What is an example of a Weak Limit that is not Limit I was wondering if there was a standard example of weak limit that is not a limit (in the categorical sense). I have been thinking of the problem, and it seems like weak limits are  limits most of the time, so I was wondering if a simple example exists.
Any examples are greatly enjoyed.
 A: Have to admit I had no idea previously about weak limits, but thanks to nLab weak limits, maybe we will be able to understand the following example: every non-empty (aka inhabited) set is a weak limit (namely, a weak final object) in the category of sets. But only sets with just one element are final objects; that is, true limits.
Why? Because, given any non-empty set $S$, there is always a map $f: A \longrightarrow S$ from any other set $A$ (pick any element $s\in S$ and send all elements in $A$ to $s$, for instance). But the map $f$ is unique only when $S$ has just one element, obviously.
(Ha! They're funny those weak limits.  :-D   )
A: In algebraic topology a cofibered pair $(X,A)$ is often defined by means of a weak pushout:
$\begin{array}{ccc}
A\times\left\{ 0\right\}  & \rightarrow & X\times\left\{ 0\right\} \\
\downarrow &  & \downarrow\\
A\times\mathbb{I} & \rightarrow & X\times\mathbb{I}\end{array}$
Looking at weak limits as objects of a category you end up with the limits as terminal objects. Dually looking at weak colimits as objects you end up with the colimits as initial objects.
