I'm trying to understand the difference of span and product. So far every example of span I know of is also a product. E.g. Wikipedia has an example:
- Any object yields the trivial span $A = A = A$; formally, the diagram $A ← A → A$, where the maps are the identity.
The example is also a product, by virtue of having an object $W=A$ with morphisms (the identity) to all three objects.
The only way for a span $X$ to not be a product seem to have an object $W$ with arrows to all objects of the diagram except of $X$. E.g.:
$$ \begin{array}{} \begin{align} & \; W \\ \swarrow & \quad \searrow \\ X₁ ← & X → X₂ \end{align} \end{array} $$
To me the diagram seem to be useless outside of category theory, because usually if the object $W$ has enough info to get to $X₁$ and $X₂$, it's also enough to get to $X$. But mathematicians wouldn't create a whole separate concept out of something extra rare, so there have to exist examples of spans that are not products. What are they?
Span is any diagram of shape $• ← • → •$ while product is a very special span – it is such span that any other span with the same end-objects uniquely factorizes through it. For example, the trivial span is almost never a product: $A × A$ is often a different object than $A$.