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2022-06-17
 

Monomorphism

   
  In Category Theory, a monomorphism is a morphism $f:X\to Y$ for which  the following condition holds:  For any two morphisms  $g_1, g_2: Z\to X$  the identity $f\circ g_1 = g\circ g_2$ induces $g_1 = g_2$.  
   

As an Injective Function

 Monomorphisms are the categorial way to describe an injective  function, in that an injective function on sets is always a monomorphism.    

Let $f:X\to Y$ be an injective function for the sets $X$ and $Y$. This  in particular means that there is a function $h:f(X)\to X$ such that  $h(f(x)) = x$ for all $x\in X$.    Now lets have another set, $Z$, and two functions $g_1, g_2: Z\to X$  such that $f\circ g_1 = f\circ g_2$ which means in particular that  $$f(g_1(z)) = f(g_2(z))$$ for all $z\in Z$. Therefore we get for each $z$  

     \begin{align*}   g_1(z) &= h(f(g_1(z)))\\   &= h(f(g_2(z))) \\   &= g_2(z)\,.  \end{align*}    This means that $g_1=g_2$, and hence that $f$ is a monomorphism.    

To Note

 The categorial way to define a monomorphism cannot talk about the  elements of the category's objects, because they have none, but it  nevertheless manages to define an injective function.