Assertion: isomorphism | /r/HasProperty | natural
Facets:
value: for such an isomorphism
type: OTHER
value: in the sense
type: OTHER
Source sentences:
text: These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab op × Ab op × Ab
text: The theorems explicitly construct isomorphisms that are the natural choice for such an isomorphism; in all cases, the isomorphism constructed by the theorem is effectively the only possible thing to write down.
text: However, in the absence of additional constraints (such as a requirement that maps preserve the chosen basis), the map from a space to its dual is not unique, and thus such an isomorphism requires a choice, and is "not natural".
text: But all the isomorphisms are natural, coming from a universal property, so all the diagrams will commute because of this, and everything will work out fine.
text: We are kind of cheating here; there is a diagram, we can’t see it because of this trick of using to mean an isomorphism, the isomorphism is natural, but the naturality does not immediately imply that the diagram we need to commute, commutes.