Definition:Product of Morphisms
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Definition
Let $\mathbf C$ be a metacategory.
Let $A, A'$ and $B, B'$ be pairs of objects admitting binary products:
- $\begin{xy}\xymatrix@R-1em@C+1em@L+3px{
A
&
A \times A' \ar[l]_*+{p_1} \ar[r]^*+{p_2}
&
A'
\\
B
&
B \times B' \ar[l]_*+{q_1} \ar[r]^*+{q_2}
&
B'
}\end{xy}$
Let $f: A \to B$ and $f': A' \to B'$ be morphisms.
The product morphism of $f$ and $f'$, denoted $f \times f'$, is the unique morphism making the following diagram commute:
- $\begin{xy}\xymatrix@+1em@L+3px{
A \ar[d]_*+{f}
&
A \times A' \ar[l]_*+{p_1} \ar[r]^*+{p_2} \ar@{-->}[d]^*+{\hskip{1.3em} f \times f'}
&
A' \ar[d]^*+{f'}
\\
B
&
B \times B' \ar[l]^*+{q_1} \ar[r]_*+{q_2}
&
B'
}\end{xy}$
Thus we see that $f \times f'$ is the morphism $\gen {f p_1, f' p_2}$.
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Also see
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 2.6$