Canonical map

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a canonical map, also called a natural map, is a <a href="/facts/Map_(mathematics)/FABF1l2f">map</a> or <a href="/facts/Morphism/Kdpuyz3S">morphism</a> 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 <a href="/facts/Canonical_isomorphism/pj8KgkxU">canonical isomorphism</a> is a canonical map that is also an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> (i.e., <a href="/facts/Inverse_function/Tg5nnXlu">invertible</a>). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical example, see <a href="/facts/Prestack/2byjNpEA">prestack</a>.
For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.

Canonical map open-in-new

Canonical map