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Excuse me if this sounds silly. Does anybody know why injections and surjections are sometimes denoted symbolically as $f:V\hookrightarrow W$ and $g:V\twoheadrightarrow W$? How do the arrows $\hookrightarrow$ and $\twoheadrightarrow$ convey the meanings of injections and surjections?

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2 Answers

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I like to think $\hookrightarrow$ is reminiscent of the $\subset$ symbol, which is apt since if you have $i\colon A \hookrightarrow B$, then $A \simeq \operatorname{Im}i \subset B$. An injective map is kinda like picking out a subset. I think that Quillen used ↣ for injections (monomorphisms really) instead. For surjective maps $\twoheadrightarrow$ looks like two things being stacked, so like a quotient.

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The curved arrow makes me think of an inclusion, which is appropriate to figure an injection ; the double arrow seems to mean that the mapping really reaches the arrival domain, which is appropriate to figure a surjection.

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