---- > [!definition] Definition. ([[isomorphism]]) > Let $\mathsf{C}$ be a [[category]]. A morphism $f \in \text{Hom}_{\mathsf{C}}(A,B)$ is called an **isomorphism** if it has a (two-sided) inverse under composition: there exists $g \in \text{Hom}_{\mathsf{C}}(B,A)$ such that $gf=1_{A} \text{ and } fg=1_{B}.$ > If $f$ has an inverse, it is unique and denoted $f^{-1}$. ^definition > [!justification] Uniqueness requires a brief justification. Suppose $f$ has inverses $g_{1},g_{2}$. We know $g_{1}f=1_{A}=g_{2}f$; now apply $g_{1}$ on (the right of) both sides and use [[associative|associativity]] to obtain $g_{1}=g_{2}$. ^justification > [!basicproperties] > - Each identity $1_{A}$ is an [[isomorphism]] and is its own inverse; >- If $f$ is an [[isomorphism]], then $f ^{-1}$ is too and further $(f ^{-1})^{-1}=f$; >- If $f \in \text{Hom}_{\mathsf{C}}(A,B)$ and $g \in \text{Hom}_{\mathsf{C}}$ are [[isomorphism|isomorphisms]], $gf$ is too with $(gf)^{-1}=f ^{-1} g ^{-1}$. ^properties > [!basicexample] > - The isomorphisms in $\mathsf{Set}$ are precisely the [[bijection|bijections]]. >- It is possible for identities to be the *only* isomorphisms in a [[category]]. Take, for example, [[category#^basic-example-2|the final example here]] where $\text{Obj}(\mathsf{C})=\mathbb{Z}$ and morphisms are induced by the relation $\leq$. The morphisms can 'only go in one direction'. >- It is possible for *every* morphisms in a [[category]] to be an isomorphism. See [[groupoid]]. ^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