Projective models for simplicial and variable-module diagrams

Free diagrams and evaluation. For a fixed constructed category $\mathcal{C}$, let $F_j\text{\hspace{0.17em}⁣}:\mathcal{C}\to {\mathcal{C}}^J$ be the free diagram, $(F_{j}X{)}_t={\coprod }_{u\text{\hspace{0.17em}⁣}:j\to t}X$, left adjoint to evaluation at $j$; the adjunction is verified pointwise and is enriched because the coproducts are. Generate projective cofibrations and acyclic cofibrations by $F_j$ applied to the boundary and horn free generators of [F1]. The right lifting classes are exactly the objectwise trivial fibrations and objectwise fibrations by this explicit adjunction. Generating domains are small because evaluation creates sequential colimits and the original domains are small.

The projective model structure. Small-object factorizations exist by the same construction as in [F1]. A relative cell of the acyclic generating set is objectwise a composite of pushouts of coproducts of acyclic cofibrations in $\mathcal{C}$, hence is objectwise acyclic: the left lifting property against fibrations is preserved by these operations, and the model axiom of $\mathcal{C}$ identifies this class with the acyclic cofibrations. Defining weak equivalences and fibrations objectwise, the identical factor-and-retract argument of [F1] proves the projective diagram model structure. Projective cofibrations are objectwise cofibrations by their cell and retract construction, and cotensors of diagrams are objectwise, so the cotensor corner of an objectwise fibration has the objectwise conclusions of [F1]; transposing against a projective cofibration proves the diagram simplicial corner axiom exactly as in [F1]. No diagram-model existence theorem is imported.

Variable module sections. Let $B\text{\hspace{0.17em}⁣}:J\to \mathrm{s}\mathrm{C}\mathrm{R}\mathrm{i}\mathrm{n}\mathrm{g}$ be a strict ring diagram. A module section is a family $M_j\in {\mathrm{s}\mathrm{M}\mathrm{o}\mathrm{d}}_{B_j}$ with transition maps $M_j\to {\mathrm{R}\mathrm{e}\mathrm{s}}_{B_j}^{B_t}{M}_t$ for $j\to t$, satisfying ordinary composition. The free section at $j$ has $(F_{j}M{)}_t={⨁}_{u\text{\hspace{0.17em}⁣}:j\to t}{B}_t{\otimes }_{B_j}M$, with transitions induced by tensor associativity and composition of $u$; these canonical maps give the required coherence. The adjunction $\mathrm{Hom}(F_{j}M,N)\cong {\mathrm{Hom}}_{B_j}(M,N_j)$ follows by evaluating at ${\mathrm{i}\mathrm{d}}_j$, with inverse from the transition maps. Generate the model using $F_j$ applied to the actual $B_j$-module generating maps; smallness follows by evaluation, and for an acyclic generator the evaluation at $t$ is a coproduct of extensions $B_t{\otimes }_{B_j}(-)$, which are left Quillen because restriction creates fibrations and weak equivalences; hence it is an acyclic cofibration. The same small-object, retract and factorization axioms follow, with objectwise weak equivalences and fibrations.

The corner and the cone adjunction. The pointwise $B_j$-cotensor action is constant in the simplicial variable and is respected by the section transitions, so the corner argument of step 2.1 makes the section category simplicial with the same corner property. A compatible cone $B_j\to B_0$ gives an enriched adjunction from sections to $B_0$-modules: the left functor takes the colimit of the extended modules $B_0{\otimes }_{B_j}{M}_j$ with their transition maps, and the right functor restricts a $B_0$-module to the $B_j$. Writing these functors as $L$ and $R$, restriction commutes with cotensors, $R(N^{\mathrm{\Delta }[n]})=(RN{)}^{\mathrm{\Delta }[n]}$. Applying the ordinary colimit/tensor adjunction to $N^{\mathrm{\Delta }[n]}$ in each degree gives $\mathrm{M}\mathrm{a}\mathrm{p}(LM,N{)}_n\cong \mathrm{M}\mathrm{a}\mathrm{p}(M,RN{)}_n$; these bijections respect simplicial operators and enriched composition by their naturality and the pointwise cotensor formulas. Restriction creates fibrations and weak equivalences objectwise, so the adjunction is Quillen and its derived enriched mapping comparison is supplied by [F3]; in the fixed-base case this is the ordinary colimit versus constant-diagram adjunction. This constructs strict projective diagrams and their derived mapping and colimit adjunctions, without claiming that every homotopy-coherent cartesian section has a strict representative or that this projective derived colimit agrees with every independently specified infinity-categorical model.