In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram: $A \rightrightarrows_{d^0}^{d^1} B \to^d C$, together with maps $A \leftarrow^t B \leftarrow^sC$ satifying:
- $d^0 \circ t = id_B$
- $d^1 \circ t = s \circ d$
- $d \circ s = id_C$
Now, the definition of $s,t$ feel a lot like a "contracting homotopy", which probably is what motivates their terminology, but I would like to make this more precise. That is, I would like to interpret this definition simplicially, as saying something about the nerve of the relevant diagrams. Can someone show me how this works?
I'm not firm on necessarily using nerves, but I'd like an explicit connection with simplicial/topological things. A test of how good the connection is might be: can you extend the definition of contractible coequalizer to other diagrams/colimits? I'm open to overly-sophisticated technology, so if this has a particularly nice phrasing in some sort of $\infty$ categorical world, let me know. I'm also open to any philosophical musings about how the topological interpretation of a coequalizer fits into the Beck monadicity theorem.