Product (category theory)

From Maths
(Redirected from Categorical product)
Jump to: navigation, search
(Unknown grade)
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
This needs fleshing out with things like notation, compared to coproduct and such
Note: see product and coproduct compared for a definition written in parallel with a coproduct definition. This demonstrates how close the concepts are.


Given a pair of objects [ilmath]A[/ilmath] and [ilmath]B[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] a product (of [ilmath]A[/ilmath] and [ilmath]B[/ilmath]) is a[1]:

  • Wedge [ilmath]\xymatrix{ A & S \ar[l]_{p_A} \ar[r]^{p_B} & B}[/ilmath] (in [ilmath]\mathscr{C} [/ilmath]) such that:
    • for any other wedge [ilmath]\xymatrix{ A & X \ar[l]_{f_A} \ar[r]^{f_B} & B}[/ilmath] in [ilmath]\mathscr{C} [/ilmath]
      • there exists a unique arrow [ilmath]X\mathop{\longrightarrow}^mS[/ilmath] (called the mediating arrow) such that the following diagram commutes:
[ilmath]\xymatrix{ & A \\ X \ar[ur]^{f_A} \ar[dr]_{f_B} \ar[r]^m & S \ar[u]_{p_A} \ar[d]^{p_B} \\ & B}[/ilmath]
Diagram of the product of [ilmath]A[/ilmath] and [ilmath]B[/ilmath]


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