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37 lines
1.3 KiB
37 lines
1.3 KiB
<div class="date">20220617</div>


<h1 class="title">Monomorphism</h1>




<div class='definition'>


In Category Theory, a <b>monomorphism</b> 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 $g_1 = g_2$.


</div>




<h2 id="injective">As an Injective Function</h2>


Monomorphisms are the categorial way to describe an injective


function, <b>in that an injective function on sets is always a monomorphism.</b>




<p>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 $z$


</p>






$$\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.




<h2 id="note">To Note</h2>


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.


