An inverse system is a tuple $(X_i,\varphi_{ij},I)$ where
- $(I,\preceq)$ is a directed poset,
- $\{X_i\}_{i \in I}$ is a collection of topological spaces,
- $\varphi_{ij} : X_i \to X_j$ is a continuous map whenever $i \succeq j$
such that $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ whenever $i \succeq j \succeq k$.
We know that each inverse system has an inverse limit $(X,\varphi_i)$ where
- $X$ is a topological space,
- $\varphi_i: X \to X_i$ are continuous,
satisfying the universal property. Furthermore, the inverse limit is unique up to homeomorphism.
Question: What would happen with the inverse limit if we omit the condition $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ for $i \succeq j \succeq k$?
- Would the inverse limit still exist?
- Would the inverse limit still be unique (up to homeomorphism)?
I was curious why we needed this condition in the first place and could not see when we used it explicitly. Could you please help me with this question?
Thank you!
Best Answer
Nothing about the existence and uniqueness of the inverse limit relies on the assumption that $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$. However, omitting this assumption does not actually give any greater generality. Indeed, note that given $X$ with maps $\varphi_i:X\to X_i$ satisfying $\varphi_{ij}\varphi_i=\varphi_j$ whenever $i\succeq j$, we have $$\varphi_{jk} \varphi_{ij}\varphi_i=\varphi_{jk}\varphi_j=\varphi_k=\varphi_{ik}\varphi_i$$ whenever $i\succeq j\succeq k$. In other words, the image of the map $\varphi_i$ must be contained in the subset $Y_i\subseteq X_i$ consisting of elements $x$ such that $\varphi_{jk} \varphi_{ij}(x) = \varphi_{ik}(x)$ whenever $i\succeq j\succeq k$. This means we can restrict the inverse system to the $Y_i$ instead of the $X_i$ (exercise: check that $\varphi_{ij}(Y_i)\subseteq Y_j$) without changing what an inverse limit of the system is. When we restrict to the $Y_i$, the equations $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ are true.
To put it another way, the assumption that $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ is essentially inherent in the condition $\varphi_{ij}\varphi_i=\varphi_j$ in the definition of the inverse limit. You could have elements of $X_i$ on which $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ is not true if you really wanted to for some reason, but those elements are irrelevant to the inverse limit.
(Note that the all above comments also apply to the assumption that $\varphi_{ii}$ is the identity map on $X_i$, which you omitted but is also part of the definition of an inverse system.)
From the perspective of Kevin Carlson's answer, dropping the condition $\varphi_{jk} \varphi_{ij} = \varphi_{ik}$ means that you are not really talking about a limit indexed by the poset $I$, but rather a different category (namely, the category freely generated by $I$ as a directed graph). In practice, limits indexed by that different category pretty much never come up naturally and do not have any special properties to differentiate them from arbitrary limits, so they are not discussed separately from general limits the way that inverse limits are.