To start, an object is simply any mathematical construct, be it a group, a space, a manifold, or anything else that can be defined mathematically. While we tend to think of an object as a tangible "thing," mathematical objects are no less real, even if they are less tangible. For example, a ball is a sphere. An equation can be written that, when plotted in three-dimensional space, will produce a sphere. Although the ball can be touched and the mathematical construct is intangible, is the graphed sphere any less real than the ball? And, by extension, aren't all surfaces, lines, curves, shapes, and other objects described by mathematical equations equally real?
If two objects can be related to one another through some sort of consistent mathematical relationship, this relationship is called a "morphism," or a map. Thinking about it, the term "map" is not unreasonable so much as it is unexpected. The typical image that comes to mind when hearing the word "map" is of a street map, not a mathematical relationship.
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