Equivalent ways of computing total homotopy fibres
Let \({\mathcal{C}}\) be a pointed ((∞,1)-)category with all finite limits. An \(n\)-cube in \({\mathcal{C}}\) is a functor \(A : {\mathcal{P}}(T) \to {\mathcal{C}}\), where \(T\) is an \(n\) element set and \({\mathcal{P}}(T)\) is the poset of all its subsets. The total (homotopy) fiber of the cube is the fiber \({\mathrm{tfib}}(A)\) of the induced morphism \(A(\emptyset) \to \lim_{S \neq \emptyset} A(S)\).
One can also compute the total fiber by taking fibers of all the edges of the cube in a fixed direction and then taking total fiber of the resulting cube, more explicitly, if \(A : {\mathcal{P}}(T) \to {\mathcal{C}}\) is a cube, and \(t \in T\), we can form a \((|T|-1)\)-dimensional cube \({\mathcal{P}}(T\setminus\{t\}\}) \to {\mathcal{C}}\) given by \(S \mapsto {\mathrm{fib}}(A(S) \to A(S\cup\{t\}))\), and we claim that its total fiber is the total fiber of \(A\).
Iterating this, we can compute total fibers by taking fibers repeatedly reducing the dimension of the cube by one each time.
More generally picking any \(T' \subset T\) we can take total fibers of the cubes \({\mathcal{P}}(T \setminus T') \to {\mathcal{C}}\) given by \(S \mapsto A(S \cup S')\) and the total fiber of the resulting cube, namely the functor \({\mathcal{P}}(T')\to{\mathcal{C}}\) given by \(S' \mapsto {\mathrm{tfib}}(A(- \cup S'))\), is the same as the total fiber of \(A\).
To prove this last claim, we’ll characterize the total fiber in a different way.
Let \(\phi_T : {\mathcal{C}}\to {\mathrm{Fun}}({\mathcal{P}}(T), {\mathcal{C}})\) be the functor that sends \(X\) to the cube with \(X\) in the initial corner and zeros elsewhere, \(\phi_T(X)(\emptyset) = X\) and \(\phi_T(X)(S) = 0\) for \(S \neq \emptyset\). This \(\phi_T\) is left adjoint to the total fiber functor \({\mathrm{tfib}}: {\mathrm{Fun}}({\mathcal{P}}(T),{\mathcal{C}}) \to {\mathcal{C}}\).
We need to show that \({\mathrm{Nat}}(\phi_T(X), A) \cong {\mathcal{C}}(X, {\mathrm{tfib}}(A))\). For that, notice both sides are equivalent to the fiber above \(0\) in the induced map \({\mathcal{C}}(X, A(\emptyset)) \to \lim_{S \neq \emptyset} {\mathcal{C}}(X, A(S))\).
Now it is easy to prove the claim above. It is easy to check that \(\phi_T\) is the composite \[{\mathcal{C}}\xrightarrow{\phi_{T'}} {\mathrm{Fun}}({\mathcal{P}}(T'),{\mathcal{C}}) \xrightarrow{\phi_{T \setminus T'}} {\mathrm{Fun}}({\mathcal{P}}(T \setminus T'), {\mathrm{Fun}}({\mathcal{P}}(T')),{\mathcal{C}}) \xrightarrow{\cong} {\mathrm{Fun}}({\mathcal{P}}{T}, {\mathcal{C}}),\] so that the right adjoint is \({\mathrm{tfib}}_{T} \cong {\mathrm{tfib}}_{T'} \circ {\mathrm{tfib}}_{T \setminus T'}\) (where we’ve decorated the total fiber functor to indicate the domain of the cubes it acts on).