Suppose we have the diagram of spectra $$A \to B \to C$$ Then taking the limit of this diagram is essentially taking the limit over the Nerve of the category $$\cdot \to \cdot \to \cdot$$ which I believe is exactly $\Delta^2$.
$\Delta^2$ has a subcomplex $i:\partial \Delta ^2 \hookrightarrow \Delta^2 $. So given this functor from $\Delta^2$ to Spectra, we can compose with the inclusion $i$ and get a functor from $\partial \Delta ^2$ to Spectra. My question is, What is the limit of this functor over $\partial \Delta ^2$? In particular I would like to be able calculate the homotopy groups of this limit.
The limit over $∂Δ^2$ is simply the homotopy equalizer (alias ∞-equalizer) of the maps given by the image $f\colon A→C$ of 0→2 respectively the composition of the images of 0→1→2. Up to homotopy, the latter composition can be taken to be $f$. Thus, we are interested in the homotopy equalizer of $f$ and $f$.
This homotopy equalizer can be computed as the homotopy fiber of the map $$A⨯C=A⊕C→C⨯C=C⊕C$$ whose component $A→C⨯C$ is the map $(f,f)\colon A→C⨯C$, whereas the component $C→C⨯C$ is the diagonal map.
The homotopy groups of this homotopy fiber can be computed in the standard manner, for example, using the long exact sequence associated to this fiber sequence.