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I've stumbled upon the definition of exact sequence, particularly on Wikipedia, and noted the use of $\hookrightarrow$ to denote a monomorphism and $\twoheadrightarrow$ for epimorphisms.

I was wondering whether this notation was widely used, or if it is common to define a morphism in the general form and indicate its characteristics explicitly (e.g. "an epimorphism $f \colon X \to Y$").

Also, if epimorphisms and monomorphisms have their own special arrows, are isomorphisms notated by a special symbol as well, maybe a juxtaposition of $\hookrightarrow$ and $\twoheadrightarrow$?

Finally, are there other kinds of morphisms (or more generally, relations) that are usually notated by different arrows depending on the type of morphism, particularly in the context of category theory?


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Note that there are categories where an arrow that is both monic and epic may fail to be an isomorphism. For example, the map $x \to \exp 2 \pi x i$, $[0, 1) \to S^1$ is both monic and epic, but not an isomorphism in the category of topological spaces and continuous functions. – Zhen Lin Feb 2 '11 at 14:15
You're right. I probably should have said "bimorphisms", which I understand is common terminology for monic and epic morphisms. – Abel Feb 2 '11 at 15:56
This isn't obviously categorical (though I think it can be phrased as such) but often immersions of manifolds are notated with the same inclusion hook arrow except that the hook part comes down and crosses the main horizontal shaft. – Aaron Mazel-Gee Feb 2 '11 at 17:41
up vote 12 down vote accepted

Some people use those notations, some don't. Using $\hookrightarrow$ to mean that the map is a mono is not a great idea, in my opinion, and I much prefer $\rightarrowtail$ and use the former only to denote inclusions. Even when using that notation, I would say things like «consider the epimorphism $f:X\twoheadrightarrow Y$».

In some contexts (for example, when dealing with exact categories) one uses $\rightarrowtail$ and $\twoheadrightarrow$ to denote that the map is not only a mono or an epi, but that it has certain special properties (for example, that it is a split mono, a cofibration, or what not)

Denoting isomorphisms by mixing $\twoheadrightarrow$ and $\rightarrowtail$ is something I don't recall seeing.

You will find that there are no rules on notation, and that everyone uses pretty much whatever they like---the only important thing is that when you use what you like you make it clear to the reader.

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I agree with everything Mariano says, especially about $\hookrightarrow$ and $\rightarrowtail$ only being for inclusions. I wanted to add that I usually denote an isomorphism by $\xrightarrow{\sim}$, which kind of evokes $\simeq$ and $\cong$. Another type of "morphism" that often gets a special arrow is a natural transformation, namely $\alpha:F\Rightarrow G$ for a natural transformation between the functors $F,G:\mathcal{A}\rightarrow\mathcal{B}$ (see here).

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It may be worth noting that these arrows can mean different things in different contexts. For example, in model categories, one often uses either $\hookrightarrow$ or $\rightarrowtail$ to indicate that a map is a cofibration, and $\twoheadrightarrow$ to indicate a fibration. If a map is a weak eqivalence, we denote this by placing a $\sim$ over the relevant arrow.

This could potentially lead to ambiguity. For example, in the category of topological spaces, the standard model structure declares fibrations to be Serre fibrations. However, Serre fibrations need not be surjective (that is, an epimorphism in $\mathbf{Top}$). In many cases, one of the pairs cofibration/monomorphism or fibration/epimorphism will coincide, and in all cases, a weak equivalence will become an isomorphism in the homotopy category, so there is generally no confusion.

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