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Why call on an onto function surjective?

Jared
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    I'm not convinced you'll get any decent answers here, but this question is very much related: https://math.stackexchange.com/questions/240913/category-theory-without-codomains?rq=1 Indeed, if you only allow surjections, you are drastically changing Hom-sets in many categories. It goes without saying that category theory is very much established and people are unlikely/unwilling to change it. – Mathematician 42 Mar 06 '18 at 12:24
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    Also, I very much object that every function should be surjective. I think that Qiaochu Yuan's answer to the linked question makes some good points and I can image more objections other than those listed there. – Mathematician 42 Mar 06 '18 at 12:26
  • The human language of mathematics has evolved much like any other human language: through use and repetition and so on. "Surjective" is an old term (from before the beginning of my 40-year-so-far mathematical career), and is probably not going to be supplanted any time soon by your new suggested term "right total". Demonstrate to the mathematical community that your suggestions are actually mathematically useful, though... that's what will make them catch on. – Lee Mosher Mar 06 '18 at 12:32
  • Do you have a specific question you wish to learn the answer to, or do you just want to rant? –  Mar 06 '18 at 13:39

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Why use the word "surjective" which rhymes with injective?

Your suggested alternatives, right-totality and left-totality, also rhyme, and furthermore understanding them depends on the order of writing things, which is not canonical. (There's no reason why you shouldn't have $Y \leftarrow X : f$, and in fact when writing about category theory you see notations like this pop up now and then. Also note that there are languages written right to left in general.) If you speak a language with influence from romance languages you might recognize "sur" as meaning something like "onto", which provides easy intuition. In the end it doesn't matter too much which words we use; it is easy to get used to these concepts.

Also while on the topic why are functions commonly defined as triplets?

This is actually not, typically, how functions are defined in ZFC (which is often the implied background of mathematics). We sometimes use the triple of the definition specifically because we want to associate the codomain to the function; the statement that "a function is surjective" then makes sense without explicating a codomain, and it also has real implications (like the existence of a section).

If one wants to specify that the image of their function is a subset of some other set, why can't they just write $\mathrm{img}(f)\subseteq Y$?

Because it's more work.

Why purposefully restrict the expressive power of ones notation by defining a notion of codomain and requiring composition adhere to rules formulated by it

Because it's more convenient in practice. In practice, it's more convenient to keep the functional notation $f: X \to Y$ even when $f$ does not surject onto $Y$, and when you have a function $g: Y' \to Z$ with $Y' \subseteq \mathrm{img}(f)$ you just write $g \circ f$ for the intended composition, even if it is not strictly speaking correct.

Ultimately this is the answer to all your questions: it doesn't really matter that much. All notation has advantages and disadvantages. Mathematicians have settled on what they have settled on, often with decent reasons. Your suggestions would provide easier notation only in marginal situations, and make things worse slightly more often than that. High-level serious proofs would rarely, if ever, be affected.

Mees de Vries
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  • @Jared, I misunderstood, I thought you meant that "left-totality" should replace "injective". So your comment is actually that "surjective" and "injective" are more like "total" and "functional" respectively, than like each other? In that case I suggest you study some abelian categories. – Mees de Vries Mar 06 '18 at 12:34
  • "why not write $(f,Y)$?" -- because it doesn't matter, and $(f, X, Y)$ reads as a closer proxy to $f: X \to Y$ than $(f, Y)$. We are not programming; there are no resources being spent. Our unimportant definitions can be redundant. – Mees de Vries Mar 06 '18 at 12:36
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    I think you meant to write "right inverse" instead of "left inverse" for surjective functions. – Tobias Kildetoft Mar 06 '18 at 12:39