# Colimits of cosimplicial rings

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

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 $R_{n+1}$.