Canonical map

From Wikipedia, the free encyclopedia

In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention).

A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps.

A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical example, see prestack.

For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.[1]



  1. ^ Buzzard, Kevin. "Grothendieck Conference Talk".
  2. ^ Vialar, Thierry (2016-12-07). Handbook of Mathematics. BoD - Books on Demand. p. 274. ISBN 9782955199008.