Initial object

Object

context ${\bf C}$ … category
definiendum $I:\mathrm{Ob}_{\bf C}$
for all $X:\mathrm{Ob}_{\bf C}$
postulate $\exists_!i.\ i:{\bf C}[I,X]$

Discussion

Alternative definitions

The initial object of ${\bf C}$ can be characterized by the initial morphism $\langle I,\mathrm{id}_\bullet\rangle$ from $\bullet:\mathrm{Ob}_{\bf 1}$ to the (unique) functor $U$ mapping to the discrete category ${\bf 1}$, which only has a single object. Because then $U(g)=f$ is trivially true for all $g:\mathrm{Mor}_{\bf C}$ and $f:\mathrm{Mor}_{\bf 1}$ (the latter is necessarily the identity), the initial morphisms definition reduces to the statement that ${\bf C}[I,X]$ has only one term:

$\forall X:\mathrm{Ob}_{\bf C}.\ \exists_!(g:{\bf C}[I,X]).\ true$

Reference

Wikipedia: Initial and terminal objects

Parents

Context

Categories

Requirements

Category theory

Element of

Initial morphism

Terminal object