The strict simplicial algebra adjunctions underlying the cotangent construction

Everything is degreewise and simplicial. All four constructions and the maps between them are performed degreewise and the face and degeneracy maps are induced by those of $A,B,C,I,M$; the induced maps preserve addition, scalar action, multiplication and the identity where required, so each construction lands in the category named and the constructions are natural in degree. Every category appearing is locally small: morphisms are compatible degreewise functions between fixed sets. Hence it suffices by [F2] to exhibit natural hom-set bijections for each adjunction.

Adjunction 1. An augmented $B$-algebra map $\theta \text{\hspace{0.17em}⁣}:B{\otimes }_{A}C\to D$ restricts to an augmented $A$-algebra map $f=\theta \circ (1\otimes -)\text{\hspace{0.17em}⁣}:C\to D$, and conversely $f$ determines the unique $B$-algebra map $b\otimes c\mapsto b\cdot f(c)$; the balanced tensor relation is respected because $f$ is $A$-linear, and multiplicativity, unit and augmentation compatibility hold for $\theta$ exactly when they hold for $f$. The two prescriptions are inverse natural bijections.

Adjunction 2. For an augmented $B$-algebra $D$ with section $s\text{\hspace{0.17em}⁣}:B\to D$ and augmentation $\pi \text{\hspace{0.17em}⁣}:D\to B$, the kernel $I(D)$ is a $B$-module through $s$ and is closed under multiplication; every $d\in D$ has the unique expression $d=s(\pi (d))+(d-s(\pi (d)))$ with the second summand in $I(D)$, so the displayed map $B\oplus I(D)\to D$, $(b,x)\mapsto s(b)+x$, is a bijective $B$-module map in each degree and is compatible with the simplicial operators. Multiplying two expressions uses $s(b)s(c)=s(bc)$ and $s(b)y=by$ to compute $(bc,by+cx+xy)$, so the formula makes $B\oplus I(D)$ a unital commutative $B$-algebra and the bijection an isomorphism of augmented $B$-algebras; the construction is natural in $D$. Conversely, a nonunital $B$-algebra map $\phi \text{\hspace{0.17em}⁣}:I\to I(D)$ extends uniquely by $(b,x)\mapsto s(b)+\phi (x)$ to an augmented $B$-algebra map $K(I)\to D$, and restriction to $I$ inverts this, giving the natural bijection and the equality $IK(I)=I$.

Adjunction 3. The $B$-submodule generated by all products $xy$ is a simplicial submodule of $I$, because every simplicial operator preserves the multiplication and the $B$-action. A $B$-linear map $\phi \text{\hspace{0.17em}⁣}:I\to M$ is multiplicative into the zero-multiplication algebra $Z(M)$ exactly when it annihilates every product $xy$, and that happens exactly when $\phi$ factors uniquely through the quotient $Q(I)=I\mathrm{/}(xy)$. These factorizations are inverse natural hom-set bijections, so $Q$ is left adjoint to $Z$.

Weak equivalences. A morphism of augmented $B$-algebras restricts to a morphism of the kernels, and the isomorphism of step 1.3 decomposes the underlying simplicial abelian group of an augmented algebra as the direct sum $s(B)\oplus I$ of simplicial abelian groups. The associated Moore complex of a direct sum is the direct sum of the Moore complexes, and its map is the identity on the $B$ summand and the induced map on $I$; therefore ${\pi }_{\ast }(B\oplus I)\cong {\pi }_{\ast }B\oplus {\pi }_{\ast }I$ compatibly, so the functor $I$ and its two-sided inverse $K$ preserve and reflect weak equivalences as defined in [F1].

Conditional Quillen clause. Assume now that model structures on the four categories exist with fibrations and weak equivalences created in the underlying simplicial sets, and give each category that model structure. The right adjoints of 1 and 3 leave the underlying simplicial-set map unchanged in each degree, hence preserve fibrations and trivial fibrations, and so give Quillen adjunctions by [F3]. For 2, the underlying simplicial set of $K(I)=B\oplus I$ is $B\times I$; a lifting problem for $K(I)\to D$ relative to a fibration of augmented algebras projects to a lifting problem for $I\to \ker D$ with the zero simplex in the $B$-coordinate, and conversely a lifting problem for $I$ extends by the zero simplex in the $B$-coordinate, so $I$ preserves and reflects fibrations and trivial fibrations; step 2.1 gives the corresponding preservation and reflection of weak equivalences. Hence 2 is a Quillen equivalence, and each of the three adjunctions is a Quillen adjunction, under the stated existence hypothesis and no more.

No hidden imports. The constructions are strict algebraic ones, and the proof imports no model-category existence, derived mapping-space or descent theorem; the only model-category statement made is the conditional one just proved. No choice principle is used: all constructions are degreewise formulas and all hom-set bijections are given by explicit inverse formulas.