TITLE: What is the inverse limit of $...\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}$ (multiplying by all positive integers)?
QUESTION [5 upvotes]: According to a modified answer of this question, the direct limit of the sequence
$$
\mathbb{Z}\xrightarrow{1}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{4}...
$$
in the category of abelian groups is $\mathbb{Q}$. What is the inverse limit of the system
$$
...\xrightarrow{4}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{1}\mathbb{Z},
$$
i.e. is this an abelian group known under a different name? What about the inverse limit of
$$
...\xrightarrow{5}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{1}\mathbb{Z}
$$
(i.e. multiplying by the primes)?

REPLY [3 votes]: The problem is somewhat more obvious if you take the diagram
$$
...\xrightarrow{4}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{1}\mathbb{Z}$$
and replace it with the isomorphic diagram
$$
...\xrightarrow{1}6\mathbb{Z}\xrightarrow{1}2\mathbb{Z}\xrightarrow{1}\mathbb{Z}\xrightarrow{1}\mathbb{Z}$$
and so you're taking the nested intersection of a sequence of subsets of $\mathbb{Z}$.