What is the intuitive difference between these two simplicial subdivision functors?

Question

$\newcommand{\sset}{\mathsf{sSet}}\newcommand{\poset}{\mathsf{Poset}}\newcommand{\p}{\mathscr{P}^{\mathsf{nd}}}\newcommand{\N}{\mathcal{N}}\newcommand{\sd}{\operatorname{sd}}$Following this paper, I encountered two different subdivision functors on $\sset$, and while there is a clear mathematical difference in their definitions it is rather unclear to me how these functors are "actually" different. I use different notation than what appears in the paper.

---

Definitions:

Let $\N:\poset\to\sset$ denote the (restriction of) usual nerve functor, where posets are viewed as categories in the standard way, given by $\N(K)_n=\poset([n],K)$ where $[n]=[0<1<2<\cdots<n]$ for each $n\in\Bbb N_0$. We consider the functor $\p:\sset\to\poset$ which, on objects, maps the simplicial set $X$ to the set of its nondegenerate simplices (in any and all degrees) ordered by the face relation $\sigma\preceq\tau$ iff. there exists an injection $f$ in $\Delta$ such that $X_f(\tau)=\sigma$. On arrows, this functor can be defined and proven to be a functor using repeated applications of the Eilenberg-Zilber lemma.

I consider the first "naive" subdivision functor $S:\sset\to\sset$ which is defined to be the composite $\N\p$. Then there is the second subdivision functor, which is the one that is actually considered the simplicial subdivision, $\sd:\sset\to\sset$ which could be defined as the left Kan extension of $S\circ\Delta^\bullet$ along $\Delta^\bullet$ or as the unique cocontinuous functor agreeing with $S$ on the $\Delta^n$ or as the coend $X\mapsto\int^{n\in\Delta}X_n\otimes S\Delta^n$ or as $\sd X=\varinjlim_{(n,\sigma)\in\mathrm{el}(X)}S\Delta^n$. These all agree up to isomorphism. This is left adjoint with right adjoint $\Gamma(X)_n:=\sset(S\Delta^n,X)$

By easy abstract nonsense, $\sd\Delta^n\cong S\Delta^n$ naturally in $n$. In fact there is a canonical comparison natural transformation $\pi:\sd\implies S$ which can be shown to always be surjective; $\pi_X$ is an isomorphism if $X=\N(Q)$ for some poset $Q$. It can be shown (a bit more care than is usually given is required) that $|S\Delta^n|\cong|\Delta^n|$ naturally in $n$ and it follows that $|\sd(X)|\cong|X|$ naturally in the simplicial set $X$.

---

The actual question: $\pi$ surjects, but what is stopping it from being injective in general? How does $\sd$ differ from $S$ in "what it is actually doing"? The functor $S$ handles data about chains of subfaces of the n.d. simplices and the functor $\sd$ handles data about chains of face maps glued together when the images of two simplices under these maps agree; loosely, $\sd$ is also bundling equivalence classes of chains of subfaces of the n.d. simplices and both are fairly reasonable (to my mind) attempts at a notion of subdivision; however, only one has succeeded and earned its title. Jardine refers to $S$ as a "classifying space" functor instead.

Usually when trying to understanding something simplicial, it helps to take geometric realisations and view what is happening, but since $|\sd|\cong|\cdot|$ and in all the simple examples I've tried, $|S|\cong|\cdot|$ as well, I can't really see what simplices $\sd$ has that $S$ does not. The question could be restated as: what obstructs $\pi$ from being an isomorphism?

I would appreciate any insights or good examples as to how $\sd$ differs from $S$. For example, that might mean showing $S$ fails to be cocontinuous (for all I know, $S$ is cocontinuous) with some explanation as to why. Of course, having a cocontinuous functor is very useful, but I wonder if there are intuitions/motivations for $\sd$ that are less centred on pure convenience.

Answer 1

Lemma 2.2.11 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf shows that the surjection you denote $sd(X) \to S(X)$ is an isomorphism if and only if $X$ is a non-singular simplicial set, i.e., if and only if each non-degenerate simplex in $X$ is embedded. It may be interesting to determine for which $X$ the initial non-singular simplicial set under $sd(X)$ is precisely $S(X)$. In the book above, we called this the desingularization of $sd(X)$.

Answer 2

Let us compare what both functors do on $X=S^2=\Delta^2/\partial \Delta^2$.

There are only two nondegenerate simplices, so the first functor sends it to an interval. The second functor sends it to the barycentric subdivision of $\Delta^2$ modulo its boundary, i.e. a 2-sphere.

The map between the two sends the boundary to one endpoint of the interval and the new inner point in the barycentric subdivision of $\Delta^2$ to the other point. It looks a bit like the height on the usual visualization of the 2-sphere.