Initial (category theory)

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See initial and final compared (category theory) for this definition and its dual side by side
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Created to move towards nets and generalised convergence/limits.
  • Demote once examples and co are added

Definition

Let [ilmath]\mathcal{C} [/ilmath] be a category and let [ilmath]S\in\text{Ob}(\mathcal{C})[/ilmath] be any object of [ilmath]\mathcal{C} [/ilmath]. Then [ilmath]S[/ilmath] is initial in [ilmath]\mathcal{C} [/ilmath] if[1]:

  • For all [ilmath]A\in\text{Ob}(\mathcal{C})[/ilmath]
    • there exists a unique morphism: [ilmath]\xymatrix{S \ar[r] & A} [/ilmath]

Examples

See also

References

  1. ↑ An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition

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Category Theory
Overview of the concepts of Category Theory
Key objects
[ilmath]\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }[/ilmath]
Typical morphism types
(see diagram on right)
Key objects
[ilmath]\text{Initial} [/ilmath], [ilmath]\text{Final} [/ilmath] (compared)
Primitive constructs
Key constructs
Important examples
Trivial category examples
Common categories
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