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initial_object [2014/09/28 19:25] nikolaj |
initial_object [2014/12/04 22:34] nikolaj |
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| ===== Initial object ===== | ===== Initial object ===== | ||
| - | ==== Collection ==== | + | ==== Object ==== |
| | @#55CCEE: context | @#55CCEE: ${\bf C}$ ... category | | | @#55CCEE: context | @#55CCEE: ${\bf C}$ ... category | | ||
| | @#FFBB00: definiendum | @#FFBB00: $I:\mathrm{Ob}_{\bf C}$ | | | @#FFBB00: definiendum | @#FFBB00: $I:\mathrm{Ob}_{\bf C}$ | | ||
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| ==== Discussion ==== | ==== Discussion ==== | ||
| === Alternative definitions === | === 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. Since then, by definition, $U(g)=f$ is true for all $g:\mathrm{Mor}_{\bf C}$ and $f:\mathrm{Mor}_{\bf 1}$, the initial morphisms definition reduces to the statement that ${\bf C}[I,X]$ has only one term: | + | 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: |
| $\forall X:\mathrm{Ob}_{\bf C}.\ \exists_!(g:{\bf C}[I,X]).\ true$ | $\forall X:\mathrm{Ob}_{\bf C}.\ \exists_!(g:{\bf C}[I,X]).\ true$ | ||
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| ==== Parents ==== | ==== Parents ==== | ||
| === Context === | === Context === | ||
| - | [[Category]] | + | [[Categories]] |
| === Requirements === | === Requirements === | ||
| [[Category theory]] | [[Category theory]] | ||
