Example of category in which direct limit does not exist

Question

I am trying to understand concept of direct limit.I am looking for

An example of a category and directed sequence whose direct limit does not exist.

I am unable to find one please help.

Answer 1

Hint: In a posetal category, a direct limit of a chain $x_0 \le x_1 \le x_2 \le \cdots$ (considered as a directed diagram $x_0 \to x_1 \to x_2 \to \cdots$ of morphisms in the category) is exactly a supremum of the chain.

Answer 2

An initial object in a category $\mathcal{C}$ is a direct limit (to be more precise, it is the direct limit of the unique functor $\emptyset\rightarrow\mathcal{C}$, where $\emptyset$ is the empty category), so to find an example for your question it suffices to find a category without initial object.

The first one that comes to my mind is $\mathbf{Fld}$, the category of fields. To see that this category does not have a initial object, just ask yourself what should be the characteristic of an eventual "initial field". In general, I think that it is nice and useful to try to figure out what kinds of limits and colimits are admitted in $\mathbf{Fld}$.