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

Question

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 $i ≤ j ≤ k$.

Answer 1

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 $i<j$. 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 $j$.

So, yes, they are equivalent.