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My basic intuition for limits/colimits was "limits suck up, colimits suck down". Now, having seen colimits used in presheaf categories, algebraic geometry, and topology, I have much clearer intuition of colimits as being "glueing things together". What, intuitively, do limits mean?

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See this answer for some comments on inverse limits. –  Arturo Magidin Mar 24 '12 at 21:00
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There is an identical question on mathoverflow: mathoverflow.net/questions/23268/geometric-intuition-for-limits I have also answered there. –  Martin Brandenburg Mar 25 '12 at 8:59
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up vote 6 down vote accepted

Limits can also be used to glue things together, but in a somewhat different way. For example, consider the inverse system with objects $\mathbb{Z} / p^n \mathbb{Z}$ (for $n \in \lbrace 1, 2, 3, \ldots \rbrace$) and morphisms the canonical reduction maps. The inverse limit $\varprojlim_n \mathbb{Z} / p^n \mathbb{Z}$ is the additive group of $p$-adic integers, $\mathbb{Z}_p$. Unlike any of the objects that went into making it, $\mathbb{Z}_p$ is infinite, and I'd argue that $\mathbb{Z}_p$ is something made by gluing together the groups $\mathbb{Z} / p^n \mathbb{Z}$.

Now, whereas a colimit $\varinjlim_j X_j$ is equipped with morphisms to the colimit (i.e. $X_j \to \varinjlim_j X_j$), limits are equipped with morphisms from the limit (i.e. $\varprojlim X_j \to X_j$). Perhaps this is a helpful picture: with colimits, you glue parts together to make a whole, whereas with limits, you reconstruct an object from its shadows.

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"with colimits, you glue parts together to make a whole, whereas with limits, you reconstruct an object from its shadows." This made it click for me, thanks! –  Yuri Sulyma Mar 27 '12 at 0:20
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Think for a moment to products in $\mathbf{Set}$ (or other concrete category): a product of the family $\langle A_i \rangle_{i \in I}$ is given by the set $\prod_{i \in I} A_i$ is the set of functions of type $f \colon I \to \bigcup_{i \in I} A_i$ such that for each $i \in I$ we have that $f(i) \in A_i$: these are $I$-ple of elements of the $A_i$. Clearly every function $f \in \prod_{i \in I} A_i$ can be intuitively be thought as the result of a gluing together elements of the $A_i$ via concatenations (they are string of elements of the $A_i$).

Because limits are all restrictions (i.e. equalizers) of products [by a theorem] clearly all the limits in $\mathbf{Set}$ (but also many other $\mathbf{Set}$-based categories) are obtained as objects whose elements are gluing of elements of other objects in the category itself.

From another perspective limits are gluing things together but forward, as colimits are gluing things backward. Remember that both limits and colimits serve to represent family of morphisms indexed by objects of a category via just one morphism in the category, meaning that with limits we represents special families of kind $\langle \sigma_i \colon A \to B_i \rangle_{i \in I}$ via a (necessarily unique) morphism $\sigma \colon A \to \varprojlim_{i \in I} B_i$, while with colimits we represents special families of type $\langle \tau_i \colon A_i \to B\rangle_{i \in I}$ via a (necessarily unique) morphism $\tau \colon \varinjlim_{i \in I} A_i \to B$. [Here by special families I mean families of morphisms that make commute certain diagrams]. So both limits and colimits are just a way to talk about different special families using just the data of the category (i.e. objects and morphisms, family of morphisms aren't properly data of the category).

So both limits and colimits are objects the serve to gluing families of arrows in one arrow.

Hope this answers, at least partially, your question.

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