As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself with understanding the case in which $I$ is the opposite category of a non-directed poset and $A$ is the category of abelian groups.


Now, my question is, how much of the theory of derived limits for inverse systems as presented for instance in Mardesic's *Strong Shape and Homology* or Jensen's *Les foncteurs derivées de lim* is preserved without that directedness assumption?


My interest lies mainly in the following results:


1. Flasque Goblot's Theorem: If $\mathbf{G}$ is an inverse system of abelian groups with surjective bonding morphisms of cofinality $\aleph\_n$, then $\lim^k \mathbf{G} = 0$ for all $k \geq n+1$.
2. Cofinality Theorem: If $C \subseteq I$ is cofinal in the indexing poset, then $\lim^n \mathbf{G} \cong \lim^n ( \mathbf{G} \restriction C)$.
3. $\lim^n \mathbf{G} \cong \check{H} ((I, \tau(I) ), \mathcal{G})$ where $\tau(I)$ is the topology of downwards-closed subsets of $I$ and $\mathcal{G}$ is the sheaf on it given by $U \mapsto \lim (\mathbf{G} \restriction U ) $.


Since the questions are quite general maybe I should narrow it down by saying that I am mostly concerned with diagrams of countably generated abelian groups with surjective morphisms and that regarding question 1 I think I might need the case $cof(I) \leq \aleph\_0 $ as the base case of an induction.



Non-cofiltered derived limits
Let $F:C\to Ab$ be the constant functor on an Abelian group $A$. Then $${\lim}^n F = H^n(BC, A),$$ where $BC$ is the classifying space or simplicial nerve of $C$. Take $C$ to be a finite poset with the homotopy type of $S^n$, e.g. the poset of faces of the usual CW structure with two cells in each dimension up to $n$. Then you get a nontrivial ${\lim}^n F$. This gives you an example where 1 doesn’t hold.


