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Given two sets $X$ and $Y$ and two equivalence relations $\sim_X$ and $\sim_Y$ defined on $X$,$Y$ respectively, then a function (a binary relation which is functional and is left-total) is called morphism if two equivalent arguments $x$ and $y$ on $X$ with respect to $\sim_X$ are mapped to equivalent values on $Y$ with respect to $\sim_Y$.

This definition is so abstract and I cannot deeply understand it.

what are some nice examples of morphisms? (I prefer not to discuss about some abstract algebra concepts if it's possible)

  • Basically, there is a well-defined map $X/\sim_X \to Y/\sim_Y$. Many intuitive examples come from algebra. Think about computing modulo $n$ for instance. One of the simplest example is to take $\sim$ as the usual $=$, so a morphism here is a function. – Qi Zhu Jan 29 '20 at 15:34
  • @QiZhu, can we define a morphim for a function like $f(x)=e^x$ and say two arguments in $x,y$ in $D_{f}$ are equivalent under $=$ and are mapped to a same value with respect to $=$ of $R(f)$ ?(means $e^x=e^y$) where $R(f)$ denotes the range of $f(x)$. –  Jan 29 '20 at 15:42
  • Of course. Your condition says $x=y$, then $f(x) = f(y)$. – Qi Zhu Jan 29 '20 at 15:43
  • @QiZhu, do you have another examples? –  Jan 29 '20 at 15:57
  • As I've mentioned, any function $X/\sim_X \to Y/\sim_Y$ induces morphisms in your sense. So e.g. think about maps $\mathbb{Z}/n \mathbb{Z} \to G$. – Qi Zhu Jan 29 '20 at 16:55

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