Is there a standard arrow or symbol to represent homomorphisms $R\to S$ between two structures? I would prefer not to write out the word homomorphism every time I need to say something is a homomorphism!

I believe in category theory the arrows are assumed to be homomorphisms?...but not in ring theory, group theory etc. So really looking for a symbol in the latter contexts.


Edit based on current answers: (To clarify im not looking for non-standard abbreviations such as "hom" or "homo" and i'm not looking for symbols/arrows for isomorphisms, surjections, injections etc.)

  • $\begingroup$ Use the abbreviation hom and do "By "hom" we shall mean the word homomorphism.. $\endgroup$ – Shine On You Crazy Diamond Dec 6 '17 at 20:52
  • $\begingroup$ If the homomorphism is surjective then $\twoheadrightarrow$ is often used, and if the homomorphism is injective then the $\hookrightarrow$ is often used. $\endgroup$ – ÍgjøgnumMeg Dec 6 '17 at 20:52
  • $\begingroup$ If you do $\xrightarrow{\sim} $ That means isomorphism. So it is presumed that the reader knows what the homs are. Alternatively inline: $\phi : A \simeq B$. $\endgroup$ – Shine On You Crazy Diamond Dec 6 '17 at 20:53
  • $\begingroup$ It's more of an editor problem. Good writing implies you make complete sentences. Personally, I mage shorcuts like hmm which automatically expand to homomorphismand things like that. $\endgroup$ – Bernard Dec 6 '17 at 20:57

When the context is clear about the structures of $A$ and $B$, and they are of the same type, then the standard meaning of an arrow $A\to B\ $ is $\ $that it's a homomorphism.

  • $\begingroup$ is this in a category theory context? as most books i have seen in group/ring theory contexts seem to use the word 'homomorphism' each time they talk about one and use $\to$ for an arbitrary map. $\endgroup$ – vkan Dec 6 '17 at 21:52
  • $\begingroup$ Well, yes, but in a sentence like 'Let $A,B$ be groups and consider $f:A\to B$', $f$ is implicitly meant to be a homomorphism. If you want to emphasize it, you can make a note about this notation at the beginning of your writing. $\endgroup$ – Berci Dec 6 '17 at 23:25
  • $\begingroup$ I think it would make sense for the standard arrow to represent a homomorphism as i can't think of a context when you would want a map between two structures of the same type to not be a homomorphism. However at the same time perhaps there are contexts when you do not want the standard arrow to represent a homomorphism....for example i wonder if there is a context in which one would consider a map between rings that preserves the abelian group structure but not the multiplication. $\endgroup$ – vkan Dec 9 '17 at 11:13

Usually, you can say "arrow" or "map" when you have defined what the arrows in your category are. Say, for example, you are working on the category of groups. Then saying "...let $G\to H$ be an arrow/map and consider..." is reasonable, as long as it doesn't lead to doubt. In general, be there two (or more) different different categories at hand, authors are consistent with the letters used for various objects of the different categories. For example, if you are considering algebras and coalgebras, it is reasonable to stick to $A,A'$ and variants for algebras, and to $C,C'$ and variants for coalgebras. A good way to learn how to do this is to read some papers where this happens!

To be concrete, take an editorial look (nevermind the mathematical content) of the paper Minimal models in Homotopy Theory by Baues and Lemaire. They consistently use $V$ for vector spaces, $A$ or $B$ for algebras, $C$ for coalgebras, $M$ for modules, $L$ for Lie algebras. Thus, when an arrow between two letters appear, you will automatically understand what kind of arrow this is, or if need be, the authors will clarify this.

  • $\begingroup$ im not sure this answers my question but thanks for your answer anyways. $\endgroup$ – vkan Dec 6 '17 at 21:10

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