My book asserts that, for several common categories, such Vec, Grp, Top, etc., the forgetful functor is not injective on the category's class of objects, $\mathcal{Ob}$.
I'm looking for examples of this assertion for the forgetful $U_\mathbf{Vec} = U$ on Vec.
(By Vec I mean the category of vector spaces over R, with the R-linear transformations as morphisms.)
And a second, related question:
Is there a $X\in \mathcal{Ob}_\mathbf{Set}$ such that
- |X| > 1, and
- $U(V\;) = X$ for exactly one $V\, \in \mathcal{Ob}_\mathbf{Vec}\;\;$?
P.S. One reason for my difficulties with this question is that the common modern mathematical practice of declaring superficially different entities as "essentially the same" (as in "a vector space $V$ and its double dual $V^{**}$ are essentially the same", or "all singletons are essentially the same") makes it difficult for me to settle questions regarding injectivity, which hinge crucially on knowing when two things are different (e.g. $V, W \in \mathcal{Ob}_\mathbf{Vec}$, on the one hand, and $U(V\;), U(W\;)\in\mathcal{Ob}_\mathbf{Set}$ on the other).
Edit: added a link to the book I referenced, and changed Vect to Vec to match the book's notation.