Embedding is defined to be a one-to-one structure preserving mapping.
My question is if the one-to-one condition is really critical.
Like if linear mappings from high-dimensional space to low-dimensional space such as principal component analysis (PCA), could be technically considered as an embedding. Or perhaps another technical term should be used for them. The reason I ask is that in other disciplines, say in machine learning, the term embedding is used a lot for dimensionality reduction methods, that are not one to one. (See the abstract of this paper for example.)
Yes, an embedding is by definition injective. If you change the definition, then it's not what anyone calls an embedding anymore.
It's like asking if the condition "every element has an inverse" is "really critical" in the definition of a group... Sure, you can consider "groups" where not all elements have inverses. Well, except nobody calls that a "group", but a monoid.
If your "embedding" is not injective, then it's not an "embedding" at all. You (and people from other fields of math or science) are free to call something else "embedding", but at least in the areas of math I know, an embedding is injective.