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[parent] monomorphisms of category of sets (Theorem)
Theorem 1   Every monomorphism in the category of sets is an injection.
Proof. Assume $ f \colon A \to B$ is a monomorphism. Then, by definition of monomorphism, given any two maps $ g,h \colon C \to A$, if $ f \circ g = f \circ h$, then $ g = h$. Suppose $ x$ and $ y$ are two elements of $ A$ such that $ f(x) = f(y)$. Let $ C$ be a set with one element, let $ g$ be the map which sends this one element to $ x$ and let $ h$ be the map which sends this one element to $ y$. Because $ f(x) = f(y)$, we have $ f \circ g = f \circ h$. Since $ f$ is a monomorphism, $ g = h$, so $ x = y$. This implies that $ f$ is injective. $ \qedsymbol$
Theorem 2   Every injection is a split monomorphism.
Proof. Assume $ f \colon A \to B$ is injection. If $ A$ is empty, the result is trivial, so we assume that $ A$ is not empty; let $ z$ be an element of $ A$. Set
$\displaystyle g = \{ (f(x),x) \mid x \in A \} \cup \{(x,z) \mid x \in B \land (\forall y \in A) x \neq f(y)\} $
We claim that $ g$ is a function from $ B$ into $ A$. Suppose that $ x$ is an element of $ B$. If $ x \neq f(y)$ for any $ y \in A$, then we have exactly one element of $ g$ with $ x$ as the first element, namely $ (x,z)$. If $ x = f(y)$ for some $ y \in A$, then we the pair $ (x,y)$ with $ x$ as first element; were there another pair with $ x$ as first element, then we would have $ (f(x_1),x_1) = (f(x_2),x_2)$ but, as $ f$ is an injection, $ f(x_1) = f(x_2)$ would imply $ x_1 = x_2$, so this would not be a distinct pair. Hence $ g$ is a function. Furthermore, by construction $ g \circ f (x) = x$ for all $ x \in A$, so $ f$ is a split monomorphism. $ \qedsymbol$

Note that the second theorem is stronger than a simple converse to the first theorem -- it states that an injection is not just a monomorphism, but that it is actually a split monomorphism. In particular, this means that, in the category of sets, all monomorphisms are actually split monomorphisms.



"monomorphisms of category of sets" is owned by rspuzio. [ full author list (2) ]
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Cross-references: converse, simple, split monomorphism, implies, maps, injection, category of sets, monomorphism
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This is version 4 of monomorphisms of category of sets, born on 2007-02-18, modified 2008-04-30.
Object id is 8927, canonical name is MonomorphismsOfCategoryOfSets.
Accessed 503 times total.

Classification:
AMS MSC: 18-00 (Category theory; homological algebra :: General reference works )
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