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I am working on an assignment for my advanced math class. I have the following question:

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The question goes on but I am having trouble understanding what this part of the question is asking.

I understand that I am determining G composed of something. I do not know what the function $1_{x}$ is. Can anyone advise as to what this function is? Is it some kind of identity function?

I have looked through the assigned textbook and have not been able to determine what this symbol means.

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1 Answer 1

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$1_X$ is a common notation for the identity map from $X$ to $X$.

In a very general setting where such a thing would make sense, called a category, it is simply defined to be an arrow $1_X:X\to X$ with the property that for any $F:Z\to X$ and any $G:X\to Y$, we have $1_X\circ F=F$ and $G\circ 1_X=G$. Note that these compositions make sense, because $1_X$ goes from $X$ to $X$, $F$ goes from $Z$ to $X$, and $G$ goes from $X$ to $Y$.

In general, if $K$ and $H$ are functions, the composite function "$K\circ H$" only makes sense if $K$'s domain is the same object as $H$'s codomain; i.e. $K$ "starts at" the exact place that $H$ "ends". Note that the domain and codomain of $1_X$ are both the object $X$.

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Great response. Exactly what I was looking for. Thank you. –  Doug Ramsey Feb 24 '13 at 20:04
    
Glad I could help! –  Zev Chonoles Feb 24 '13 at 20:06

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