Difference between morphism, bundle, and generalized element

Question

What's the difference between

• A morphism with codomain $X$
• A generalized element of $X$
• A bundle over $X$?

They all seem to refer to a morphism with codomain $X$, but the different words are used in different situations.

Answer 1

Yes, there is no formal difference. It happens frequently in category theory that objects which seem distinct, once sufficiently generalized, turn out to be special cases of the same general concept. A morphism $f:A\to X$ may also be thought of as a generalized element of $X$, indexed by or shaped like $A$, as if $A=S^1$ in a category of topological spaced, $f$ corresponds to a circle drawn in $X$. But $f$ may also be thought of as a bundle over $X$, especially if there is some nice notion of fiber in the given category and the fibers are nicely related.