Clarify the usage of symbols for “-morphism”.

morphismnotationterminology

I'm confused about the notation for "-morphism"! Even one author uses them differently in different books.

For example, Some books use "$\simeq$" to represent both "path homotopic" and "homeomorphic" but some use $\sim$ to represent "homotopic" and "equivalence relation".

And for the concept "isomorphism" some authors prefer using $\cong$ while others tend to use $\approx$.

$\cong$ sometimes also refers to homeomorphisms…… Besides, it seems that there is no symbols for "monomorphism" and "epimorphism"…

My questions:

  1. Is there a standard of how to use these notations?

  2. What should I notice when I want to use these symbols, should I clarify the meaning or just use them?

  3. And also, does there exist symbols for "monomorphism" and "epimorphism" (I guess no)? It will be better if someone knows sources that clarify this issue.

Thanks for your help

Best Answer

$1.$ Is there a standard of how to use these notations?

I do not think that there is an overall standard convention. However, most authors tend to clarify which symbol there are going to use for a specific notion.
For example, I have never seen someone using $\approx$ for an isomorphism but both $\cong$ and $\simeq$. On the other hand I have also seen $\simeq$ for equivalence of categories and in the same text $\cong$ for isomorphisms of objects; clearly distinct notions cleary separated by symbols. When it comes down to choosing a symbol to use its ultimately a matter of taste.

$2.$ What should I notice when I want to use these symbols, should I clarify the meaning or just use them?

Be aware of commonly used symbols, chose the one you would like to use and clarify what you mean when you use a particular symbol. This is especially important when two concepts appear which are sometimes denoted by the very same symbol (as you gave some examples for this in your question).
Sadly, as noted by Shaun in the comments too, authors tend to not be that strict with themselves. Anyway, I have seen in at least on two occasions authors declaring their usage of symbols explictly: first, in E. Riehl's "Category Theory in Context" (right away in the preface) and, second, in P. Aluffi's "Algebra: Chapter $0$" (when introducing injectivity and surjectivity). So there are positive examples.

$3.$ And also, does there exist symbols for "monomorphism" and "epimorphism" (I guess no)? It will be better if someone knows sources that clarify this issue.

There are not symbols like $\cong$ as monos and epis are more of directed notions. Anyway, when the information of direction is encoded within an arrow commonly $\rightarrowtail$ \rightarrowtail (or $\hookrightarrow$ \hookrightarrow, although more for embeddings whenever this makes sense) is used for monos while $\twoheadrightarrow$ \twoheadrightarrow is used for epis (and in a similiar manner $A\overset{\sim}\rightarrow B$ for isomorphisms). Obviously $A\hookrightarrow B$ is quite different from $B\hookrightarrow A$ but, on the other hand, $A\cong B$ and $B\cong A$ refer to the same relation.

Related Question