Question: Would it be feasible to teach undergraduate math students a "map"-centric view early on? If so, how early on?
Now that I'm preparing for a phd program, I'm also reflecting on my undergraduate education and wishing that I had adopted a so-called "map"-centric point of view earlier on. Here are toy-examples:
A point $x \in X$ is a map $f:\{\bullet\} \hookrightarrow X$
Or maybe the product as the cartesian product $A \times B$ equipped with canonical projections vs. $A \sqcup B$ (and injections.)
Either way, these are things of a categorical "flavor" that probably wouldn't have really profoundly affected the math at an early stage, but did clarified some things for me later on (paths in $X$ are continuous maps $[0,1] \to X$, resp. the difference between direct product/sum in algebra.)
My question I guess is not really should this be done? But rather can this be done successfully?
An answer to my question would be either someone who has personally taken up this point of view (un)successfully or arguments for why it can(not) be done.