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# System prepended metadata

title: Monomorphism

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# Monomorphism

## Definition
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 Injective Function
**Monomorphisms are the categorial way to describe an injective function.** It is a bit more general, but for functions on sets, an injective function 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


\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 a category object, because it has none, but it nevertheless manages to define a surjective function.

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