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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.

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    $\begingroup$ Sorry of "map-centric" point of view is not clear here. If asked to provide something more concrete, I would be glad to. here is the article by Tom Leinster that offers some arguments for "rethinking set theory." $\endgroup$
    – Andres Mejia
    Sep 23, 2018 at 21:40
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    $\begingroup$ It's not clear to me what map centric means (is it commonly understood). Also, what course does it apply to? Are you assuming math major undergrads or science/engineering. And (assuming math majors), what level of skill/smarts (best of the best? average? bottom tenth)? $\endgroup$
    – guest
    Sep 23, 2018 at 23:23
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    $\begingroup$ One thought is I would be somewhat hesitant of assuming that you can introduce all kinds of abstractions earlier because they make sense to you NOW but would have been tough from the start. I see this mental blinders a lot on here. $\endgroup$
    – guest
    Sep 23, 2018 at 23:25
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    $\begingroup$ Finally it is not clear to me what the advantages of being more map centric are. $\endgroup$
    – guest
    Sep 23, 2018 at 23:25
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    $\begingroup$ The category theorists agenda. $\endgroup$
    – user5402
    Sep 24, 2018 at 12:10

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Yes!

See the text "Conceptual Mathematics", which aims to teach set theory notions, among other things, from a categorical perspective to complete novices.

It's an excellent text!

https://www.amazon.ca/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X

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