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I am really bummed out to find that the term "strict monomorphism" is already used to mean something else.

Can anybody console me with the knowledge that there is another name I can use for a monomorphism that is not an isomorphism?

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I have observed that "proper monomorphism" is sometimes used to mean "a monomorphism that is not an isomorphism".

For example, if $A$ is a subset of $B$ and if $A\neq B$, then we write that "$A$ is a proper subset of $B$". We know in the category of sets, for example, that if there is an monomorphism from $A$ into $B$, then $A$ can be identified with a subset $C$ of $B$. In this case, $C$ is a proper subset of $B$ if and only if $f$ is a "monomorphism that is not an isomorphism".

However, I could be incorrect as I do not have a definite source for this terminology.

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    MacLane uses the term _proper monic_ in [Categories for the Working Mathematician](http://books.google.com/books?id=MXboNPdTv7QC&pg=PA129&dq=%22proper+monic%22&hl=en&ei=QgdzTu_yBLD14QSNlrXEDQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDQQ6AEwAQ#v=onepage&q=%22proper%20monic%22&f=false) but I did not find a definition of this term in his book.2011-09-16
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    @Martin: Nice catch. That's canonical enough for me2011-09-16
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You could go with "non-invertible monomorphism", which is really just a restatement of "monomorphism that isn't an isomorphism". But as far as I know, there is no "special" or "reserved" name for such maps.

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    Let it be hereby known that such maps will thusforth and forevermore be known officially as "rigid monomorphisms", since they were heretofore unnamed and all that jazz.2011-09-16
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    By the way, I was using all that archaic-sounding razzamatazz so it sounds more official. Make sure to use this word that I coined right in front of you right here right now.2011-09-16
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    Of course I will. Heaven forbid a mathematical word that conveys any information about what it means!2011-09-16