<-- Go Back Last Updated: 11/01/2025

Initial Object**

An object \(\iota\) of a category \(\mathcal{C}\) is called 'initial' if, for any other object \(x\) of \(\mathcal{C}\), there is a unique arrow from \(\iota\) to \(x\), i.e. \(\forall x\in\mathcal{C}_0:\exists! \iota_x\in\mathcal{C}_1:\mathrm{dom}(\iota_x)=\iota\land\mathrm{cod}(\iota_x)=x\) or (in terms of hom-sets) \(\forall x\in\mathcal{C}_0:\mathrm{Hom}_\mathcal{C}(\iota,x)=\{\iota_x\}\). If a category has initial objects they are all isomorphic to each other.