# Equivalent definitions of inverse limit (Atiyah-Macdonald, chapter 10)

In chapter 10 of Atiyah-Macdonald's book, an inverse system of topological abelian groups is defined as follows: a pair $$((A_n),(\theta_{n}))$$ where $$(A_{n})$$ is a sequence of topological abelian groups and $$(\theta_{n})$$ is a sequence of morphisms $$\theta_{n+1}\colon A_{n+1} \longrightarrow A_{n}$$. With no more restrictions, the inverse limit of an inverse system is defined as

$$\varprojlim A_{n} := \left\{(a_{n}) \in \prod_{n \in \mathbb{N}}A_{n} : \theta_{n+1}a_{n+1} = a_{n} \text{ for all } n \in \mathbb{N} \right\}$$

Is this definition equivalent to the usual definition of inverse limit, taking $$(I,<)$$ as the usual poset defined by the natural numbers? I'm aware that A-M ignores the projection morphisms, but what about the prerequisites in the usual definition of inverse system? That is,

1) $$f_{ii}$$ is the identity on $$A_{i}$$.

2) $$f_{ik} = f_{ij}f_{jk}$$ for all $$i ≤ j ≤ k$$.

The $$f_{ij}$$ are the actions on morphisms of a (contravariant) functor, namely $$f_{ij}$$ is the image of $$i\leq j$$ of the (directed) poset viewed as a category. $$\theta$$ is not a functor at all. It is just a sequence of morphisms (continuous group homomorphisms in this case). $$\theta$$, however, induces a (contravariant) functor from $$(\mathbb N,\leq)$$ by defining $$f_{ii}=id_{A_i}$$ and $$f_{ij}=\theta_{i+1}\circ f_{(i+1)j}$$ for $$i. You can easily show $$f_{ik}=f_{ij}\circ f_{jk}$$ for $$i\leq j\leq k$$. In other words, we define $$f_{i(i+1)}=\theta_{i+1}$$ and $$f_{ij}$$ in general is generated by considering all composable sequences. It's easy to show that $$\theta_{i+1}(a_{i+1})=a_i$$ implies $$f_{ij}(a_j)=a_i$$ for all appropriate $$i$$ and $$j$$.