---- > [!definition] Definition. ([[epimorphism]]) > Let $\mathsf{C}$ be a [[category]]. A morphism $f \in \text{Hom}_{\mathsf{C}}(A,B)$ is called an **epimorphism** or **right-cancellable** if for all objects $Z$ and all $\beta',\beta'' \in \text{Hom}(B,Z)$ we have $\beta' \circ f= \beta'' \circ f \implies \beta' = \beta''.$ > > That is, an epimorphism is a monomorphism in the [[opposite category]] $\mathsf{C}^{\text{op}}$. ^definition > [!basicexample] > In $\mathsf{Set}$, the definition looks like this: A function $f:A \to B$ is called an **epimorphism** if for all sets $Z$ and all $\beta',\beta'':B \to Z$ we have $\beta' \circ f = \beta'' \circ f \implies \beta' = \beta''.$ > Recall in this special case the [[characterization of injectivity and surjectivity in Set|characterization]] in terms of [[surjection|surjectivity]]. > > >[!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