Definition
Isomorphism
Let $\mathcal{C}$ be a category (Definition~Category).
A morphism
$$
f : X \to Y
$$
is called an isomorphism if there exists a morphism
$$
g : Y \to X
$$
such that
$$
g \circ f = \operatorname{id}_X
\qquad
f \circ g = \operatorname{id}_Y.
$$
The morphism $g$ is called the inverse of $f$ and is denoted
$$
f^{-1}.
$$
If such a morphism exists we say that $X$ and $Y$ are isomorphic and write
$$
X \cong Y.
$$
This notion uses the identity morphisms and composition introduced in
Definition~Category.
Depends on
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nf2c6bdf5["Category"]
n136199b7["Isomorphism"]:::current
n45ea5662["Uniqueness of Inverses"]
n30fdee11["Corollary"]
n384e2b5d["Functors Preserve Isomorphisms"]
nf2c6bdf5 --> n136199b7
n136199b7 --> n45ea5662
n136199b7 --> n30fdee11
n136199b7 --> n384e2b5d
click nf2c6bdf5 "../objects/f2c6bdf5.html" "_self"
click n45ea5662 "../objects/45ea5662.html" "_self"
click n30fdee11 "../objects/30fdee11.html" "_self"
click n384e2b5d "../objects/384e2b5d.html" "_self"