Cone (category theory)

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Note: the definitions for cone and cocone are very similar and contrast each other well, see the page cone and cocone compared for the definitions compared side by side


Given two objects [ilmath]A[/ilmath], [ilmath]B[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath], a cone[1] is:

  • Another object, [ilmath]X[/ilmath] from [ilmath]\mathscr{C} [/ilmath], coupled with two arrows also from [ilmath]\mathscr{C} [/ilmath] as follows:
[ilmath]\xymatrix{ & A\\ X \ar[ur] \ar[dr] & \\ & B}[/ilmath]
Diagram of a cone

This is an instance of a wedge (a wedge to [ilmath]A[/ilmath] and [ilmath]B[/ilmath])

See also

  • Cocone - another kind of wedge but with arrows from [ilmath]A[/ilmath] to [ilmath]X[/ilmath] and from [ilmath]B[/ilmath] to [ilmath]X[/ilmath] rather than outwards from [ilmath]X[/ilmath]
  • Product and coproduct compared - parallel definitions of a product and coproduct, which are special wedges
    • A product is a special instance of a cone


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