Maurer–Cartan classes classify fixed-carrier codifferential deformations

By Complete coderivations form a filtered differential graded Lie algebra, $\mathfrak{g}$ is a complete filtered differential graded Lie algebra. Since $D$ and $R$ have degree one and $D^2=0$, direct expansion gives $(D+R{)}^2=DR+RD+R^2=dR+[R,R]\mathrm{/}2$. Thus the displayed assignment is an object bijection, with inverse subtraction of $D$.

By Coalgebra automorphisms congruent to the identity are exactly coderivation exponentials, every allowed $G$ has the unique form $G=\exp X$ with $X\in {\mathfrak{g}}^0$. The conjugation identity in Exponential and logarithm in a noncommutative complete filtered algebra gives $G(D+R)G^{-1}=e^{{\mathrm{ad}}_X}(D+R)$. Since $[X,D]=-dX$, subtracting $D$ gives precisely the displayed action. Each term has increasing coefficient order. Conjugation preserves coderivations, their square-zero condition, and reduction to $D$, so this is an action on the object set. The group law is composition, transported through exp and log; no unproved BCH construction is needed.

Let $s:\mathfrak{g}\to \mathfrak{g}[1]$ have degree $-1$. The shift lemma in the statement has $q_1(sx)=-s(dx)$ and $q_2(sx,sy)=(-1{)}^{\mathrm{\mid }x\mathrm{\mid }}s[x,y]$, with no higher operations. Hence the shifted curvature at $sR$ is $-s(dR+[R,R]\mathrm{/}2)$, and its gauge vector with parameter $s\xi$ is $s(-d\xi +[\xi ,R])$. Therefore the exact path equation of Completed polynomial gauge paths for filtered Maurer–Cartan elements is $R^{\mathrm{\prime }}=-d\xi +[\xi ,R]$ for compatible polynomial coefficients $R(t)$ and $\xi (t)$, with . The scalar variable has degree zero, and algebraic integration divides $t^j$ by $j+1$ in the usual way.

Fix any such variable parameter $\xi (t)$ and work in the complete degree-zero algebra and write . Starting at $U_0=V_0=\mathrm{id}$, iterate $U_{a+1}(t)=\mathrm{id}+{\int }_0^t\xi (u)U_a(u)du$ and $V_{a+1}(t)=\mathrm{id}-{\int }_0^t{V}_a(u)\xi (u)du$. Multiplication by $\xi$ raises coefficient order by one, so successive differences have order at least $a+1$. Every quotient consequently stabilizes after finitely many iterations as a polynomial. Completeness of $E^0$ gives unique solutions $U^{\mathrm{\prime }}=\xi U$, $V^{\mathrm{\prime }}=-V\xi$, $U(0)=V(0)=\mathrm{id}$. Uniqueness follows from the same integral equation: a difference in $F^a$ must lie in $F^{a+1}$, and the filtration is separated.

Differentiating gives $(VU{)}^{\mathrm{\prime }}=-V\xi U+V\xi U=0$, and its initial value is the identity, so $VU=\mathrm{id}$. Similarly $Z=UV-\mathrm{id}$ satisfies $Z^{\mathrm{\prime }}=\xi Z-Z\xi$ and $Z(0)=0$. Its integral equation is in degree zero, raises coefficient order and forces $Z$ into every $F^a$, so $UV=\mathrm{id}$. Thus $V=U^{-1}$ as completed polynomial endomorphisms.

In the completed tensor target put $H=\mathrm{\Delta }U-(U\otimes U)\mathrm{\Delta }$. The coderivation equation for $\xi$ and the product rule give $H^{\mathrm{\prime }}=(\xi \otimes 1+1\otimes \xi )H$ and $H(0)=0$. Its integral equation forces successive coefficient orders and hence $H=0$. The same argument applied to $U(1_C)-1_C$ gives zero, using $\xi (1_C)=0$; and $(ϵU{)}^{\mathrm{\prime }}=ϵ\xi U=0$ gives $ϵU=ϵ$. Together with step 3.1 these prove that $U(t)$, and in particular $U(1)$, is an actual continuous coaugmented counital coalgebra automorphism congruent to the identity modulo $I$. All equations are quotientwise polynomial equations, not analytic-time claims.

For an initial degree-one deformation $K_0=D+R_0$, set $K(t)=U(t)K_{0}V(t)$. Differentiation gives $K^{\mathrm{\prime }}=[\xi ,K]$, and $K(0)=K_0$. Consequently $R(t)=K(t)-D$ satisfies $R^{\mathrm{\prime }}=-d\xi +[\xi ,R]$, and $K(t{)}^2=U{K}_0^{2}V=0$. Any other completed polynomial solution with the same parameter and initial value differs by $Z^{\mathrm{\prime }}=[\xi ,Z]$, $Z(0)=0$, whose degree-one integral equation raises order and forces $Z=0$ by separatedness of $E^1$. Every elementary shifted gauge path is therefore this actual conjugation path for its own parameter and initial value. Its endpoints are conjugate by the allowed automorphism $U(1)$, and any finite chain of such paths remains conjugate by composition.

Conversely suppose an allowed $G$ carries $D+R_0$ to $D+R_1$. Step 2.1 gives $X=\log G\in {\mathfrak{g}}^0$. Set $U(t)=\exp (tX)$; its series is polynomial modulo each coefficient order, and step 2.1's coalgebra exponential theorem applies coefficientwise in $t$. The path $U(t)(D+R_0)U(t{)}^{-1}-D$ has parameter $\xi (t)=X$, is Maurer–Cartan for every $t$, and joins $R_0$ to $R_1$. Step 2.2 identifies it as an elementary shifted gauge path. Thus every allowed equivalence lies in the gauge relation.

Step 5.1 proves gauge-equivalent elements give equivalent deformations, and step 6.1 proves the converse. Combining this equivalence of relations with the object bijection in step 1.1 proves the claimed bijection on quotient classes, with the full equivalence group and displayed action of step 2.1.

For the tensor carrier, work first at each coefficient quotient $K_n=B\mathrm{/}{I}^n$, on $M_n=A{\otimes }_k{K}_n$ and its tensor coalgebra; its length-$r$ component is $M_n^{{\otimes }_{K_n}r}\cong A^{{\otimes }_{k}r}{\otimes }_k{K}_n$. The suspension and desuspension maps extend $K_n$-linearly, and the sign calculations in [F7] use only the grading, tensor Koszul signs, and multilinear composition; they use no division or other field-specific operation. They therefore apply to arbitrary $K_n$-multilinear Taylor maps, not only scalar extensions of maps over $k$. The same calculation identifies every positive-arity coderivation with the native unsuspended A∞ operations and every coaugmentation- and counit-preserving coalgebra map with the native A∞ morphism, including its composition signs. A filtration-preserving continuous $B$-linear map on the coefficient completion induces compatible $K_n$-linear maps on these quotients; conversely, every compatible family of $K_n$-linear quotient maps defines a filtration-preserving continuous $B$-linear map on the inverse limit. Hence the quotientwise correspondences pass to all continuous completed operations and maps. Reduction to the fixed structure and to the identity is unchanged by the invertible suspension signs. For the symmetric carrier, [F8] is exactly the degree-one coderivation definition; adjoining the coaugmentation and extending a coderivation by zero on it preserves its equation. Its finite unshuffle formulas extend through the coefficient quotients, with all integer denominators invertible over the characteristic-zero base. Thus the objects and full equivalences in step 7.1 are precisely the two asserted fixed-carrier homotopy-algebra deformation problems in their stated conventions.