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I have a naive question on category theory. It is about inverse and inductive limits.

We have the notion of inverse system in a category $C$ which can be seen as a contravariant functor $F:I \to C$, where $I$ is a directed poset. The inverse limit of F is then just the limit of $F$ in the categorical sense. It is a universal arrow $(r, u)$ from the diagonal functor $\Delta:I \to C$ to $F$ (e.g. like a product for instance).

My question is the following: Would it make sense to look at the colimit of an inverse system (like a coproduct)? Does it exists in general and would it be an interesting object?

I have also the dual question for the limit of an inductive system.

Thanks in advance!

Regards,
Moustik

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The interesting part of a directed set is (co)limits in the direction of the set; limits of functors $I^\text{op} \to \mathcal{C}$ or colimits of functors $I \to \mathcal{C}$ are the notions that have special importance.

You can consider limits of functors $I \to \mathcal{C}$ or colimits of functors $I^\text{op} \to \mathcal{C}$, but I don't think there's much special about them. Off hand, the only feature I can think of is that such (co)limits are connected.

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    Ok thanks! I have to meditate examples (e.g. profinite Galois groups, or p-adic integers ring) to feel that this is the limit we have to take and not the co-limit to get the object of interest. – 2017-01-31
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    @Moustik: $\mathbb{N}$-indexed limits are probably the most dramatic examples! The colimit of a functor $\mathbb{N} \to \mathcal{C}$ is interesting, but the limit of such a functor is simply its first term! Similarly, the colimit of a functor $\mathbb{N}^\text{op} \to \mathcal{C}$ is the first term. – 2017-01-31
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    Indeed! Very enlightening thanks Hurkyl! – 2017-02-01