Non-cofiltered derived limits

Question

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.

Answer 1

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.