Difference between revisions of "Product (category theory)"

Latest revision as of 23:32, 29 February 2016

Note: see product and coproduct compared for a definition written in parallel with a coproduct definition. This demonstrates how close the concepts are.

Definition

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

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

References

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