===== Initial object =====
==== Object ====
| @#55CCEE: context     | @#55CCEE: ${\bf C}$ ... category |
| @#FFBB00: definiendum | @#FFBB00: $I:\mathrm{Ob}_{\bf C}$ |
| @#FFFDDD: for all     | @#FFFDDD: $X:\mathrm{Ob}_{\bf C}$ |
| @#55EE55: postulate   | @#55EE55: $\exists_!i.\ i:{\bf C}[I,X]$ |

==== 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: [[http://en.wikipedia.org/wiki/Initial_and_terminal_objects|Initial and terminal objects]]
==== Parents ====
=== Context ===
[[Categories]]
=== Requirements ===
[[Category theory]]
=== Element of ===
[[Initial morphism]]
=== Related ===
[[Terminal object]]