I call Segal subdivision the endofunctor of simplicial objects in a category $\mathcal{C}$ induced by the doubling endofunctor of $\Delta^{op}$ sending $x_0<\cdots<x_n$ to $x_0<\cdots <x_n<x_n'<\cdots<x_0'$. I am calling this Segal subdivision as in Weibel's K-book chapter IV exercise 3.10. Here is Segal's original paper, where what I am discussing is the content of the first appendix.
It turns out that if we consider a simplicial space $A$ and denote its Segal subdivion by $sub(A)$, this yields homotopy equivalent spaces after geometric realization. I am not managing to complete the proof provided in the paper. If somebody knows of a ressource with (or is willing to write up) a more detailed proof this would be greatly welcome and would (obviously) solve my problem.
However, not wanting to ask too much, I do have a more specific question in mind, which I am hoping will be enough to help me. Why is the proof so unsimplicial? What I mean is the the homotopy equivalence does not at arise at the simplicial level i.e $n$ simplicies aren't sent to $n$ simplicies (unless I severly missunderstood the two maps). Similarly, what fails with general simplicial obejcts, why must we work with simplicial spaces?
Both of these restrictions are making me feel like my practice with simplicial stuff will be useless, and that this result doesn't "belong to" simplicial theory in some sense.
Thank you to anyone with the time and motivation to answer my querry, any help is appreciated.
I have after scouring the internet found an alternative proof which is much more "simplicial" and in my opinion more satisfying.
It can be found as lemma 5.2 in these slides by Jardine. This proof in addition to being more in line with my personal taste lends itself better to potential generalizing and philosophizing, instead of feeling like a coincidence.