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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?

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As for references: Read Quillen, he's excellent and I'm pretty sure he's the one who introduced simplicial rings in the first place. Try "Homology of Rings" (or something like that) and "Homotopical Algebra". –  Aaron Mazel-Gee Oct 24 '12 at 23:06
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It is the union of the $R_n$ modulo the ideal generated by the relation that every element in $R_n$ is identified with its $n+1$ images in $R_{n+1}$.

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