---- > [!definition] Definition. ([[monomorphism]]) > Let $\mathsf{C}$ be a [[category]]. A morphism $f \in \text{Hom}_{\mathsf{C}}(A,B)$ is called a **monomorphism** or **left-cancellable** if for all objects $Z$ of $\mathsf{C}$ and all morphisms $\alpha',\alpha'' \in \text{Hom}_{\mathsf{C}}(Z,A)$, $f \circ \alpha' = f \circ \alpha'' \implies \alpha'=\alpha''.$ > > ^definition > [!basicexample] > In $\mathsf{Set}$, the definition looks like this: A function $f: A \to B$ is called a **monomorphism** if for all sets $Z$ and all functions $\alpha', \alpha'': Z \to A$, $f \circ \alpha'=f \circ \alpha'' \implies \alpha' = \alpha''.$ > Recall in this special case the [[characterization of injectivity and surjectivity in Set|characterization]] in terms of [[surjection|injectivity]]. > > >[!basicnonexample] Warning. > > In general, though, [[monomorphism]] + [[epimorphism]] does not imply [[isomorphism]]. For example, in [[category#^basic-example-2|this example]] with $\mathbb{Z}$ and $\leq$, every morphism is both a [[monomorphism]] and an [[epimorphism]]... but the only [[isomorphism|isomorphisms]] are the identities! > ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02