I'm studying the Appendix A to this paper, where they introduce the notions of geometric realization of an acyclic cetgory and polyhedral CW complexes.
I did already study some algebraic topology, and also some combinatorial toplogy: to be precise, before approaching this topic, I studied the simplicial complex associated to a poset and in general some notions concernig "poset topology" (like the nerve lemma, some results concerning geometric lattices and shellability). This is just to give you some background: I'm pretty new to these topics, but I'm trying to increase my knowledge.
Whith this background in mind, I didn't really have problems with the first definitions presented in the Appendix: the authors give the definition of the geometric realization $ \| \mathscr{C} \| $ of an acyclic category $\mathscr{C}$.
Then, for a fixed $x \in \text{Ob}_\mathscr{C}$, they define the slice category $ \mathscr{C} / x $ as follows:
- The objects of $ \mathscr{C} / x $ are the morphisms $ \varphi \in \text{Mor}_\mathscr{C} $ ending in $x$.
- Given $ \varphi, \, \psi \in \text{Ob}_{\mathscr{C}/x} $, for any morphism $\zeta \in \text{Mor}_\mathscr{C} $ such that $ \varphi = \psi \circ \zeta $ there is a morphism $ \mu_\zeta : \varphi \to \psi $ in $\text{Mor}_{\mathscr{C}/x}$
Then the functor $$ J_x: \mathscr{C}/x \to \mathscr{C} $$ that sends every $ \varphi \in \text{Ob}_{\mathscr{C}/x} $ to the source object of $\varphi$ in $\mathscr{C}$ and every morphism $\mu_\zeta \in \text{Mor}_{\mathscr{C}/x}$ to the mophism $\zeta \in \text{Mor}_\mathscr{C}$ induces a continuos function on the geometric realizations $$ j_x: \| \mathscr{C}/x \| \to \| \mathscr{C} \| $$
The part that leavs me dubious is Lemma A.4:
Lemma: The map $j_x$ is injective in the interior of each simplex of $\| \mathscr{C}/x \| $ that has $id_x$ as a vertex.
They give the following proof:
The identity $id_x$ is a vertex of the simplex $\Delta[\gamma]$ with $ \gamma = ( \mu_{\zeta_1}, \dots, \mu_{\zeta_d}) $ if and only if the target object of $\mu_{\zeta_d}$ is $ t(\mu_{\zeta_d}) = id_x$, i.e. $t(\zeta_d) = x$. Now, $J_x$ maps every sequence $ \gamma = ( \mu_{\zeta_1}, \dots, \mu_{\zeta_d})$ of composable morphisms in $\mathscr{C}/x$ with $t(\mu_{\zeta_d}) = id_x$ to the sequence $(\zeta_1, \dots , \zeta_d )$ of composable morphisms of $\mathscr{C}$, with $t(\zeta_d) = x $. The evident injectivity of this map implies injectivity of $j_x$ on the open cell $ int(\Delta[\gamma]) \subseteq \mathscr{C}/x$.
My problem with this lemma is that I don't see the need of the request that $id_x$ be a vertex. With the exact same proof, I might say that $J_x$ sends a face $( \mu_{\zeta_1}, \dots, \mu_{\zeta_d})$ to the face $(\zeta_1, \dots , \zeta_d )$ in a way that is clearly injective by the definition of $\mathscr{C}/x$. What am I missing here? Why do I need that $id_x$ be a vertex of a face to be able to say that $j_x$ is injective on that face?
Consider a category with three objects $\{s, t, x\}$ and morphisms generated by $f: s \to t$, $g_1: t \to x$, and $g_2: t \to x$. Set $h_1 = g_1 \circ f$ and $h_2 = g_2 \circ f$.
Then, we have two distinct 1-simplices in $\mathscr{C}/x$ spanned by $(f, g_1, h_1)$ and $(f, g_2, h_2)$, both forgetting to $f$ in $\mathscr{C}$, which shows that the map on simplices is not injective.
In other words, given only a morphism $f: s \to t$ in $\mathscr{C}$, there may be multiple ways to extend it to a triangle with third vertex $x$. However, if the sequence of composable morphisms ends in $\mathrm{id}_x$, then all the maps to $x$ along the sequence are forced.