The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings
$R_0 \rightarrow R_1 \rightarrow R_2 \rightarrow \cdots$
is just the usual colimit of the underlying sets equipped with the obvious ring structure (because this is a filtered diagram).
What about the colimit of a diagram of rings that looks like follows:
$R_0 \rightarrow R_1 \rightrightarrows R_2 \quad\text{three arrows}\quad R_3 \quad\text{four arrows}\quad R_4 \quad\cdots$ (sorry for the poor typesetting). What does the colimit of such a diagram look like? In examples I care about this diagram comes as the coface maps of a cosimplicial object. Is there a name for such a "colimit of a cosimplicial ring"?
More generally, does anyone have any references or pointers to works that talk about colimits of such diagrams of coface maps of cosimplicial objects?