# What is the inverse limit of $...\to\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}$ (multiplying by all positive integers)?

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)?

• The zero group? If $(a_1,a_2,...)$ is an element then $a_1=2a_2$, $a_2=3a_3$, $a_3=4a_4$, ... If one of the components is zero then all the ones before are zero. If one component is zero, then all the ones before are non-zero. If all components are non-zero, then the first component is $2\times 3\times 4\times ...$. Only the zero sequence satisfies all this.
– OR.
Nov 14, 2013 at 16:22

$$...\xrightarrow{4}\mathbb{Z}\xrightarrow{3}\mathbb{Z}\xrightarrow{2}\mathbb{Z}\xrightarrow{1}\mathbb{Z}$$
$$...\xrightarrow{1}6\mathbb{Z}\xrightarrow{1}2\mathbb{Z}\xrightarrow{1}\mathbb{Z}\xrightarrow{1}\mathbb{Z}$$
and so you're taking the nested intersection of a sequence of subsets of $\mathbb{Z}$.