In which Category Inverse Limit does not exist?

Question

We know that in category of $\mathbb{Set}$ the inverse limit is the direct product. But I am looking for specific category in which inverse limit does not exist. Any comments would be highly appreciated.

Answer 1

Remember that every partial order can be viewed as a category (with a unique morphism $A\rightarrow B$ whenever $A\le B$. Consider $(\mathbb{N}, <)$ as a category $\mathcal{N}$, and think about $\mathcal{N}^{op}$; the whole category forms an inverse system, but it has no inverse limit since there is no largest natural number.

(If you don't require inverse systems to be directed, things are simpler: consider the two-object category with no non-identity morphisms.)

Answer 2

Try the category of finite sets and the following inverse system:

• $ I=-\Bbb N$
• $A_{n}= \{0,1,\ldots,-n\}$ for $n\in-\Bbb N$
• $f_{i,j}\colon A_i\to A_j$ given by $f_{i,j}(x)=\begin{cases}x,&x\in A_j\\0,&x\notin A_j\end{cases}$ for $i,j\in-\Bbb N$, $i\le j$

Then ${\lim\limits_{\leftarrow}}_{i\in I}A_i$ does not exist because it would have to be infinite.