposition . The corestriction exists if and only if contains the image of . In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of . More generally, one can consider corestriction of a morphism in general categories with images.[2] The term is well known in category theory, while rarely used in print.[3]

Andreotti[4] introduces the above notion under the name coastriction, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if is a surjection of sets (that is a quotient map) then Andreotti considers the composition , which surely always exists.

References

  1. "Elementary set theory - is there a word for restricting the codomain of a function?".
  2. nlab, Image, https://ncatlab.org/nlab/show/image
  3. (Definition 3.1 and Remarks 3.2) in Gabriella Böhm, Hopf algebroids, in Handbook of Algebra (2008) arXiv:0805.3806
  4. paragraph 2-14 at page 14 of Andreotti, A., Généralités sur les categories abéliennes (suite) Séminaire A. Grothendieck, Tome 1 (1957) Exposé no. 2, http://www.numdam.org/item/SG_1957__1__A2_0
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