I have the category $C$, where:
objects are nonempty sets with one fixed element $Obj = \{(A,a)$, where $A$-nonemty sets, $a\in A\}$,
morphisms are $Mor=\{ f:(A,a)\rightarrow (B,b)$; where $f$ – is a mapping from $A$ to $B$ and $f(a)=b\}$.
How can I prove that for every two elements of $C$ there exists their product but not the coproduct?
I know both definitions but still don't know how to prove it. Is there any standard way of proof?
Best Answer
The product of $(A,a)$ and $(B,b)$ in the category of pointed sets is $(A \times B,<a,b>)$ with the same projections and mediating morphism as in $\mathcal {Set}$. You can easily prove this, basically by noticing that projections and mediating morphism respect the basepoints $a,b$ and $<a,b>$.
Regarding the coproduct:
The coproduct od $(A,a)$ and $(B,b)$ in the category of pointed sets is $(A \bigsqcup B/\sim ,*)$ where $\sim$ is the equivalwence relation identifying $<a,0>$ with $<b,1>$ and calling them "*". The injections and mediating morphism are modified from $\mathcal {Set}$ accordingly. For ex. the injection $i_A$ takes $a$ to * and any other $p$ to $<p,0>$.