Wedge (category theory)

(Redirected from Cone and cocone compared)
Note: the page Cone and cocone compared redirects here as this is basically a discussion page on how the two differ (which exists because of the page Product and coproduct compared)

Definition

For a pair of objects [ilmath]A[/ilmath] and [ilmath]B[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] we define[1]:

Wedge to the pair [ilmath]A[/ilmath], [ilmath]B[/ilmath] Wedge from the pair [ilmath]A[/ilmath], [ilmath]B[/ilmath]
is an object [ilmath]X[/ilmath] in [ilmath]\mathscr{C} [/ilmath] together with a pair of arrows (also from [ilmath]\mathscr{C} [/ilmath]) as follows:
[ilmath]\xymatrix{ & A \\ X \ar[ur] \ar[dr] & \\ & B}[/ilmath] [ilmath]\xymatrix{A \ar[dr] & \\ & X \\ B \ar[ur] & }[/ilmath]
(AKA: cone) (AKA: cocone)