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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.

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up vote 2 down vote accepted

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.

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@chanler. You accepted, but to my regrets I made a mistake in saying that weak products in posets are lower bounds (also the dual of it -concerning coproducts - is wrong). I was not really comparing limit and weak limit, but cones and limiting cones there. In a poset arrows in a homset are automatically unique so there cannot be any distinction between limit and weak limit there. In an edit I took away the mistake. If you withdraw your acceptance then I have full understanding for that. My apologies. – drhab Jan 14 '14 at 11:35

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 )

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