Isomorphism (category theory)

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Definition

Given a pair of arrows, [ilmath]A\mathop{\longrightarrow}^fB[/ilmath] and [ilmath]B\mathop{\longrightarrow}^g A[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath], we say that [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are isomorphisms if:

• [ilmath]g\circ f=\text{Id}_A[/ilmath] and [ilmath]f\circ g=\text{Id}_B[/ilmath]

We may also call [ilmath]f[/ilmath] and [ilmath]g[/ilmath] an inverse pair of isomorphisms. For clarity I say again: both [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are themselves isomorphisms

See also

• Types of category arrows
• Bimorphism