Monoidal fibration is left derivable

Proof. Fix a span \(\alpha \colon I \xleftarrow {f} K \xrightarrow {g} J\) in \(\Span (\Fin )\). Under the identification \(C_I^{\otimes } \simeq C^I\), the cocartesian transport functor along \(\alpha \) is given by \[ \alpha _!\colon C^I \xrightarrow {f^*} C^K \xrightarrow {g_{\otimes }} C^J, \qquad (x_i)_{i \in I} \mapsto \left (\bigotimes _{k \in g^{-1}(j)} x_{f(k)}\right )_{j \in J}. \] The product category \(C^I\) again carries weak equivalences and cofibrations componentwise, by Lemma 6.6.7. It therefore suffices, by Proposition 6.6.3, to show that \(\alpha _!\) preserves componentwise weak equivalences between componentwise cofibrant objects.

This is clear for the restriction functor \(f^*\). For the functor \(g_{\otimes }\), let \(\{u_k\colon x_k \to y_k\}_{k \in K}\) be a componentwise weak equivalence between componentwise cofibrant tuples. Then every object \(y_k\) is in particular tensor-cofibrant, and the \(j\)-th component of \(g_{\otimes }(u)\) is the map \[ \bigotimes _{k \in g^{-1}(j)} x_k \xrightarrow {\bigotimes _{k \in g^{-1}(j)} u_k} \bigotimes _{k \in g^{-1}(j)} y_k, \] which is a weak equivalence by the left derivability assumption. Thus \(\alpha _!\) admits an absolute left derived functor \[ \bL \alpha _!\colon C^I[(W^I)^{-1}] \to C^J[(W^J)^{-1}]. \]

It remains to verify compatibility with composition. Consider composable spans \[ \alpha \colon I \xleftarrow {f} K \xrightarrow {g} J \qquadtext {and} \qquad \beta \colon J \xleftarrow {h} L \xrightarrow {k} M. \] We show that the canonical comparison map \[ \bL \beta _! \circ \bL \alpha _! \to \bL (\beta \circ \alpha )_! \] is an isomorphism. Let \(X=(X_i)_{i\in I}\) be componentwise cofibrant. Every component of \(\alpha _!(X)\) is a finite tensor product of cofibrant objects, and hence is tensor-cofibrant. Choose any componentwise weak equivalence \[ q\colon P\xrightarrow {\sim }\alpha _!(X) \] with \(P\) componentwise cofibrant. Such a map is obtained by factoring the morphism from the initial object to \(\alpha _!(X)\) in \(C^J\), and no functorial choice is required. By the defining property of total left derived functors, we have \[ (\bL \beta _!\circ \bL \alpha _!)(\gamma _I X) \simeq \bL \beta _!(\gamma _J\alpha _!(X)) \simeq \gamma _M\beta _!(P). \] The structure transformation for \(\bL \beta _!\) at \(\alpha _!(X)\) is represented under this identification by the map \[ \gamma _M\beta _!(q)\colon \gamma _M\beta _!(P)\longrightarrow \gamma _M\beta _!\alpha _!(X). \] This map is an isomorphism. Indeed, restriction along \(h\) preserves weak equivalences, and each component after applying \(k_{\otimes }\) is a tensor product of weak equivalences whose sources are cofibrant and whose targets are tensor-cofibrant; this is a weak equivalence by Definition 20.1.3. The composite structure transformation used to construct the comparison in Observation 20.1.11 is therefore an isomorphism at \(\gamma _I X\). The structure transformation for \(\bL (\beta \circ \alpha )_!\) is also an isomorphism there, since \(X\) is componentwise cofibrant. Hence the comparison map is an isomorphism on the image of every componentwise cofibrant object of \(C^I\). By part (1) of Corollary 2.2.9, every object of \(C^I[(W^I)^{-1}]\) is isomorphic to the image of such an object, so the comparison map is an isomorphism everywhere. We conclude that \(p_C\) is left derivable with respect to \(\{W^I\}_{I \in \Fin }\). □