HTT, Remark 4.2.4.5

Question

In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the following:

Let $\mathbf{A}$ be a combinatorial simplicial model category, $S$ a small simplicial set, and $\phi:\mathfrak{C}[S]\to \mathcal{C}$ a DK-equivalence of small simplicial categories. Then the map $$\theta:N((\mathbf{A}^\mathcal{C})^\circ)\to \operatorname{Fun}(S,N(\mathbf{A}^\circ))$$is an equivlence of $\infty$-categories.

Here for a simplicial model category $\mathbf{B}$, we denoted by $\mathbf{B}^\circ $ its full simplicial subcategory spanned by the fibrant-cofibrant objects. Skimming through the proof, it looks like $\mathbf{A}^\mathcal{C}$ is endowed with the projective model structure. However, in Remark 4.2.4.5, he asserts that the claim also holds for the injective model structure. As far as I could tell, the proof does not easily generalize to the injective model structure. (If we replace "projective" by "injective" in Appendix A.3.4, the argument in the proof of Lemma A.3.4.10 does not hold.) Can someone explain to me why we can work with the injective model structure?

Any help/comment is appreciated. Thanks in advance.

---

Remark

The map $\theta$ is adjoint to the map $$N((\mathbf{A}^{\mathcal{C}})^{\circ})\times S\xrightarrow{1\times\psi}N((\mathbf{A}^{\mathcal{C}})^{\circ})\times N(\mathcal{C})\xrightarrow{N(\mathrm{ev})}N(\mathbf{A}^{\circ}),$$ where $\psi$ is the adjoint of $\phi$.

Answer 1

As shown by Dwyer–Kan (Function complexes in homotopical algebra, Proposition 4.8), for a simplicial model category $A$, the simplicial category $A^\circ$ is Dwyer–Kan equivalent to the simplicial category given by the hammock localization of $A$ with respect to the weak equivalences of $A$.

Applying this fact to the model category $A^C$ equipped with its projective or injective model structure, and using the fact that weak equivalences are the same for both model structures, we arrive at the desired result.

Answer 2

Let me elaborate on Dmitri Pavlov's excellent answer. (And sorry, Dmitri, for taking this long to digest your answer.)

Let us write $\theta_{\mathrm{proj}}$ for the functor $\theta$ when $\mathbf{A}^\mathcal{C}$ is equipped with the projective model structure, and define $\theta_{\mathrm{inj}}$ similarly. We want to show that $\theta_{\mathrm{inj}}$ is a categorical equivalence.

We start by recalling that if $\mathbf{M}_\Delta$ is a simplicial model category with underlying category $\mathbf{M}$, then the map $(N(\mathbf{M}^\circ),\mathrm{weq})\to (N(\mathbf{M}_\Delta^\circ),\mathrm{weq})$ is a weak equivalence of marked simplicial sets (This is 1.3.4.20 and 1.3.4.16 of Higher Algebra). (In other words, $N(\mathbf{M}^\circ_\Delta)$ is the localization of $N(\mathbf{M}^\circ)$ with respect to weak equiavlences.) Taking $\mathbf{M}_\Delta=\mathbf{A}^{\mathcal{C}}$, we deduce that the map

$$(N((\mathbf{M}^\circ)_{\mathrm{proj}}),\mathrm{weq})\to (N((\mathbf{M}_{\Delta\mathrm{proj}})^\circ, \mathrm{weq})$$

is a weak equivalence of marked simplicial sets. Now let $\mathbf{M}'\subset \mathbf{M}$ denote the full subcategory spanned by the functors taking values in $\mathbf{A}^\circ$. Using the cofibrant replacement functor, we find that the inclusion

$$(N((\mathbf{M}^\circ)_{\mathrm{proj}}),\mathrm{weq})\hookrightarrow (N(\mathbf{M}'),\mathrm{weq})$$ is a weak equivalence of marked simplicial sets. Since $\theta_{\mathrm{proj}}$ is a categorical equiavlence, this means that the map $(N(\mathbf{M}'),\mathrm{weq})\to\operatorname{Fun}(S,N(\mathbf{A}^\circ))^\natural$is a weak equiavlence of marked simplicial sets. This, in turn, implies (by repeating a similar argument) that $\theta_{\mathrm{inj}}$ is a categorical equivalence, and we are done.