Corestriction Explained
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 uniquefunction
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 French: 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.
Notes and References
- Book: Dauns . John . Hofmann . Karl Heinrich . 247487 . ix . American Mathematical Society . Memoirs of the American Mathematical Society . Representation of rings by sections . 83 . 1968.
- nlab, Image, https://ncatlab.org/nlab/show/image
- (Definition 3.1 and Remarks 3.2) in Gabriella Böhm, Hopf algebroids, in Handbook of Algebra (2008) arXiv:0805.3806
- 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
