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At the point where Lemma 1.2 is used you are already given maps $g_k: B_k\to M_k$ for $k<n$ such that $$(\alpha): \partial^M_k g_k = g_{k-1} \partial^B_k,\quad (\beta): j_k g_k = a_k\quad\text{and}\quad (\gamma): f_k g_k = b_k$$ hold, where $a_k: A_k\to M_k$ and $b_k: B_k\to N_k$ are the components of the given morphisms $A\to M$ and $B\to N$, ...

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It is hard enough to describe this model structure even with reference to the realisation–nerve adjunction, so I will not even try to do it without. As with the Joyal model structure, we take as our cofibrations all monomorphisms in $\mathbf{sSet}$. Of course, these are generated by the boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ (for ...

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I doubt that claim, given an object $X\in\mathcal{M}$, your functor $Q$ yields a factorization of the initial map $$\emptyset\stackrel{c}{\to}QX\stackrel{q_X}{\to}X$$ Where $c$ is a cofibration, and $q_x$ a weak equivalence. Applying $R$ to $QX$ yields a factorization of the terminal map  ...

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Let $\mathbf{A}$ be an abelian (or Grothendieck) category, and consider the injective model structure on the category of cochain complexes $\mathrm{Ch}(\mathbf{A})$. Then the derived category $\mathcal{D}(\mathbf{A})$ is the homotopy category $\mathrm{Ho}(\mathrm{Ch}(\mathbf{A}))$ by inverting the weak equivalences.

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Yeah, since colimits in infinity-categories are homotopy colimits.

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Geoffroy Horel pointed out that this was proved by Hinich as Theorem A.3.2 in [Deformations of sheaves of algebras].

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