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We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a point $P\in X$ by $$ \lim_{\rightarrow_{p \in U \text{ open}}} \mathcal{F}(U). $$

It seems that we get something small. We can take inverse limit of the inverse system $\{k[x]/(x)^n, n=0, 1, 2, \ldots\}$. Then we obtain $k[[x]]$ which is larger than $k[x]$. It seems that by taking inverse limit we add something to $k[x]$. It seems that taking inverse limit is to add something and taking direct limit is to get something small. Is this correct? If a system is both direct system and inverse system, how do we know that we need to take direct limit or inverse limit? Thank you very much.

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    $\begingroup$ How do we know when to add and when to multiply? There is clearly no universal answer: it depends on what we are trying to do. $\endgroup$ Nov 26 '13 at 17:27
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    $\begingroup$ Note also that e.g. for the sheaf $\mathcal{F}$ of holomorphic functions on $\mathbb{C}$, the stalk at a point is larger than any of the $\mathcal{F}(U)$'s. One often thinks of direct limits as being a generalized kind of union (when the transition maps are injective). So I'm not sure why you want to think of the direct limit as "small", although there are also examples of a direct limit of nonzero groups being zero. $\endgroup$ Nov 26 '13 at 17:32
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This is not a full answer, but some thoughts about direct and inverse limits.

I always think of the direct limit of a system $(A_i,\varphi_i)_{i \in I}$ as a union $A = \bigcup A_i$ but with compatible transition maps, i.e. an element is in this union if and only if it is compatible by those maps and the canonical injections $A_i \longrightarrow A$. In contrast to inverse limit, where I think of them not as the union but as the cartesion product (or to be exact: a subset of the cartesian product). There you don't have canonical injections, but canonical projections $A \longrightarrow A_i$ and the transition maps of the inverse system have to be compatible with this canonical projections.

In conclusion, for me a direct limit is a really tiny subset of the union of all spaces and the inverse limit is a really big subset of the cartesian product, roughly speaking.

I hope that intuition helped you!

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    $\begingroup$ I think this is a quite helpful answer. $\endgroup$ Nov 26 '13 at 17:47
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    $\begingroup$ I think a correction is warranted: The direct limit is not a subset of the union. It is a quotient of the disjoint union. $\endgroup$
    – Uday Reddy
    Nov 30 '13 at 0:29
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It is a bit strange to compare direct and inverse limits. In spite of the (very) misleading terminology, they are not variations of the same concept.

Inverse limit is a special case of the notion of limit, which searches for a universal object factorizing morphisms to a diagram. Another way to put it : limits solve problems about the representability of some contravariant functors.

Direct limit is a special case of the notion of colimit, which searches for a universal object factorizing morphisms from a diagram. Another way to put it : colimits solve problems about the representability of some covariant functors.

So, in my understanding, trying to compare direct and inverse limits in a category $\mathcal C$ is about comparing objects (and morphisms) of $\mathcal C$ and $\mathcal C^{\mathrm{op}}$, which are (a priori) very different categories...

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