I was studying Hatcher's algebraic topology text and found Hatcher's notion of what is called a "complex of spaces" introduced in the context of homotopy colimits to be confusing, and his construction of the homotopy colimit or "realization" associated to a certain simplicial set. A similar construction is carried out in Combinatorial Algebraic Topology by Kozlov, in chapter 15. I tried to write down a formalization that made sense to me. The purpose of this post is to seek constructive criticism of my way of understanding the concept ("this is not how you should think of it, a better and more natural way of thinking of it is this") and to try and seek references that would allow me to situate my understanding in context. I do not yet understand the notion of homotopy colimit, so in all probability the following viewpoint not at all relevant to understanding that construction, but perhaps there is some setting where this viewpoint is helpful and I am wondering if anyone can point me in such a direction.
Let $\mathbb{E} = \operatorname{Fam}_{Sets}(\mathbf{Top})$ be the category whose objects are families of topological spaces $\{X_\lambda\}_{\lambda \in I}$ and whose morphisms in $Hom(\{X_i\}_{i\in I},\{Y_j\}_{j\in I})$ are pairs $(f : I\to J, \{g_i\}_{i\in I})$ where each $g_i$ is a continuous map $X_i\to Y_{f(i)}$. There is a functor $p: \operatorname{Fam}_{Sets}(\mathbf{Top})\to \mathbf{Sets}$ sending $\{X_\lambda\}_{\lambda \in I}$ and the pair $(f : I\to J, \{g_i\}_{i\in I})$ to $f$. This functor is a Grothendieck fibration, making $\operatorname{Fam}_{Sets}(\mathbf{Top})$ into a fibered category over $\mathbb{B} = \mathbf{Sets}$.
Let $\mathbb{D}$ denote the full subcategory of $\Delta^{op}$ consisting of the ordinals $[0], [1]$, and $[2]$. Let $j : \mathbb{D}\to \Delta^{op}$ be the canonical inclusion.
Let $\Gamma : \Delta^{op}\to \mathbf{Sets}$ be a simplicial set. In what follows, $d_i$ and $s_j$ refer to the faces and degeneracy maps of $\Gamma$. A "complex of spaces" indexed by the $1$-skeleton of $\Gamma$ is, (I tentatively suggest), a lift $X$ of $\Gamma\circ j : \mathbb{D}\to \mathbb{B}$ along $\mathbb{E}$ (i.e. a functor $\Gamma' : \mathbb{D}\to \mathbb{E}$ such that $p\circ X = \Gamma\circ j$ on-the-nose) such that $d_i'$ = $X(d_i)$ is a Cartesian lift for each $i>0$ (so the maps $d'_1,d'_2 : X_2\to X_1$ and the map $d_1': X_1\to X_0$.
The intuition behind this construction is as follows. $X_0$ is the family of spaces of the complex, indexed by the $0$-simplices of $\Gamma_0$. $X_1$ is the same family of spaces, indexed by the first vertex of the edges in $\Gamma_1$ in the form $(X_{\sigma_0}, \omega: \sigma_0\to \sigma_1)$. The map $d'_0 : X_1\to X_0$ codes the continuous maps of the complex of spaces, and controls the domain and codomain of those maps. The commutativity condition $d_0's_0'=id_{X_0}$ here has the effect of forcing the maps associated to degenerate edges to be associated to the identity map. The commutativity condition $d_0'd_0' = d_0'd_1'$ forces the commutatity of the maps on the boundary of each $2$-cell. The other simplicial identities I haven't thought about the interpretations of yet.
If $f : I\to J$ is a set map, and $\{Y_j\}_{j\in J}$ is a family of sets indexed by $J$, denote by $f^\ast \{Y_j\}$ the object $\{Y_{f(i)}\}_{i\in I}$. It seems like in the construction of the homotopy colimit, there is a construction one carries out where one extends the definition of $X :\mathbb{D}\to \mathbb{E}$ to all of $\Delta^{op}$, giving a functor $\overline{X}$ which breaks the commuting square $p\circ X= \Gamma\circ j$ into the two commuting triangles $\overline{X}\circ j = X$ and $p\circ\overline{X} = \Gamma$.
What is done is to repeatedly pull back the object $X_{n-1}$ along $d_n$ and set $X_n = d_n^{\ast}$, setting $d_n'$ to be the Cartesian lift of $d_n$, and then filling in all other face and degeneracy maps by appealing to the simplicial identities to get conditions that the maps $d_i', s_j'$ must satisfy, and showing that a unique such $d_i', s_j'$ must exist by appealing to the universal properties of the Cartesian pullback.
Then one takes the functor $\overline{X}: \Delta^{op}\to \operatorname{Fam}_{Sets}(\mathbf{Top})$ and "forgets" that it is a family by composing with the disjoint coproduct functor $\coprod: \operatorname{Fam}_{Sets}(\mathbf{Top}) \to\mathbf{Top}$ which sends $\{X_i\}_{i\in I}$ to $\coprod X_i$. The resulting simplicial space, one takes the geometric realization of (in the sense. say, of Gelfand and Manin in Methods of Homological algebra) This geometric realization is, if I understand correctly, what Hatcher and Kozlov understand to be the realization of the diagram (modulo differences between simplicial sets and semisimplicial sets). Am I correct in saying this?
Secondly, I conjecture that part of the construction I described above (which relies only on the properties of Grothendieck fibrations) works in any Grothendieck fibration, i.e. that for any fibration $p : \mathbb{E}\to \mathbb{B}$ and for any pair of functors $X : \mathbb{D}\to \mathbb{E},\Gamma : \Delta^{op}\to \mathbb{B}$, with $p\circ X = \Gamma\circ j$, where the maps $d_i'$ of $X$ are Cartesian for $i\neq 0$, there is a functor $\overline{X}: \Delta^{op}\to \mathbb{E}$ making everything in sight commute. I am working on proving this but it is a tedious induction and will take some time, and I didn't want to dive into it before understanding the context. I find this an elegant result, as I know that $j$ is a cofibration and $p$ a fibration in the usual model structure on categories, so this solves a certain lifting problem. I would like to know if there is a source for this or a similar or more general result, or if this fits into a broader theory looking at similar approaches to lifting a functor along a fibration; because although I believe it is true I don't know the significance of it.