2.1 $A_\infty$ -algebras and morphisms

$A_\infty$ -algebras are commonly defined as sequences of multilinear maps either of the form $V^{\otimes n} \to W$ with varying degrees or of the form $V[1]^{\otimes n} \to W[1]$ , with all maps being homogeneous. In this section we adopt the latter convention, adopting the notation

\begin{align*} \mathcal{M}^\bullet (V,W) := \prod _{n=0}^\infty \hom _{\ell ^e}^\bullet (V[1]^{\otimes n}, W[1]) \end{align*}

for the graded vector space of these sequences $f = (f_n)_{n\in \mathbb{N}}$ , and we write $\overline {\mathcal{M}}(V,W) \subset \mathcal{M}(V,W)$ for the subspace of sequences with $f_0=0$ . There exist various compositions between sequences in $\mathcal{M}(V,W)$ , which are most cleanly described using the bar construction

\begin{align*} \mathsf{B} V := \left (\bigoplus _{n\in \mathbb{N}} V[1]^{\otimes n}, \Delta \right ) \end{align*}

which is a cofree coaugmented conilpotent coalgebra over $\ell$ with the decomposition coproduct $\Delta$ . The cofreeness implies that every sequence $f \in \mathcal{M}(V,V) \simeq \hom _{\ell ^e}(\mathsf{B} V,V[1])$ induces a unique coderivation $\widetilde f \colon \mathsf{B} V \to \mathsf{B} V$ via the formula (see [Reference TradlerTra08, Lemma 2.3])

\begin{align*} \widetilde f := \sum _{k\in \mathbb{N}} \sum _{i,j\geqslant 0} {\rm id}_{V[1]}^{\otimes i} \otimes f_k \otimes {\rm id}_{V[1]}^{\otimes j}, \end{align*}

and likewise every coderivation defines a sequence in $\mathcal{M}(V,V)$ by composing with the obvious projection $\unicode {x03C0} \colon \mathsf{B} V \to V[1]$ . In terms of the bar construction, $A_\infty$ -algebras are defined as follows.

The $A_\infty$ -condition implies that $\unicode {x03BC}_1^2=0$ , so that $\unicode {x03BC}_1$ makes $A[1]$ into a cochain complex. We say an $A_\infty$ -algebra is compact if the cohomology with respect to $\unicode {x03BC}_1$ is finite dimensional over $\mathbb{C}$ .

Given $V,W\in \operatorname {\mathsf{grMod}} \ell ^e$ , every sequence $f \in \mathcal{M}(V,W) \simeq \hom _{\ell ^e}(\mathsf{B} V,W[1])$ induces a cohomomorphism $\widehat f \colon \mathsf{B} V \to \mathsf{B} W$ between the associated coalgebras via the formula

\begin{align*} \widehat f := \sum _{k\in \mathbb{N}} \sum _{n_1,\ldots ,n_k\in \mathbb{N}} f_{n_1} \otimes \cdots \otimes f_{n_k}, \end{align*}

and every cohomomorphism between bar-coalgebras can be described in this way. These lifts are used to define (pre-)morphisms of $A_\infty$ -algebras.

The morphism condition implies that $f_1\colon A[1] \to B[1]$ is a morphism of cochain complexes. A morphism $f\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ is called a quasi-isomorphism if $f_1$ is a quasi-isomorphism.

By lifting pre-morphisms to cohomomorphisms, one obtains an associative composition between (pre-)morphisms which we will write as

\begin{align*} \diamond \colon \mathcal{M}^0(B,C) \times \mathcal{M}^0(A,B) \to \mathcal{M}^0(A,C),\quad g \diamond f := g \circ \widehat f. \end{align*}

With this composition, $A_\infty$ -algebras form a category $\mathsf{Alg}^\infty$ with morphism spaces ${{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ . We remark that the composition of two quasi-isomorphisms is again a quasi-isomorphism.

Definition 2.4. A unit for an $A_\infty$ -algebra $A = (A,\unicode {x03BC})$ is an injective map $\unicode {x1D7D9} \colon \ell [1] \to A[1]$ such that

\begin{align*} \begin{aligned} \unicode {x03BC}_2(\unicode {x1D7D9}(l),a) = la,\quad \unicode {x03BC}_2(a,\unicode {x1D7D9}(l)) = (-1)^{|a|+1} al \quad &\textrm{for all}\; l\in \ell [1],\ a\in A[1]\\[5pt] \unicode {x03BC}_n(\ldots ,\unicode {x1D7D9}(l),\ldots ) = 0 \quad &\textrm{for all}\; n\neq 2, \end{aligned} \end{align*}

and a triple $(A,\unicode {x03BC},\unicode {x1D7D9})$ is called a unital $A_\infty$ -algebra.

If $A$ and $B$ are unital $A_\infty$ -algebras, with units $\unicode {x1D7D9}_A$ and $\unicode {x1D7D9}_B$ , then one can consider the subset ${{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm un}}(A,B) \subset {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ of morphisms $f$ satisfying

\begin{align*} \begin{aligned} f_1 \circ \unicode {x1D7D9}_A &= \unicode {x1D7D9}_B,\\[5pt] f_{i+j+1} \circ ({\rm id}^{\otimes i} \otimes \unicode {x1D7D9}_A \otimes {\rm id}^{\otimes j}) &= 0\quad \textrm{for all}\; i+j \gt 0, \end{aligned} \end{align*}

which are called unital $A_\infty$ -morphisms. The composition of two unital morphisms is again unital, so unital $A_\infty$ -algebras again form a category $\mathsf{Alg}^{\infty ,{\rm un}}$ . Finally, we consider a weakening of the unital condition in cohomology.

Definition 2.5. A weak unit, or cohomological unit, is a map $\unicode {x1D7D9}\colon \ell [1] \to A[1]$ for which the image is $\unicode {x03BC}_1$ -closed and which satisfies the following relations in $\unicode {x03BC}_1$ -cohomology:

\begin{align*} \unicode {x03BC}_2([\unicode {x1D7D9}(l)],[a]) = [la],\quad \unicode {x03BC}_2([a],[\unicode {x1D7D9}(l)]) = (-1)^{|a|+1}[al]. \end{align*}

One can likewise define a category of cohomologically unital $A_\infty$ -algebras, with morphisms consisting of $A_\infty$ -morphisms $f$ for which $[f_1(\unicode {x1D7D9}(l))] = [\unicode {x1D7D9}(l)]$ .

2.2 Homotopies

$A_\infty$ -algebras are a model for homotopical algebra, and it is therefore natural to consider morphisms up to homotopy. There are various definitions, ranging from algebraic [Reference Lefèvre-HasegawaLH03, Reference KellerKel01] to more continuous variants [Reference Flajolet and SedgewickFS09], which lead to equivalent theories of homotopy. Below we use a variation of the piecewise-smooth model in [Reference Flajolet and SedgewickFS09, § 4.2.1].

For each $f\in \mathsf{PC}^\infty ([0,1])$ , the derivative ${\partial x}/{\partial t}$ is by definition well defined on the smooth locus, and therefore extends to a unique equivalence class in $\mathsf{PC}^\infty ([0,1])$ if it is bounded. We can therefore consider the DG algebra of piecewise differential forms

\begin{align*} \Omega ^\bullet _{[0,1]} := \{ x+y{\rm d} t \mid x,y \in \mathsf{PC}^\infty ([0,1]),\ x\textrm{continuous},\ \tfrac {\partial x}{\partial t}\; \textrm{bounded}\}, \end{align*}

with the usual wedge product and differential ${\rm d} \colon x+y {\rm d} t \mapsto {\partial x}/{\partial t} {\rm d} t$ . Then we consider, for any $A_\infty$ -algebra $(A,\unicode {x03BC})$ , the tensor product $A_\infty$ -algebra

\begin{align*} \big (\Omega ^\bullet _{[0,1]}\otimes A,\ \unicode {x03BC}^\otimes \big ). \end{align*}

where the $A_\infty$ -algebra structure is defined as in [Reference LodayLod11, Proposition 3.7] by

(3) \begin{align} \unicode {x03BC}_1^\otimes &= {\rm d} \otimes {\rm id} + {\rm id} \otimes \unicode {x03BC}_1, \end{align}

(4) \begin{align} \unicode {x03BC}_k^\otimes &= (-\cdots -) \otimes \unicode {x03BC}_k, \end{align}

where $(-\cdots -)$ denotes the multiplication on $k$ elements in $\Omega ^\bullet _{[0,1]}$ , and maps are applied along the isomorphism $(\Omega _{[0,1]}^\bullet \otimes A[1])^{\otimes k} \cong (\Omega _{[0,1]}^\bullet )^{\otimes k} \otimes A[1]^{\otimes k}$ observing the usual Koszul sign conventions. The tensor product fits into a diagram of three quasi-isomorphisms

\begin{align*} A \xrightarrow {\ a \mapsto 1\otimes a\ } \Omega ^\bullet _{[0,1]} \otimes A \xrightarrow {(\mathsf{ev}_0, \mathsf{ev}_1)} A \times A \end{align*}

where $\mathsf{ev}_t((x+y{\rm d} t)\otimes a) = \lim _{t'\to t} x(t') \cdot a$ . These maps make $\Omega ^\bullet _{[0,1]} \otimes A$ into a path-space object as in [Reference Flajolet and SedgewickFS09, Definition 4.2.1], which leads to the following notion of a homotopy.

To make the condition more explicit we can write $H \in \overline {\mathcal{M}}^0(A,\Omega ^\bullet _{[0,1]}\otimes B)$ as $H = H_x + H_y {\rm d} t$ for some coefficient functions $t\mapsto H_x^t\in \overline {\mathcal{M}}^0(A,B)$ and $t\mapsto H_y^t \in \overline {\mathcal{M}}^{-1}(A,B)$ . Then, as observed in [[Reference Flajolet and SedgewickFS09, (4.2.41)], the map $H$ is a homotopy if and only if the following two conditions hold:

(5) \begin{align} H_x^t &\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B) & \textrm{for all}\; t\in [0,1], \textrm{and} \end{align}

(6) \begin{align} \frac {\partial }{\partial t} H_x^t &= \unicode {x03BC}_B \circ (\widehat H_x^t \otimes H_y^t \otimes \widehat H_x^t) + H_y^t \circ \widetilde {\unicode {x03BC}}_A & \textrm{for almost all}\; t\in [0,1]. \end{align}

It follows from a general argument [Reference Flajolet and SedgewickFS09, Proposition 4.2.37] that homotopy defines an equivalence relation between $A_\infty$ -morphisms, which is compatible with the composition. We give a direct proof below, which will also apply in the analytic setting.

Proof. It is clear that $f\sim f$ for any $f\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ via the constant homotopy $H^t = f$ . Likewise, if $f\sim g$ via a homotopy $H = H_x + H_y {\rm d} t$ then $G^t = H^{1-t}_x - H_y^{1-t} {\rm d} t$ defines a homotopy $g\sim f$ . Now if $f\sim g$ and $g\sim h$ via homotopies $H = H_x + H_y{\rm d} t$ and $G= G_x + G_y{\rm d} t$ , then we consider the pre-morphism $G*H \in \overline {\mathcal{M}}^0(A,\Omega ^\bullet _{[0,1]}\otimes B)$ with

\begin{align*} (G*H)^t = \left\{\begin{array}{l@{\quad}l} H_x^{2t} + 2 H_y^{2t}{\rm d} t & t\leqslant 1/2\\[5pt] G_x^{2t-1} + 2 G_y^{2t-1} & t\gt 1/2. \end{array} \right.\end{align*}

We remark that this is well defined: the coefficient functions are piecewise-smooth, $(G*H)_x$ is continuous since $H_x^1 = g=G_x^0$ by assumption, and the derivative of $(G*H)_x$ is bounded by the derivatives of $G_x$ and $H_x$ . By construction, $(G*H)_x^t \in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ for all $t\in [0,1]$ , showing (5). The identity (6) holds for almost all $t \in [0,1/2)$ since

\begin{align*} \begin{aligned} \frac {\partial }{\partial t} (G*H)_x^t &= 2\cdot \left(\unicode {x03BC} \circ \left(\widehat H_x^{2t} \otimes H_y^{2t} \otimes \widehat H_x^{2t}\right) + H_y^{2t} \circ \widetilde {\unicode {x03BC}}\right) \\[5pt] &= \unicode {x03BC} \circ \left({\widehat {(G*H)}}_x^t \otimes (G*H)_y^t \otimes {\widehat {(G*H)}}_x^t\right) + (G*H)_y^t \circ \widetilde {\unicode {x03BC}}, \end{aligned} \end{align*}

and a similar computation shows that the identity holds for almost all $t\in (1/2,1]$ . Hence $G*H$ is a homotopy between $f = \mathsf{ev}_0 \circ (G*H)$ and $h = \mathsf{ev}_1\circ (G*H)$ .

We say that two morphisms $f,g$ in $\mathsf{Alg}^\infty$ are homotopy inverse if $f\diamond g$ and $g\diamond f$ are homotopic to the identity. In this case $f$ and $g$ are said to be homotopy equivalences and their source/target $A_\infty$ -algebras are homotopy equivalent.

2.3 Minimal models

Recall that an $A_\infty$ -algebra $(A,\unicode {x03BC})$ is called minimal if $\unicode {x03BC}_1=0$ . It is often convenient to replace an $A_\infty$ -algebra by a homotopy equivalent minimal one: a minimal model.

Definition 2.11. A minimal model for $A$ is a minimal ${\rm H}(A) \in {\mathsf{Alg}^\infty }$ with quasi-isomorphisms $P = P_A \in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,{\rm H}(A))$ and $I = I_A \in {{\rm Hom}}_{\mathsf{Alg}^\infty }({\rm H}(A),A)$ such that:

(7) \begin{align} P\diamond I = {\rm id}_{{\rm H}(A)}\quad \textrm{and}\quad I\diamond P \sim {\rm id}_A. \end{align}

It is well known that every $A_\infty$ -algebra admits a minimal model, which can be constructed explicitly using Kadeishvili’s homotopy transfer formula [Reference KadeishviliKad80] applied to the cohomology algebra of $A$ .

Given $A_\infty$ -algebras $A,B\in {\mathsf{Alg}^\infty }$ with a fixed choice of minimal models ${\rm H}(A)$ and ${\rm H}(B)$ , we obtain for every morphism $f\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ an induced map

\begin{align*} {\rm H}(f) := P_B \diamond f \diamond I_A \in {{\rm Hom}}_{\mathsf{Alg}^\infty }({\rm H}(A),{\rm H}(B)). \end{align*}

It is well known that if $f$ is a quasi-isomorphism, then ${\rm H}(f)$ is an isomorphism. The inverse at the level of minimal models can be constructed using a ‘tree formula’ that we recall below.

Note that there is a unique tree in $\mathcal{O}(1) = \mathcal{O}(1,0)$ and that for $n\gt 1$ each tree $T\in \mathcal{O}(n)$ is uniquely determined by the ordered list of rooted subtrees $T_1,\ldots ,T_k$ starting in the internal node of $T$ connected to the root.

For any pre-morphism $f\in \overline {\mathcal{M}}^0(A,B)$ and map $g_1 \colon B[1]\to A[1]$ we define a map for every $T\in \mathcal{O}(n)$ as follows: for the unique tree $T\in \mathcal{O}(1)$ we set $g_T = g_1$ and for general $T\in \mathcal{O}(n)$ we set

\begin{align*} g_T = -g_1 \circ f_k \circ (g_{T_1} \otimes \cdots \otimes g_{T_k}), \end{align*}

where $T_1,\ldots ,T_k$ are the subtrees starting in the internal node of $T$ connected to the root. The following result is classical, and can be easily verified using the recursive formula.

Proposition 2.13. Let $f\in \overline {\mathcal{M}}^0(A,B)$ be a pre-morphism such that $f_1\colon A[1]\to B[1]$ admits an inverse $g_1\colon B[1]\to A[1]$ . Then the pre-morphism $g\in \overline {\mathcal{M}}(B,A)$ defined by

\begin{align*} g_n = \sum _{T\in \mathcal{O}(n)} g_T \end{align*}

is an inverse to $f$ . If $f$ is, moreover, an $A_\infty$ -morphism, then $g$ is again an $A_\infty$ -morphism.

Given a quasi-isomorphism $f$ , the composition $g = I_A \diamond {\rm H}(f)^{-1}\diamond P_B$ is a homotopy inverse to $f$ . To see this, note that there are identities

(8) \begin{align} g\diamond f=I_A \diamond {\rm H}(f)^{-1}\diamond P_B \diamond f \,\sim \, I_A \diamond {\rm H}(f)^{-1}\diamond \underbrace {P_B \diamond f \diamond I_A}_{{\rm H}(f)} \diamond P_A \,\sim \, I_A \diamond P_A \sim {\rm id}_A, \end{align}

(9) \begin{align} f\diamond g=f \diamond I_A \diamond {\rm H}(f)^{-1}\diamond P_B \,\sim \, I_B \diamond \underbrace {P_B \diamond f \diamond I_A}_{{\rm H}(f)} \diamond {\rm H}(f)^{-1} \diamond P_B \,\sim \, I_B \diamond P_B \sim {\rm id}_B, \end{align}

induced by the homotopies $I_A\circ P_A \sim {\rm id}_A$ and $I_B\circ P_B \sim {\rm id}_B$ . Hence, every quasi-isomorphism is also a homotopy equivalence.

2.4 Analytic algebras and morphisms

We now consider the category $\operatorname {\mathsf{grNMod}} {\ell ^e}$ of graded normed $\ell ^e$ -bimodules, by which we mean pairs $(V,{{\left \|\cdot \right \|}}_V)$ of a bimodule $V\in \operatorname {\mathsf{grMod}} \ell ^e$ and a norm ${{\left \|\cdot \right \|}}_V \colon V\to \mathbb{R}$ on the underlying ungraded vector space such that for any $l,r\in \ell$ the multiplication $v \mapsto l v r$ is bounded in ${{\left \|\cdot \right \|}}_\ell$ . Between any $V,W \in \operatorname {\mathsf{grNMod}} {\ell ^e}$ there is a graded normed vector space

\begin{align*} \hom _{\ell ^e}^{{\rm cont}}(V,W) := \{ f\in \hom _{\ell ^e}(V,W) \mid {\left \|f\right \|}_{{\rm op}} \lt \infty \} \end{align*}

of bounded morphisms with respect to the operator norm ${\left \|f\right \|}_{{\rm op}} := \sup _{\|v\|_V=1} \|f(v)\|_W$ . The shift naturally extends to a functor $[1]\colon \operatorname {\mathsf{grNMod}} {\ell ^e} \to \operatorname {\mathsf{grNMod}}{\ell ^e}$ , and for any pair $V,W\in \operatorname {\mathsf{grNMod}} {\ell ^e}$ the tensor product $V\otimes W$ is again normed via the projective tensor norm

\begin{align*} \|z\|_\pi := \inf \{\textstyle \sum _i \|v_i\|_V \|w_i\|_W \mid \textstyle z = \sum _i v \otimes _{\mathbb{C}} w \}, \end{align*}

where the infimum is over all presentations of an element $z\in V\otimes W$ as a sum of pure tensors in the tensor product $V\otimes _{\mathbb{C}} W$ over $\mathbb{C}$ . If $f_i\colon V_i \to W_i$ are bounded bimodule morphisms between $V_1,V_2,W_1,W_2 \in \operatorname {\mathsf{grNMod}}{\ell ^e}$ , the tensor product $f_1\otimes f_2$ has operator norm

\begin{align*} {\left \|f_1\otimes f_2\right \|}_{{\rm op}} = {\left \|f_1\right \|}_{{\rm op}} {\left \|f_2\right \|}_{{\rm op}} \lt \infty , \end{align*}

and is therefore a bounded bimodule map $V_1\otimes V_2 \to W_1\otimes W_2$ . Henceforth, we will drop the subscripts on the above norms to reduce clutter.

Now given a normed $\ell$ -bimodule $V\in \operatorname {\mathsf{grNMod}} {\ell ^e}$ and $r \in \mathbb{R}$ with $r \gt 0$ we consider the following normed version of the bar construction:

\begin{align*} \mathsf{B}(V,r) := (\mathsf{B} V, {{\left \|\cdot \right \|}}_r),\quad {\left \|\textstyle \sum v_n\right \|}_r := \sup _{n\in \mathbb{N}} {\left \|v_n\right \|}r^{-n}, \end{align*}

noting that the supremum in the definition of ${{\left \|\cdot \right \|}}_r$ is finite for any finite sum in $\mathsf{B} V$ . Bounded maps out of these normed $\ell$ -bimodules are characterised by the following lemma.

Proof. (i) $\implies$ (ii). Without loss of generality we may assume that $f_0=0$ . Let $C\gt 0$ be such that ${\left \|f_n\right \|} \lt C^n$ for all $n\geqslant 1$ , and fix some $r \lt C^{-1}$ . Then the map $f \colon \mathsf{B}(V,r) \to W[1]$ satisfies, for every $v = \sum _{n\geqslant 1} v_n \in \mathsf{B}(V,r)$ ,

\begin{align*} {\left \|f(v)\right \|} \leqslant \sum _{n\geqslant 1} {\left \|f_n(v_n)\right \|} \lt \sum _{n\geqslant 1} C^n {\left \|v_n\right \|} \leqslant \sum _{n\in \mathbb{N}} (Cr)^n {\left \|v\right \|}_r = \frac {{\left \|v\right \|}_r}{1-Cr}, \end{align*}

where the final equality for the geometric series holds because $Cr \lt 1$ . It follows that $f$ has operator norm bounded by $(1-Cr)^{-1} \lt \infty$ .

(ii) $\implies$ (i). If $f \colon \mathsf{B}(V,r) \to W[1]$ is bounded in the operator norm by some constant $K\gt 1$ , then for each $n\geqslant 1$ and $v_n \in V[1]^{\otimes n}$ the map $f_n$ satisfies

\begin{align*} {\left \|f_n(v_n)\right \|} = {\left \|f(v_n)\right \|} \leqslant K {\left \|v_n\right \|} r^{-n} \leqslant (K/r)^n {\left \|v_n\right \|}. \end{align*}

Hence, picking any $C\gt K/r$ it follows that ${\left \|f_n\right \|} \lt C^n$ for all $n\geqslant 1$ .

(i) $\implies$ (iii). Let $C\gt 0$ be such that ${\left \|f_n\right \|} \lt C^n$ for all $n\geqslant 1$ , fix $r'\gt 0$ and let $K = \min \{r',1\}$ . Then assuming $f_0=0$ and expanding the formula for $\widehat f$ , we find that, for every $v = \sum _{n\in \mathbb{N}} v_n \in \mathsf{B} V$ ,

\begin{align*} \begin{aligned} {\left \|\widehat f(v)\right \|}_{r'} &\leqslant \sum _{n\geqslant 1} \sum _{n_1+\cdots +n_k = n} {\left \|f_{n_1}\otimes \cdots \otimes f_{n_k}(v_n)\right \|} (r')^{-k} \\[5pt] &\leqslant \sum _{n\in \mathbb{N}} \sum _{n_1+\cdots +n_k = n} C^n K^{-k} {\left \|v_n\right \|} \\[5pt] &\leqslant \sum _{n\in \mathbb{N}} ({\#}\textrm{ partitions of n})\cdot \left (\frac CK\right )^n {\left \|v_n\right \|}. \end{aligned} \end{align*}

The number of partitions of $n$ is bounded by the number of ordered partitions (also known as compositions), of which there are $2^{n-1}$ , and it is therefore, in particular, bounded by $2^n$ . Picking $r\lt {K}/{2C}$ , the geometric series with terms $({2rC}/{K})^n$ again converges, which shows that

\begin{align*} {\left \|\widehat f(v)\right \|}_{r'} \leqslant \sum _{n\in \mathbb{N}} \left (\frac {2rC}{K}\right )^n {\left \|v\right \|}_r = \frac {K}{K-2rC} {\left \|v\right \|}_r. \end{align*}

As $r$ can be chosen independently of $v$ , it follows that $\widehat f \colon \mathsf{B}(V,r) \to \mathsf{B}(V,r')$ is bounded.

(i) $\implies$ (iv). Let $C\gt 1$ be such that ${\left \|f_n\right \|}\lt C^n$ for all $n\in \mathbb{N}$ , fix $r'\gt 0$ and let $K=\min \{r',1\}$ . Then assuming $f_0=0$ and expanding the formula for $\widetilde f$ yields, for every $v=\sum _{n\in \mathbb{N}} v_n \in \mathsf{B} V$ ,

\begin{align*} \begin{aligned} {\left \|\widetilde f(v)\right \|}_{r'} &\leqslant \sum _{n\in \mathbb{N}} \sum _{i+k+j=n} {\left \|({\rm id}^{\otimes i} \otimes f_k \otimes {\rm id}^{\otimes j})(v_n)\right \|} (r')^{-(i+j+1)} \\[5pt] &\leqslant \sum _{n\in \mathbb{N}} \sum _{i+k+j=n} C^k K^{-(i+j+1)}{\left \|v_n\right \|} \\[5pt] &\leqslant \sum _{n\in \mathbb{N}} n^3 \left (\frac CK\right )^n{\left \|v_n\right \|}. \end{aligned} \end{align*}

Picking $q\gt 0$ such that $n^3 \lt q^n$ for all $n$ and an $r\lt {K}/{qC}$ , it again follows that

\begin{align*} {\left \|\widetilde f(v)\right \|}_{r'} \leqslant \sum _{n\in \mathbb{N}} \left (\frac {rqC}{K}\right )^n {\left \|v\right \|}_r = \frac {K}{K - rqC}{\left \|v\right \|}_r, \end{align*}

and hence $\widetilde f$ is bounded when considered as a linear map $\mathsf{B}(V,r) \to \mathsf{B}(V,r')$ .

(iii), (iv) $\implies$ (ii). For any $r'$ , the projection $\unicode {x03C0}\colon \mathsf{B}(V,r') \to V[1]$ is obviously bounded, so if a map $\widehat f$ or $\widetilde f$ is bounded it follows that $f = \unicode {x03C0} \circ \widehat f = \unicode {x03C0} \circ \widetilde f$ is again bounded.

We define the space of analytic sequences of multilinear maps between $V$ and $W$ as

\begin{align*} {\mathcal{A}}(V,W) = \bigcup _{r\gt 0} \hom _{\ell ^e}^{{\rm cont}}(\mathsf{B}(V,r), W[1]) \subset \mathcal{M}(V,W), \end{align*}

and note that its elements are sequences $f = (f_n)_{n\in \mathbb{N}}$ satisfying one of the equivalent conditions in the above lemma. As before, we write $\overline {\mathcal{A}}(V,W) \subset {\mathcal{A}}(V,W)$ for the sequences with $f_0=0$ .

Given two analytic $A_\infty$ -algebras $A$ and $B$ , we say that a pre-morphism $f$ is analytic if it lies in the subspace $f\in {\mathcal{A}}^0(A,B)$ . Lemma2.16 implies that the composition of analytic pre-morphisms is well defined.

Lemma 2.21. For any three analytic $A_\infty$ -algebras $A,B,C$ , the composition restricts to a map

\begin{align*} \diamond \colon {\mathcal{A}}^0(B,C) \times {\mathcal{A}}^0(A,B) \to {\mathcal{A}}^0(A,C). \end{align*}

Proof. Given $f\in {\mathcal{A}}^0(A,B)$ and $g\in {\mathcal{A}}^0(B,C)$ , it follows from Lemma2.16 that there exist $r_1,r_2\gt 0$ such that the maps

\begin{align*} g \colon \mathsf{B}(B,r_1) \to C[1],\quad \widehat f \colon \mathsf{B}(A,r_2) \to \mathsf{B}(B,r_1) \end{align*}

are bounded. Then the composition $g \diamond f=g \circ \widehat f$ is also bounded.

We conclude that analytic $A_\infty$ -algebras again form a category $\mathsf{Alg}^{\infty ,{\rm an}}$ , with morphism spaces

\begin{align*} {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(A,B) := \overline {\mathcal{A}}(A,B) \cap {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B), \end{align*}

with an obvious forgetful functor ${\mathsf{Alg}^{\infty ,{\rm an}}} \to {\mathsf{Alg}^\infty }$ .

2.5 Analytic homotopies

To generalise the definition of homotopies to analytic $A_\infty$ -algebras we introduce a norm on the path object. The algebra $\mathsf{PC}^\infty ([0,1])$ is bounded for the essential supremum norm ${{\left \|\cdot \right \|}}_{\infty }$ on $L^\infty ([0,1])$ , and likewise $\Omega ^\bullet _{[0,1]}$ is bounded in the norm

\begin{align*} {\left \|x+y{\rm d} t\right \|} = {\left \|x\right \|}_\infty + {\left \|\tfrac {\partial x}{\partial t}\right \|}_\infty + {\left \|y\right \|}_\infty . \end{align*}

For $A\in {\mathsf{Alg}^{\infty , {\rm an}}}$ there is an induced tensor norm on $\Omega ^\bullet _{[0,1]}\otimes A$ and one checks easily that this is an analytic $A_\infty$ -algebra, using (3) and (4). The following is then a natural generalisation of homotopy between analytic $A_\infty$ -morphisms.

We will use an explicit sufficient condition for a homotopy to be analytic in terms of the coefficient functions. Suppose $H = H_x + H_y {\rm d} t$ , then for all $k\geqslant 1$ we have a bound

\begin{align*} {\left \|H_k\right \|} \leqslant {\left \|H_{x,k}\right \|}_\infty + {\left \|\tfrac {\partial }{\partial t} H_{x,k}\right \|}_\infty + {\left \|H_{y,k}\right \|}_\infty , \end{align*}

where ${{\left \|\cdot \right \|}}_\infty$ denotes the essential supremum of the operator norms over the interval, and bounds

\begin{align*} {\left \|H_{x,k}\right \|}_\infty \leqslant {\left \|H_k\right \|},\quad {\left \|\tfrac {\partial }{\partial t}H_{x,k}\right \|}_\infty \leqslant {\left \|H_k\right \|},\quad {\left \|H_{y,k}\right \|}_\infty \leqslant {\left \|H_k\right \|}. \end{align*}

Hence $H$ is analytic if and only if there is a $C\gt 0$ such that for each $k\geqslant 1$ the functions $H_{x,k}$ , $\tfrac {\partial }{\partial t} H_{x,k}$ , and $H_{y,k}$ are bounded almost everywhere by $C^k$ . We use this to prove the following.

Proof. Given $f\in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(A,B)$ , the constant homotopy $H=1\otimes f$ has norms ${\left \|(1\otimes f)_k\right \|} = {\left \|f_k\right \|}$ and is therefore analytic. Hence $f\sim _{{\rm an}} f$ always holds. Likewise, if $f\sim _{{\rm an}} g$ with analytic homotopy $H = H_x + H_y {\rm d} t$ , then the coefficient functions of the homotopy $G^t = H_x^{1-t} - H_y^{1-t} {\rm d} t$ have norms

\begin{align*} {\left \|G_{x,k}\right \|}_\infty = {\left \|H_{x,k}\right \|}_\infty ,\quad {\left \|\tfrac {\partial }{\partial t} G_{x,k}\right \|}_\infty = {\left \|\tfrac {\partial }{\partial t} H_{x,k}\right \|}_\infty ,\quad {\left \|G_{y,k}\right \|}_\infty = {\left \|H_{y,k}\right \|}_\infty . \end{align*}

Hence $G$ is again analytic and provides an analytic homotopy $g\sim _{{\rm an}} f$ . Finally, suppose $f\sim _{{\rm an}} g$ and $g\sim _{{\rm an}} h$ via analytic homotopies $H$ and $G$ , then we have

\begin{align*} \begin{gathered} {\left \|(G*H)_{x,k}\right \|}_\infty = \max \{{\left \|H_{x,k}\right \|}_\infty ,{\left \|G_{x,k}\right \|}_\infty \},\quad {\left \|(G*H)_{x,k}\right \|}_\infty=2 \max \{{\left \|\tfrac {\partial }{\partial t}H_{x,k}\right \|}_\infty ,{\left \|\tfrac {\partial }{\partial t}G_{x,k}\right \|}_\infty \},\\[5pt] {\left \|(G*H)_{x,k}\right \|}_\infty=2 \max \{{\left \|H_{y,k}\right \|}_\infty ,{\left \|G_{y,k}\right \|}_\infty \}.\quad \end{gathered} \end{align*}

Since the coefficient functions of $H$ and $G$ are essentially bounded by $C^k$ for some $C\gt 0$ , the coefficient functions of $G*H$ are essentially bounded by $(2C)^k$ . Hence, $G*H$ is an analytic homotopy $f\sim _{{\rm an}} h$ .

The relation $\sim _{{\rm an}}$ is again compatible with $\diamond$ , because the compositions used in Lemma2.10 are analytic. Hence we obtain an equivalence relation on morphisms in $\mathsf{Alg}^{\infty ,{\rm an}}$ which is compatible with composition. In particular, we can speak of analytic homotopy inverse maps and analytic homotopy equivalences as before.

2.6 Analytic minimal models

We define an analogue of a minimal model in the analytic setting by requiring all the structures involved to be analytic.

Definition 2.24. An analytic minimal model for $A\in {\mathsf{Alg}^{\infty ,{\rm an}}}$ is a minimal ${\rm H}(A) \in {\mathsf{Alg}^{\infty ,{\rm an}}}$ with quasi-isomorphisms $P \in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(A,{\rm H}(A))$ and $I \in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}({\rm H}(A),A)$ such that:

(10) \begin{align} P\diamond I = {\rm id}_{{\rm H}(A)}\quad \textrm{and}\quad I\diamond P \sim _{{\rm an}} {\rm id}_A. \end{align}

In contrast to the non-analytic case, not every analytic $A_\infty$ -algebra has an analytic minimal model. Indeed, it is in general not even possible to find chain maps $I_1$ and $P_1$ providing a bounded splitting of its cohomology. Nonetheless, the construction in [Reference TuTu14] shows that the homotopy transfer formula can be used to construct analytic minimal models in geometric settings, which we will see in § 5.

We conclude by generalising some results about (homotopy) inverses from § 2.3.

Proof. By assumption there exists a common constant $C\gt 1$ such that ${\left \|g_1\right \|} \lt C$ and ${\left \|f_n\right \|} \lt C^n$ for all $n\geqslant 1$ . We now claim that for any tree $T \in \mathcal{O}(n,d)$ the norm of $g_T$ is bounded by ${\left \|g_T\right \|} \lt C^{2n+2d-1}$ . This follows by induction: for the base case $T\in \mathcal{O}(1) = \mathcal{O}(1,0)$ , we have

\begin{align*} {\left \|g_T\right \|} = {\left \|g_1\right \|} \lt C=C^{2+0-1}, \end{align*}

while for $n\gt 1$ any tree $T\in \mathcal{O}(n)$ determined by subtrees $T_i \in \mathcal{O}(n_i,d_i)$ such that $\sum n_i = n$ and $\sum d_i = d-1$ , we have

\begin{align*} {\left \|g_T\right \|} \leqslant {\left \|g_1\right \|}{\left \|f_k\right \|}{\left \|g_{T_1}\otimes \cdots \otimes g_{T_k}\right \|} \lt C \cdot C^k \cdot C^{2\sum n_i+2\sum d_i - k} = C^{2n+2d -1}, \end{align*}

assuming the claim holds for all $n_i\lt n$ and $d_i \lt d$ ; this shows the identity for each $T$ . To derive the analytic bound we note that any $T\in \mathcal{O}(n)$ has valency $\geqslant 3$ at internal nodes, and therefore has at most $n-1$ internal nodes. Hence, $\mathcal{O}(n)$ consists of the subsets $\mathcal{O}(n,d)$ with $d=1,\ldots , n-1$ internal nodes. Any $T\in \mathcal{O}(n,d)$ has $n+d$ total vertices, and is therefore bounded by the Catalan number $\mathcal{C}_{n+d-1} = {1}/{n+d}\binom {2n+2d}{n+d}$ , which counts the number of all planar rooted trees with $n+d$ vertices (see e.g. [Reference FukayaFuk09, § I.3.21]). The cardinality of $\mathcal{O}(n)$ is therefore bounded by

\begin{align*} |\mathcal{O}(n)| \leqslant \sum _{d=1}^{n-1} \mathcal{C}_{n+d-1} \leqslant (n-1) \mathcal{C}_{2n-2} = \tfrac {n-1}{2n-1}\binom {4n-4}{2n-2} \leqslant \sum _{j=0}^{4n-4} \binom {4n-4}{j} = 2^{4n-4} \leqslant 16^n. \end{align*}

We therefore find the following bound on the norm of $g_n$ :

\begin{align*} {\left \|g_n\right \|} \leqslant \sum _{d=0}^{n-1} \sum _{T\in \mathcal{O}(n,d)} \|g_T\| \lt \sum _{d=0}^{n-1} \sum _{T\in \mathcal{O}(n,d)} C^{2n+2d-1} \leqslant |\mathcal{O}(n)| \cdot C^{4n} \lt (16C^4)^n. \end{align*}

Because the constant $16C^4$ is independent of $n$ , it follows that $g$ is analytic.

Note that if the $A_\infty$ -algebras $A$ and $B$ are finite dimensional, then the continuity of $g_1$ is automatic. The above now implies that the morphism between minimal models induced by an analytic quasi-isomorphism is invertible in the analytic category.

If $f$ is an analytic quasi-isomorphism between compact analytic $A_\infty$ -algebras admitting minimal models, then $g = I_A \diamond {\rm H}(f)^{-1} \diamond P_B$ is an analytic homotopy inverse, since the homotopies (8) and (9) are again analytic. Hence for such analytic $A_\infty$ -algebras, analytic quasi-isomorphisms and analytic homotopy equivalences are the same.

Finally, we show that the existence of minimal models only depends on the analytic homotopy equivalence class of an $A_\infty$ -algebra.

Proof. By assumption there are maps $f \in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(A,B)$ and $g\in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(B,A)$ which are homotopy inverse. Then the composition $g\diamond f \in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}^{{\rm an}}(A,A)$ is a homotopy equivalence, and hence a quasi-isomorphism. If $A$ admits a minimal model, it follows from Lemma2.26 that $g\diamond f$ admits an analytic homotopy inverse $T = I_A \diamond {\rm H}(g\diamond f)^{-1} \diamond P_A$ . We see that

\begin{align*} (P_A\diamond T\diamond g) \diamond (f\diamond I_A) = P_A \diamond I_A \diamond {\rm H}(g\diamond f)^{-1} \diamond {\rm H}(g\diamond f) = {\rm id}_{{\rm H}(A)}, \end{align*}

so that $P_A\diamond T\diamond g$ is left-inverse to $f\diamond I_A$ . To show it is homotopy right-inverse, first note that $g\diamond f \sim _{{\rm an}} {\rm id}_A$ implies that $T$ is homotopic to the identity via

\begin{align*} T=I_A \diamond {\rm H}(g\diamond f)^{-1} \diamond P_A \sim _{{\rm an}} I_A \diamond {\rm H}(g\diamond f)^{-1} \diamond P_A \diamond g\diamond f \diamond I_A \diamond P_A \sim _{{\rm an}} = I_A \diamond P_A \sim _{{\rm an}} {\rm id}_A. \end{align*}

Since $f\diamond g\sim _{{\rm an}} {\rm id}_B$ also holds, the claim then follows from the chain of homotopies

\begin{align*} (f\diamond I_A) \diamond (P_A\diamond T\diamond g) \sim _{{\rm an}} f \diamond T \diamond g \sim _{{\rm an}} f\diamond g \sim _{{\rm an}} {\rm id}_B. \end{align*}

Hence the maps $I_B = f\diamond I_A$ and $P_B = P_A\diamond T\diamond g$ make ${\rm H}(A)$ into a minimal model for $B$ .

3.1 $A_\infty$ -bimodules

Given an $A_\infty$ -algebra $A \in {\mathsf{Alg}^\infty }$ , $A$ -bimodules and the morphisms between them can be defined via double sequences of morphisms in the graded vector space

\begin{align*} \mathcal{M}_A(M,N) := \prod _{i,j=0}^\infty \hom _{\ell ^e}(A[1]^{\otimes i} \otimes M[1] \otimes A[1]^{\otimes j}. N[1]). \end{align*}

We will use the notation $\unicode {x03C1} = (\unicode {x03C1}_{i,j})_{i,j\in \mathbb{N}}$ for homogeneous elements of $\mathcal{M}_A(M,N)$ , and when applying such a map to elements we will underline the element in $M[1]$ for clarity. As before, this space of sequences can be identified with the morphisms out of a bar construction

\begin{align*} \mathsf{B}_A M := \mathsf{B} A \otimes M[1] \otimes \mathsf{B} A = \bigoplus _{i,j=0}^\infty A[1]^{\otimes i} \otimes M[1]\otimes A[1]^{\otimes j}, \end{align*}

which is naturally a cofree cobimodule over the coalgebra $\mathsf{B} A$ . In [Reference TradlerTra08, Lemmas 3.3, 4.2] it is shown that one can lift any map $\unicode {x03C1} \in \mathcal{M}_A(M,N)$ to a cohomomorphism $\widehat {\unicode {x03C1}}\colon \mathsf{B}_AM \to \mathsf{B}_AN$ , and to a coderivation $\widetilde {\unicode {x03C1}} \colon \mathsf{B}_AM \to \mathsf{B}_A M$ if $M=N$ . These maps are explicitly given by the following formulas:

\begin{align*} \begin{aligned} \widehat {\unicode {x03C1}} &:= \sum _{i,j\geqslant 0} {\rm id}^{\otimes i} \otimes \underline {\unicode {x03C1}} \otimes {\rm id}^{\otimes j} \\[5pt] \widetilde {\unicode {x03C1}} &:= \sum _{i,j} {\rm id}^{\otimes i} \otimes \underline {\unicode {x03C1}} \otimes {\rm id}^{\otimes j} + \sum _{i,j,k\geqslant 0} {\rm id}^{\otimes i} \otimes \unicode {x03BC}_A \otimes {\rm id}^{\otimes k} \otimes \underline {{\rm id}}_{M[1]} \otimes {\rm id}^{\otimes j} \\[5pt] &\quad + \sum _{i,j,k\geqslant 0}{\rm id}^{\otimes i} \otimes \underline {{\rm id}}_{M[1]} \otimes {\rm id}^{\otimes k} \otimes \unicode {x03BC}_A \otimes {\rm id}^{\otimes j}, \end{aligned} \end{align*}

where the part of the maps landing in $M[1]$ is again underlined. These lifts are used to define $A_\infty$ -bimodule structures and their morphisms, as follows.

The main examples of bimodules which we will consider are the following; proofs that these are bimodules can be found in [Reference TradlerTra08, Lemma 5.1].

Example 3.2. The diagonal bimodule $A_\Delta = (A,\unicode {x03BD}_\Delta )$ is defined by the map

\begin{align*} \unicode {x03BD}_{\Delta ,i,j}(a,\underline b,c) = \unicode {x03BC}_{i+j+1}(a,b,c). \end{align*}

Example 3.3. The dual bimodule $A^\vee = ({{\rm Hom}}_{\mathbb{C}}(A,{\mathbb{C}}), \unicode {x03BD}_\Delta ^\vee )$ is defined by the sequence of maps

\begin{align*} \unicode {x03BD}_{\Delta ,i,j}^\vee (a, \underline b,c)(d) = (-1)^K b(\mu _{i+j+1}(c,d,a)), \end{align*}

where $K= |a|(|b| + |c| + |d|) + |b|$ is a Koszul sign.

Bimodules over an $A_\infty$ -algebra $A$ form a DG category with the following morphism complexes.

Definition 3.4. A pre-morphism between $A_\infty$ -bimodules $(M,\unicode {x03BD}_M)$ and $(N,\unicode {x03BD}_N)$ is a cochain in

\begin{align*} \hom _{A-A}^\bullet (M,N) := \left (\mathcal{M}_A^\bullet (M, N),\delta \right ),\quad \unicode {x03B4}(\unicode {x03C1}) = \unicode {x03BD}_N \circ \widehat {\unicode {x03C1}} - (-1)^{|\unicode {x03C1}|} \unicode {x03C1} \circ \widetilde {\unicode {x03BD}}_M. \end{align*}

The degree $0$ cocycles form the subspace of morphisms ${{\rm Hom}}_{A-A}(M,N) \subset \hom _{A-A}^0(M,N)$ .

Given $\unicode {x03C1} \in \hom _{A-A}(M,N)$ and $\unicode {x03C4} \in \hom _{A-A}(L,M)$ the composition is defined as $\unicode {x03C1} \diamond \unicode {x03C4} = \unicode {x03C1} \circ \widehat {\unicode {x03C4}}$ . This composition then yields a well-defined DG category $\operatorname {\mathsf{Mod}}^\infty _{A-A}$ of $A_\infty$ -bimodules.

Given a morphism $f \in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ of $A_\infty$ -algebras $A,B\in {\mathsf{Alg}^\infty }$ , there is a DG functor

\begin{align*} f^\sharp \colon \operatorname {\mathsf{Mod}}^\infty _{B-B}\to \operatorname {\mathsf{Mod}}^\infty _{A-A}, \end{align*}

mapping a $B$ -bimodule $(M,\unicode {x03BD})$ to $(M,f^\sharp \unicode {x03BD}) = (M,\unicode {x03BD} \circ (\widehat f \otimes \underline {{\rm id}}_{M[1]} \otimes \widehat f))$ and mapping a $B$ -bimodule pre-morphism $\unicode {x03C1}$ to the map $f^\sharp \unicode {x03C1}$ defined by the composition

\begin{align*} \mathsf{B}_A M = \mathsf{B} A \otimes M[1] \otimes \mathsf{B} A \xrightarrow {\widehat f \otimes \underline {{\rm id}}_{M[1]} \otimes \widehat f} \mathsf{B} B \otimes M[1] \otimes \mathsf{B} B \xrightarrow {\unicode {x03C1}} N[1]. \end{align*}

For a proof that this defines a DG functor $f^\sharp$ and that this is functorial in $f$ , see [Reference GanatraGan12, § 2.8].

For any $f \in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ , there are $A_\infty$ -bimodule morphisms $f_\Delta \in {{\rm Hom}}_{A-A}(A_\Delta ,f^\sharp B_\Delta )$ and $f_\Delta ^\vee \in {{\rm Hom}}_{A-A}(f^\sharp B_\Delta ^\vee , A_\Delta ^\vee )$ defined in components by

\begin{align*} \begin{aligned} (f_\Delta )_{i,j}(a,b,c) &= f_{i+1+j}(a,b,c),\\[5pt] (f_\Delta ^\vee )_{i,j}(a,\underline {b},c)(d) &= (-1)^K b (f_{i+1+j} (c, d, a)), \end{aligned} \end{align*}

where $a \in A^{\otimes i}$ , $c\in A^{\otimes j}$ , and $K = |a|(|b|+|c|+|d|)$ is a Koszul sign. Given any pre-morphism $\unicode {x03C1} \in \hom _{B-B}(B_\Delta ,B_\Delta ^\vee )$ there is a pre-morphism $f^*\unicode {x03C1} \in \hom _{A-A}(A_\Delta ,A_\Delta ^\vee )$ given by

\begin{align*} f^*\unicode {x03C1} \colon \mathsf{B}_A A_\Delta \xrightarrow {\widehat f_\Delta } \mathsf{B}_A f^\sharp B_\Delta \xrightarrow {\widehat {f^\sharp \unicode {x03C1}}} \mathsf{B}_A f^\sharp B_{\Delta }^\vee \xrightarrow {{\widehat f}^\vee _\Delta } A_\Delta ^\vee . \end{align*}

This defines a chain map $f^*\colon \hom _{B-B}(B_\Delta ,B_\Delta ^\vee ) \to \hom _{A-A}(A_\Delta ,A_\Delta ^\vee )$ for each $f$ , since $f^\sharp$ is a DG functor and $f_\Delta$ and $f_\Delta ^\vee$ commute with the differential. One can check that this is also functorial.

3.2 Hochschild cohomology

Given an $A_\infty$ -algebra $A$ , one can define the Hochschild cohomology with values in any $A_\infty$ -bimodule $M$ . The Hochschild complex can again be modelled using the bar construction: following Tradler [Reference TradlerTra08, Lemma 2.3], each $\unicode {x03BE} \in \mathcal{M}(A,M)$ defines a map

\begin{align*} \overline {\unicode {x03BE}} := \sum _{i,j,n\geqslant 0} {\rm id}^{\otimes i}_{A[1]} \otimes \underline {\xi _n} \otimes {\rm id}^{\otimes j}_{A[1]}, \end{align*}

which is a coderivation $\overline {\unicode {x03BE}}\colon \mathsf{B} A \to \mathsf{B}_AM$ with values in the comodule $\mathsf{B}_AM$ . As before, the underlining signifies the component mapping to the factor $M$ . Using this lift, the Hochschild cochain complex can be defined as follows.

Definition 3.5. Let $(A,\unicode {x03BC})$ be an $A_\infty$ -algebra and $(M,\unicode {x03BD})$ an $A$ -bimodule. Then the $M$ -valued Hochschild cochain complex is the complex $\mathsf{C}^\bullet (A,M) := \left (\mathcal{M}(A,M)[-1],\ {\bf b}\right )$ with differential

\begin{align*} {\bf b}(\xi ) := \unicode {x03BD} \circ \overline {\xi } - (-1)^{|\xi |} \xi \circ \widetilde {\unicode {x03BC}}. \end{align*}

The cohomology of the complex is denoted ${{\rm HH}}^\bullet (A,M)$ , and, if $M = A_\Delta$ , the complex and its cohomology are denoted as $\mathsf{C}^\bullet (A) := \mathsf{C}^\bullet (A,A_\Delta )$ and ${{\rm HH}}^\bullet (A) := {{\rm HH}}^\bullet (A,A_\Delta )$ , respectively.

The Hochschild complex is functorial in each factor: any morphisms $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ or $\unicode {x03C1}\in {{\rm Hom}}_{A-A}(M,N)$ induce respective chain maps

\begin{align*} \begin{aligned} \mathsf{C}^\bullet (B,M) &\to \mathsf{C}^\bullet (A,f^\sharp M)\quad &\unicode {x03BE} \mapsto \unicode {x03BE} \circ \widehat f,\\[5pt] \mathsf{C}^\bullet (A,M) &\to \mathsf{C}^\bullet (A,N)\quad &\unicode {x03BE} \mapsto \unicode {x03C1} \circ \overline {\unicode {x03BE}}. \end{aligned} \end{align*}

In particular, for $M= A_\Delta ^\vee$ and $\unicode {x03C1} = f_\Delta ^\vee$ the combination of these constructions yields a chain map $f^* \colon \mathsf{C}^\bullet (B,B_\Delta ^\vee ) \to \mathsf{C}^\bullet (A,A_\Delta ^\vee )$ , sending $\xi \mapsto f_\Delta ^\vee \circ \overline {\xi \circ \widehat f}$ . Explicitly, this map is given by

\begin{align*} f^*\xi (a_1,\ldots ,a_n)(a_0) = \sum _{\substack {i,j\geqslant 0\\[5pt] i+j \leqslant n}} (-1)^{K_i} \xi (\widehat f(a_{i+1},\ldots ,a_{n-j}))(f(a_{n-j+1},\ldots ,a_n,a_0,a_1,\ldots ,a_i)), \end{align*}

where $K_i = (|a_1|+\cdots +|a_i|)(|a_{i+1}|+\cdots +|a_n| + |a_0|)$ is the Koszul sign obtained from permuting the $a_i$ . This map is natural with respect to $f$ and induces a map $[f^*] \colon {{\rm HH}}^\bullet (B,B_\Delta ^\vee ) \to {{\rm HH}}^\bullet (A,A_\Delta ^\vee )$ , making the Hochschild cohomology with values in the dual bimodule functorial.

A variation of Hochschild cohomology is negative cyclic cohomology, which can be realised using Connes’ complex. An element $\unicode {x03BE} = (\xi _n)_{n\in \mathbb{N}} \in \mathsf{C}^\bullet (A,A^\vee _\Delta )$ is called cyclic if for each $n\geqslant 1$ and all $a_0,\ldots ,a_n \in A[1]$ the following relation holds:

\begin{align*} \xi _n(a_1,\ldots ,a_n)(a_0) = (-1)^{|a_0|(|a_1|+\cdots +|a_n|)} \xi _n(a_0,\ldots ,a_{n-1})(a_n) \end{align*}

One checks that the Hochschild differential preserves cyclic cochains, yielding a subcomplex

\begin{align*} \mathsf{C}^\bullet _\lambda (A) := \left (\left \{\unicode {x03BE} \in \mathsf{C}^\bullet (A,A^\vee _\Delta ) \mid \unicode {x03BE}\; \textrm{is cyclic}\right \}, {\bf b} \right ), \end{align*}

which is called Connes’ complex. The cohomology of this complex is denoted ${\rm HC}_\lambda ^\bullet (A)$ . We remark that the pullback along any morphism $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ preserves cyclic cocycles, and therefore restricts to a map $f^*\colon \mathsf{C}^\bullet _\lambda (B) \to \mathsf{C}^\bullet _\lambda (A)$ and an induced map $[f^*] \colon {\rm HC}_\lambda ^\bullet (B) \to {\rm HC}_\lambda ^\bullet (A)$ .

It is well known that Hochschild cohomology and negative cyclic cohomology are invariant under homotopy equivalences. Below we include a proof using the definition of homotopy in § 2.2, which will also apply in the analytic setting.

Proposition 3.6. Let $f,g\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,B)$ with $f\sim g$ . Then $f$ and $g$ induce the same morphisms

\begin{align*} [f^*] = [g^*] \colon {{\rm HH}}^\bullet (B,B_\Delta ^\vee ) \to {{\rm HH}}^\bullet (A,A_\Delta ^\vee ),\quad [f^*] = [g^*] \colon {\rm HC}^\bullet _\lambda (B) \to {\rm HC}^\bullet _\lambda (A). \end{align*}

Proof. If $f\sim g$ then $f = \mathsf{ev}_0 \circ H$ and $g=\mathsf{ev}_1 \circ H$ for some homotopy $H\in {{\rm Hom}}_{\mathsf{Alg}^\infty }(A,\Omega ^\bullet _{[0,1]}\otimes B)$ , and the induced maps are therefore given by

\begin{align*} [f^*] = [H^*][\mathsf{ev}_0^*],\quad [g^*] = [H^*][\mathsf{ev}_1^*]. \end{align*}

Therefore, it is sufficient to prove that the maps $\mathsf{ev}_0,\mathsf{ev}_1 \in {{\rm Hom}}_{\mathsf{Alg}^\infty }(\Omega ^\bullet _{[0,1]} \otimes B, B)$ induce the same map $[\mathsf{ev}_0^*] = [\mathsf{ev}_1^*]$ on Hochschild/negative cyclic cohomology, which we do below.

Let $\unicode {x03BE} \in Z^k\mathsf{C}(B,B^\vee _\Delta )$ be any fixed cocycle. Then the pullback along $\mathsf{ev}_t$ for any $t\in [0,1]$ is given by a cocycle $\mathsf{ev}_t^*\unicode {x03BE} \in Z^k\mathsf{C}(\Omega ^\bullet _{[0,1]} \otimes B,(\Omega ^\bullet _{[0,1]} \otimes B)^\vee _\Delta )$ , given on pure tensors of the form $\omega _i = (x_i+y_i{\rm d} t) \otimes b_i$ by

\begin{align*} (\mathsf{ev}_t^*\unicode {x03BE})_n (\omega _1,\ldots ,\omega _n)(\omega _{n+1}) = x_1(t)\cdots x_{n+1}(t) \cdot \xi _n(b_1,\ldots ,b_n)(b_{n+1}). \end{align*}

We claim that the difference $\mathsf{ev}_1^*\unicode {x03BE} - \mathsf{ev}_0^*\unicode {x03BE}$ is the ${\bf b}$ -image $\unicode {x03B6}\in \mathsf{C}^{k-1}(\Omega ^\bullet _{[0,1]} \otimes B, (\Omega ^\bullet _{[0,1]} \otimes B)^\vee _\Delta )$ defined on pure tensors $\omega _i = (x_i+y_i{\rm d} t) \otimes b_i$ as

(11) \begin{equation} \begin{aligned} \zeta _n(\omega _1,\ldots ,\omega _n)(\omega _{n+1}) &= \sum _{j=1}^n(-1)^{|\unicode {x03BE}|+|b_1|+\cdots +|b_{j-1}|} \int _0^1 x_1\cdots y_j \cdots x_{n+1} {\rm d} t \cdot \xi _n(b_1,\ldots ,b_n)(b_{n+1})\\[5pt] &\quad + (-1)^{|\unicode {x03BE}|+1+|b_1|+\cdots +|b_n|} \int _0^1 x_1\cdots x_n y_{n+1} {\rm d} t \cdot \xi _n(b_1,\ldots ,b_n)(b_{n+1}). \end{aligned} \end{equation}

For brevity we decompose the $A_\infty$ -structure defined in (3) and (4) as $\unicode {x03BC}^\otimes = {\rm d} + \unicode {x03BC}$ , using the abbreviation ${\rm d} = {\rm d} \otimes {\rm id}$ and $\unicode {x03BC} = \sum _{k\geqslant 1} (-\cdots -) \otimes \unicode {x03BC}_k$ . Then we have the following:

(12) \begin{align} {\bf b}(\unicode {x03B6}) &= {\rm d}_\Delta ^\vee \circ \unicode {x03B6} - (-1)^{|\unicode {x03B6}|} \cdot \unicode {x03B6} \circ \widetilde {{\rm d}} \end{align}

(13) \begin{align} &\quad +\unicode {x03BC}^\vee _\Delta \circ \overline {\unicode {x03B6}} -(-1)^{|\unicode {x03B6}|} \unicode {x03B6} \circ \widetilde {\unicode {x03BC}_k}. \end{align}

By inspection, the value of (12) on variables $\omega _i = (x_i+y_i{\rm d} t) \otimes b_i$ is

\begin{align*} \begin{aligned} &({\rm d}^\vee \circ \unicode {x03B6} - (-1)^{|\unicode {x03B6}|} \cdot \unicode {x03B6} \circ \widetilde {{\rm d}})(\omega _1,\ldots ,\omega _n)(\omega _{n+1})\\[5pt] &\quad = (-1)^{|\unicode {x03B6}|+|b_1|+\cdots + |b_n|} \unicode {x03B6}(x_1\otimes b_1,\ldots ,x_n\otimes b_n)({\rm d} x_{n+1}\otimes b_{n+1}) \\[5pt] &\quad \quad - \sum _{j=1}^n (-1)^{|\unicode {x03B6}|+|b_1|+\cdots +|b_{j-1}|} \cdot \unicode {x03B6}_n(x_1\otimes b_1,\ldots , {\rm d} x_j\otimes b_j , \ldots , x_n\otimes b_n)(x_{n+1}\otimes b_{n+1}) \\[5pt] &\quad = \sum _{j=1}^{n+1} \int _0^1 \left (x_0\cdots \tfrac {\partial x_j}{\partial t} \cdots x_n\right ) {\rm d} t \cdot \unicode {x03BE}_n(b_1,\ldots ,b_n)(b_{n+1}) \\[5pt] &\quad = (x_0(1)\cdots \cdots x_n(1) - x_0(0)\cdots \cdots x_n(0))\cdot \unicode {x03BE}_n(b_1,\ldots ,b_n)(b_0) \\[5pt] &\quad = (\mathsf{ev}_1^*\unicode {x03BE})_n (\omega _1,\ldots ,\omega _n)(\omega _0) - (\mathsf{ev}_0^*\unicode {x03BE})_n (\omega _1,\ldots ,\omega _n)(\omega _0). \end{aligned} \end{align*}

Conversely, the motivated reader may check that the value of (13) on pure tensors $\omega _i$ is equal to

\begin{align*} \begin{aligned} &(\unicode {x03BC}_\Delta ^\vee \circ \overline {\unicode {x03B6}} - (-1)^{|\unicode {x03B6}|} \unicode {x03B6} \circ \widetilde {\unicode {x03BC}})(\omega _1,\ldots ,\omega _n)(\omega _{n+1}) \\[5pt] &=\quad \sum _{j=1}^{n} (-1)^{|\unicode {x03BE}|+|b_1|+\cdots +|b_{j-1}|} \cdot \int _0^1 x_1\cdots y_j \cdots x_n {\rm d} t \cdot {\bf b}(\unicode {x03BE})(b_1,\ldots ,b_n)(b_{n+1}), \end{aligned} \end{align*}

which always vanishes because $\unicode {x03BE}$ is assumed to be a cocycle. We conclude that

\begin{align*} [\mathsf{ev}_1^*\unicode {x03BE}] = [\mathsf{ev}_0^*\unicode {x03BE} + {\rm d} \unicode {x03B6}] = [\mathsf{ev}_0^*\unicode {x03BE}]. \end{align*}

To see that the same is true for the cyclic classes, it suffices to show that $\unicode {x03B6}$ is cyclic whenever $\unicode {x03BE}$ is cyclic. If $\unicode {x03BE}$ is cyclic, then given homogeneous pure tensors $\omega _i$ with $\omega _j = y_j {\rm d} t \otimes b_j$ for some fixed $j\leqslant n$ and $\omega _i = x_i \otimes b_i$ for all $i\neq j$ , we have

\begin{align*} \begin{aligned} &\zeta _n(\omega _{n+1},\omega _1,\ldots ,\omega _{n-1})(\omega _n) \\[5pt] &\quad = (-1)^{|\unicode {x03BE}|+|b_{n+1}| + |b_1|+\cdots +|b_{j-1}|} \int _0^1 x_1\cdots y_j \cdots x_{n+1} {\rm d} t \cdot \xi _n(b_{n+1},b_1,\ldots ,b_{n-1})(b_n) \\[5pt] &\quad = (-1)^{|b_{n+1}| + |b_1|+\cdots +|b_{j-1}| + |b_{n+1}|(|b_1|+\cdots +|b_n|)} \int _0^1 x_1\cdots y_j \cdots x_{n+1} {\rm d} t \cdot \xi _n(b_1,\ldots ,b_n)(b_{n+1}) \\[5pt] &\quad = (-1)^{|b_{n+1}|(|b_1|+\cdots +(|b_j|+1)+\cdots +|b_n|)}\unicode {x03B6}(\omega _1,\ldots ,\omega _n)(\omega _{n+1}) \\[5pt] &\quad = (-1)^{|\omega _{n+1}|(|\omega _1|+\cdots +|\omega _n|)}\unicode {x03B6}(\omega _1,\ldots ,\omega _n)(\omega _{n+1}). \end{aligned} \end{align*}

A similar computation shows that the cyclic identity holds when $\omega _{n+1} = y_{n+1}{\rm d} t \otimes b_{n+1}$ . This shows that $\mathsf{ev}_0$ and $\mathsf{ev}_1$ induce the same classes on Hochschild and negative cyclic cohomology, as claimed.

Corollary 3.7. For an $A_\infty$ -algebra $A$ with minimal model ${\rm H}(A)$ there are isomorphisms

\begin{align*} {{\rm HH}}(A,A^\vee _\Delta ) \cong {{\rm HH}}({\rm H}(A),{\rm H}(A)^\vee _\Delta ),\quad {\rm HC}_\lambda (A,A^\vee _\Delta ) \cong {\rm HC}_\lambda ({\rm H}(A),{\rm H}(A)^\vee _\Delta ). \end{align*}

3.3 Analytic $A_\infty$ -bimodules

Given $A\in {\mathsf{Alg}^{\infty ,{\rm an}}}$ and $M\in \operatorname {\mathsf{grNMod}}{\ell ^e}$ we consider a family of norms as in § 2.4: for each $r\gt 0$ we define

\begin{align*} \mathsf{B}_A(M,r) := (\mathsf{B}_A M, {{\left \|\cdot \right \|}}_r),\quad {\left \|m\right \|}_r = \sup _{i,j} {\left \|m_{i,j}\right \|}r^{-i-j-1}, \end{align*}

where $m = \sum _{i,j} m_{i,j} \in \mathsf{B}_AM$ is a decomposition into elements $m_{i,j} \in A[1]^{\otimes i} \otimes M[1] \otimes A[1]^{\otimes j}$ . Boundedness of linear maps out of these normed bar constructions can again be characterised in various ways, analogously to Lemma2.16.

Proof. (i) $\implies$ (ii). Suppose there exists $C\gt 0$ such that ${\left \|\unicode {x03C1}_{i,j}\right \|} \leqslant C^{i+j+1}$ for all $i,j\in \mathbb{N}$ , and fix $0\lt r\lt C^{-1}$ . Then an element $m = \sum _{i,j} m_{i,j} \in \mathsf{B}_A(M,r)$ of norm ${\left \|m\right \|}_r=1$ satisfies ${\left \|m_{i,j}\right \|} \leqslant r^{i+j+1}$ for all $i,j\in \mathbb{N}$ and therefore

\begin{align*} {\left \|\unicode {x03C1}(m)\right \|} \leqslant \sum _{n\in \mathbb{N}} \sum _{i+j=n} {\left \|\unicode {x03C1}_{i,j}(m_{i,j})\right \|} \leqslant \sum _{n\in \mathbb{N}} \sum _{i+j=n} C^{i+j+1} r^{i+j+1} = \sum _{n\in \mathbb{N}} n (Cr)^{n+1}. \end{align*}

Because the power series $\sum _n n z^{n+1}$ has radius of convergence $1$ , it converges at $z = Cr \lt 1$ . In particular, the norm $\unicode {x03C1}$ is bounded.

(ii) $\implies$ (i). Let $K\gt 1$ be a constant bounding the norm of $\unicode {x03C1}\colon \mathsf{B}_A(M,r) \to N[1]$ and pick $C \geqslant {K}/{r}$ . Then for all $i,j\in \mathbb{N}$ and $m_{i,j}\in A[1]^{\otimes i} \otimes M[1] \otimes A[1]^{\otimes j}$ of norm ${\left \|m_{i,j}\right \|} = 1$ , the map $\unicode {x03C1}_{i,j}$ satisfies

\begin{align*} {\left \|\unicode {x03C1}_{i,j}(m_{i,j})\right \|} = {\left \|\unicode {x03C1}(m_{i,j})\right \|} \leqslant K {\left \|m_{i,j}\right \|} r^{-i-j-1} \leqslant \left (\frac {K}{r}\right )^{i+j+1} \leqslant C^{i+j+1}, \end{align*}

which shows that, in particular, ${\left \|\unicode {x03C1}_{i,j}\right \|} \leqslant C^{i+j+1}$ .

(i) $\implies$ (iii). Suppose there exists $C\gt 1$ such that ${\left \|\unicode {x03C1}_{i,j}\right \|} \leqslant C^{i+j+1}$ for all $i,j\in \mathbb{N}$ , and fix $r'\gt 0$ . Then we choose a constant $q\gt 0$ such that $ij \lt q^{i+j+1}$ for all $i,j\in \mathbb{N}$ , a constant $K=\min \{r',1\}$ , and a radius $r\lt {K}/{qC}$ . Then for any element $m = \sum m_{i,j} \in \mathsf{B}_A M$ of norm ${\left \|m\right \|}_r=1$ one has ${\left \|m_{i,j}\right \|} \leqslant r^{i+j+1}$ and hence

\begin{align*} \begin{aligned} {\left \|\widehat {\unicode {x03C1}}(m)\right \|}_{r'} &\leqslant \sum _{i,j\geqslant 0} \sum _{\substack {0\leqslant i\leqslant n\\[5pt] 0\leqslant j\leqslant m}} {\left \|({\rm id}^{\otimes i} \otimes \unicode {x03C1}_{n-i,m-j} \otimes {\rm id}^{\otimes j})(m_{i,j})\right \|} (r')^{-i-j-1} \\[5pt] &\leqslant \sum _{i,j\geqslant 0} \sum _{\substack {0\leqslant i\leqslant n\\[5pt] 0\leqslant j\leqslant m}} C^{i+j-i-j+1} K^{-i-j-1} r^{i+j+1} \\[5pt] &\leqslant \sum _{i,j\geqslant 0} nm \left (\frac {rC}{K}\right )^{i+j+1} \\[5pt] &\leqslant \sum _{N\geqslant 1} (N-1) \left (\frac {rq C}{K}\right )^N .\end{aligned} \end{align*}

Then the norm is bounded by the value of the power series $\sum _N (N-1) z^N$ at $z={rqC}/{K} \lt 1$ .

(i) $\implies$ (iv). This follows in an analogous way to the previous implication. We leave the details up to the reader.

(iii), (iv) $\implies$ (ii). The projection $\unicode {x03C0} \colon \mathsf{B}_A(N,r') \to N[1]$ is clearly bounded for any $r'\gt 0$ , so the boundedness of $\widehat {\unicode {x03C1}}\colon \mathsf{B}_A(M,r) \to \mathsf{B}_A(N,r')$ implies that $\unicode {x03C1} = \unicode {x03C0}\circ \widehat {\unicode {x03C1}}$ is bounded. Likewise, $\unicode {x03C1} = \unicode {x03C0} \circ \widetilde {\unicode {x03C1}}$ is bounded when $\widetilde {\unicode {x03C1}}\colon \mathsf{B}_A(M,r) \to \mathsf{B}_A(M,r')$ is bounded.

Hence, as in the algebra case we find that the union over the spaces $\hom _{\ell ^e}^{{\rm cont}}(\mathsf{B}_A(M,r),N[1])$ is given by a graded vector space of double sequences

\begin{align*} {\mathcal{A}}_A(V,W) := \left \{\; f \in \mathcal{M}_A(V,W) \;\middle |\; \exists C\gt 0 \;\textrm{such that}\; \|f_{i,j}\| \leqslant C^{i+j+1}\;\textrm{for all}\; i,j\in \mathbb{N} \;\right \}, \end{align*}

whose elements we again call analytic. In particular, we obtain a definition of analytic $A_\infty$ -bimodules over an analytic $A_\infty$ -algebra.

Example 3.11. The dual bimodule $A^\vee$ is not analytic in general, as the dual space is not endowed with a natural norm. However, one can consider the continuous dual bimodule

\begin{align*} A_\Delta ' := \left (\hom ^{{\rm cont}}_{\mathbb{C}}(A,{\mathbb{C}}), \unicode {x03BD}'_\Delta \right ), \end{align*}

where the components of $\unicode {x03BD}'_\Delta$ are defined by the same formula as the components of $\unicode {x03BD}_\Delta ^\vee$ . It is again straightforward to check that ${\left \|\unicode {x03BD}^\vee _{\Delta ,i,j}\right \|} = {\left \|\unicode {x03BC}_{i+j+1}\right \|}$ is bounded by a geometric series. If $A$ is finite dimensional, then $A'_\Delta$ is equal to $A^\vee _\Delta$ in $\mathsf{Mod}_{A-A}^\infty$ , but in general it is only a submodule.

As before, we call a (pre-)morphism $\unicode {x03C1} \in \hom _{A-A}^{\bullet }(M,N)$ between $A$ -bimodules $(M,\unicode {x03BD}_M)$ and $(N,\unicode {x03BD}_N)$ analytic if it lies in the subspace ${\mathcal{A}}_A(M,N) \subset \mathcal{M}_A(M,N)$ . It follows from Lemma3.8 that

\begin{align*} \unicode {x03B4} = \unicode {x03BD}_N \circ \widehat {(-)} - (-1)^{|\unicode {x03C1}|} (-)\circ \widetilde {\unicode {x03BD}}_M \end{align*}

is a bounded operator on this space, and hence gives rise to a complex of analytic pre-morphisms

\begin{align*} \hom _{A-A}^{\bullet ,{\rm an}}(M,N) := \left ({\mathcal{A}}_A(M,N), \delta \right ), \end{align*}

which is a subcomplex of $\hom _{A-A}^\bullet (M,N)$ . As before, we denote the subspace of analytic morphisms by ${{\rm Hom}}_{A-A}^{{\rm an}}(M,N) \subset {{\rm Hom}}_{A-A}(M,N)$ . It follows from Lemma3.8 that the composition of analytic bimodule pre-morphisms is again analytic, and therefore defines morphism complexes for a DG category $\operatorname {\mathsf{NMod}}^{\infty ,{\rm an}}_{A-A}$ of analytic bimodules over $A$ .

Given an analytic $A_\infty$ -morphism $f$ one can again consider the functor $f^*$ , and the pullback map $f^\sharp$ on the Hom-spaces between the diagonal bimodule and its continuous dual.

Lemma 3.12. Let $f\in {{\rm Hom}}^{{\rm an}}_{{\mathsf{Alg}^\infty }}(A,B)$ be an analytic morphism. Then there is a well-defined functor

\begin{align*} f^\sharp \colon \operatorname {\mathsf{NMod}}^{\infty ,{\rm an}}_{B-B} \to \operatorname {\mathsf{NMod}}^{\infty ,{\rm an}}_{A-A}, \end{align*}

which induces a pullback $f^*\colon \hom _{B-B}^{{\rm an}}(B_\Delta ,B_\Delta ') \to \hom _{A-A}^{{\rm an}}(A_\Delta ,A_\Delta ')$ .

Proof. Given $f \in {{\rm Hom}}^{{\rm an}}_{{\mathsf{Alg}^\infty }}(A,B)$ , for any $N,M\in \operatorname {\mathsf{grNMod}} {\ell ^e}$ and $\unicode {x03C1}\in {\mathcal{A}}_B(M,N)$ there exists a common constant $C\gt 0$ such that ${\left \|f_n\right \|} \lt C^n$ for all $n\in \mathbb{N}$ and ${\left \|\unicode {x03C1}_{i,j}\right \|} \lt C^{i+j+1}$ for all $i,j\in \mathbb{N}$ . Then the construction $f^\sharp \unicode {x03C1}$ has components satisfying

\begin{align*} \begin{aligned} {\left \|(f^\sharp \unicode {x03C1})_{i,j}\right \|} &\leqslant \sum _{\substack {i_1+\cdots +i_n = i\\[5pt] j_1+\cdots +j_n = j}} {\left \|\unicode {x03C1}_{n,m}\right \|} {\left \|f_{i_1}\right \|} \cdots {\left \|f_{i_n}\right \|} {\left \|f_{j_1}\right \|} \cdots {\left \|f_{j_m}\right \|} \\[5pt] &\lt \sum _{\substack {i_1+\cdots +i_n = i\\[5pt] j_1+\cdots +j_n = j}} C^{n+m+i+j+1} \\[5pt] &\leqslant ({\#} \textrm{ of partitions of $i$}) \cdot ({\#} \textrm{ of partitions of $j$})) \cdot C^{2(i+j)+1}. \end{aligned} \end{align*}

Picking some constant $q\gt 1$ such that $q^n$ bounds the number of partitions of $n$ , it follows that ${\left \|(f^\sharp \unicode {x03C1})_{i,j}\right \|} \lt q^{i+j}C^{2(i+j)+1} \leqslant (qC^2)^{i+j+1}$ , which shows that $f^\sharp \unicode {x03C1} \in {\mathcal{A}}_A(M,N)$ . If follows that $f^\sharp$ maps every analytic bimodule structure $\unicode {x03BD}$ to an analytic bimodule structure $f^\sharp \unicode {x03BD}$ , and an analytic (pre-)morphism $\unicode {x03C1} \in \hom ^{{\rm an}}_{A-A}(M,N)$ between analytic bimodules to an analytic bimodule map $f^\sharp \unicode {x03C1}$ . Hence $f^\sharp$ is a well-defined functor between categories of analytic bimodules.

It follows that $f^\sharp B_\Delta$ is an analytic bimodule, and it follows directly from the definition that the map $f_\Delta \colon A \to f^\sharp B_\Delta$ is analytic. The bimodule $f^\sharp B_\Delta '$ is likewise analytic, and it follows easily that the formula for $f^\vee _\Delta$ defines an analytic map $f_\Delta ' \colon f^\sharp B_\Delta ' \to A_\Delta '$ . Hence, if $\unicode {x03C1} \in \hom _{B-B}^{{\rm an}}(B_\Delta ,B_\Delta ')$ is an analytic bimodule map, the composition $f^*\unicode {x03C1} = f_\Delta ' \diamond f^\sharp \unicode {x03C1} \diamond f_\Delta$ is again analytic.

3.4 Analytic Hochschild cohomology

Given an analytic $A_\infty$ -algebra $(A,\unicode {x03BC})$ and an analytic $A$ -bimodule $(M,\unicode {x03BD}) \in \operatorname {\mathsf{NMod}}^{\infty ,{\rm an}}_{A-A}$ , we call a Hochschild cochain $\unicode {x03BE} \in \mathsf{C}^\bullet (A,M) = (\mathcal{M}(A,M)[-1],{\bf b})$ analytic if it lies in the subspace ${\mathcal{A}}(A,M)[-1] \subset \mathcal{M}(A,M)[-1]$ . This analytic condition can again be characterised in multiple ways.

Proof. The implication (ii) $\implies$ (i) is obvious, and hence we only check the other implication. Given $\unicode {x03BE}\in \mathsf{C}^{\bullet ,{\rm an}}(A,M)$ , there exists $C\gt 1$ such that ${\left \|\xi _n\right \|} \lt C^n$ for all $n\geqslant 1$ . Fix $r'\gt 0$ , and let $r \lt \min \{r',1/C\}$ . Then for every element $\sum _{n\in \mathbb{N}} a_n \in \mathsf{B}(A,r)$ of norm ${\left \|a\right \|}_r \leqslant 1$

\begin{align*} \begin{aligned} {\left \|\overline {\unicode {x03BE}}(a)\right \|}_{r'} &\leqslant \sum _{i,j,k\geqslant 0} \sum _{k\geqslant 0} {\left \|({\rm id}^i\otimes \underline {\xi _k}\otimes {\rm id}^{\otimes j})(a_{i+k+j})\right \|} (r')^{-i-j-1} \\[5pt] &\leqslant \sum _{i,j\geqslant 0} {\left \|\unicode {x03BE}_0(1)\right \|} {\left \|a_{i+j}\right \|} (r')^{-i-j-1} + \sum _{i,j\geqslant 0}\sum _{k\geqslant 1} C^{i+k+j} {\left \|a_{i+k+j}\right \|} \cdot (r')^{-i-j-1} \\[5pt] &\leqslant {\left \|\unicode {x03BE}_0(1)\right \|}(r')^{-1} \sum _{n\geqslant 0} (n+1) (r/r')^n + (r')^{-1}\sum _{i,j\geqslant 0}\sum _{k\geqslant 1} (Cr)^{k} \cdot (r/r')^{i+j} \\[5pt] &\leqslant \frac {{\left \|\unicode {x03BE}_0(1)\right \|}}{r'(1-(r/r'))^2} + \frac {1}{r'(1-Cr)(1-(r/r'))^2} \lt \infty . \end{aligned} \end{align*}

It follows that the operator $\overline {\unicode {x03BE}}$ is bounded.

As a corollary, it follows that the differential maps an analytic cochain $\unicode {x03BE} \in \mathsf{C}^{\bullet ,{\rm an}}(A,M)$ to a cochain

\begin{align*} {\bf b}(\unicode {x03BE}) = \unicode {x03BD} \circ \overline {\unicode {x03BE}} - (-1)^{|\unicode {x03BE}|} \unicode {x03BE} \circ \widetilde {\unicode {x03BC}} \end{align*}

which is bounded as a map $\mathsf{B}(A,r) \to M$ for some $r$ , and hence analytic by Lemma2.16. It follows that the analytic cochains form a well-defined subcomplex, which we denote by

\begin{align*} \mathsf{C}^{\bullet ,{\rm an}}(A,M) := ({\mathcal{A}}(A,M)[-1],{\bf b}). \end{align*}

For $M=A_\Delta$ we again abbreviate $\mathsf{C}^{\bullet ,{\rm an}}(A) := \mathsf{C}^{\bullet ,{\rm an}}(A,A_\Delta )$ . Similarly, we define the analytic version of Connes’ complex as the intersection

\begin{align*} \mathsf{C}^{\bullet ,{\rm an}}_\lambda (A) := \mathsf{C}^\bullet _\lambda (A) \cap \mathsf{C}^{\bullet ,{\rm an}}(A,A_\Delta '), \end{align*}

which will play the role of the negative cyclic cohomology in the analytic setting.

The analytic Hochschild complex is again functorial in each argument with respect to analytic maps: given $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}^{{\rm an}}(A,B)$ it follows directly from Lemma2.16 that there is a well-defined map

\begin{align*} -\circ \widehat f \colon \mathsf{C}^{\bullet ,{\rm an}}(B,M) \to \mathsf{C}^{\bullet ,{\rm an}}(A,f^\sharp M), \end{align*}

and likewise for any $\unicode {x03C1}\in {{\rm Hom}}_{A-A}^{{\rm an}}(M,N)$ it follows from Lemma3.13 that

\begin{align*} \unicode {x03C1} \circ \overline {-} \colon \mathsf{C}^{\bullet ,{\rm an}}(A,M) \to \mathsf{C}^{\bullet ,{\rm an}}(A,N), \end{align*}

is well defined. In particular, composition with $\widehat f$ and $f'_\Delta$ induces maps $f^*\colon \mathsf{C}^{\bullet ,{\rm an}}(B,B_\Delta ') \to \mathsf{C}^{\bullet ,{\rm an}}(A,A_\Delta ')$ and $f^*\colon \mathsf{C}^{\bullet ,{\rm an}}_\lambda (B) \to \mathsf{C}^{\bullet ,{\rm an}}_\lambda (A)$ for any analytic $A_\infty$ -morphism $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}^{{\rm an}}(A,B)$ . As before, we have a compatibility with homotopies.

Proof. If $f\sim _{{\rm an}} g$ then $f = \mathsf{ev}_0 \circ H$ and $g=\mathsf{ev}_1\circ H$ for an analytic map $H\in {{\rm Hom}}_{\mathsf{Alg}^\infty }^{{\rm an}}(A,\Omega ^\bullet _{[0,1]}\otimes B)$ . Hence, as in Proposition3.6, it suffices to show that the analytic maps $\mathsf{ev}_0$ and $\mathsf{ev}_1$ induce the same map on the analytic versions of Hochschild/negative cyclic cohomology. For this, it suffices to show that if $\unicode {x03BE} \in Z^k\mathsf{C}^{\bullet ,{\rm an}}(B,B'_\Delta )$ is an analytic cocycle then the cochain (11) is analytic. However, this is straightforward: for any fixed $n$ and elements $\omega _i = \sum _{k_i} (x_i^{k_i} + y_i^{k_i}{\rm d} t) \otimes b^{k_i}_i$ , we have

\begin{align*} \begin{aligned} {\left \|\zeta _n(\omega _1,\ldots ,\omega _n)(\omega _{n+1})\right \|} &\leqslant \sum _{k_1,\ldots ,k_{n+1}}\sum _{j=1}^n \int _0^1 x_1^{k_1}\cdots y_j^{k_j}\cdots x_{n+1}^{k_{n+1}}{\rm d} t \cdot {\left \|\xi _n(b_1,\ldots ,b_n)(b_{n+1})\right \|} \\[5pt] &\leqslant {\left \|\xi _n\right \|} \cdot \sum _{k_1,\ldots ,k_{n+1}} \sum _{j=1}^n {\left \|x_1^{k_1}\right \|}_\infty \cdots {\left \|y_j^{k_j}\right \|}_\infty \cdots {\left \|x_{n+1}^{k_{n+1}}\right \|}_\infty \cdot {\left \|b_1\right \|}\cdots {\left \|b_{n+1}\right \|}. \\[5pt] &\leqslant {\left \|\xi _n\right \|} \cdot \sum _{k_1} {\left \|(x_1^{k_1} + y_1^{k_1}{\rm d} t) \otimes b^{k_1}_1\right \|} \cdots \sum _{k_{n+1}} {\left \|(x_{n+1}^{k_{n+1}} + y_{n+1}^{k_{n+1}}{\rm d} t) \otimes b^{k_{n+1}}_{n+1}\right \|}. \end{aligned} \end{align*}

Taking the infimum over all such decompositions of $\omega _i$ into pure tensors, we see that ${\left \|\zeta _n\right \|} \leqslant {\left \|\xi _n\right \|}$ , and it follows that $\unicode {x03B6}$ is again an analytic cochain. Hence $[\mathsf{ev}_1^*\unicode {x03BE}] = [\mathsf{ev}_0^*\unicode {x03BE}]$ also holds in the analytic versions of the Hochschild and negative cyclic cohomology.

3.5 Analytic inverses for bimodule morphisms

In the remainder of this section we show the analogue of Lemma2.25 for bimodule morphisms: given $\unicode {x03C1} \in {{\rm Hom}}_{A-A}^{{\rm an}}(M,N)$ such that $\unicode {x03C1}_{0,0}$ is invertible in $\operatorname {\mathsf{grNMod}} {\ell ^e}$ , we show that it admits an analytic inverse $\unicode {x03C4} \in {{\rm Hom}}_{A-A}^{{\rm an}}(N,M)$ . To check the growth condition on $\unicode {x03C4}$ , we define it explicitly using a tree formula and a special kind of planar tree.

Figure 1.

Left: an example of a tree $T \in \mathsf{Catp}(3,4,2)$ with the root at the bottom. Right: the string diagram for the corresponding map $\unicode {x03C4}_T \colon A[1]^{\otimes 3} \otimes M[1] \otimes A[1]^{\otimes 4} \to N[1]$ .

A typical example of a caterpillar tree is given in Figure 1. There is a unique tree $T\in \mathsf{Catp}(0,0)$ which has $d=0$ internal nodes. For $n\gt 1$ one has $d\gt 1$ and every tree $T\in \mathsf{Catp}(n_1,n_2,d)$ can be uniquely decomposed into an internal node and a subtree in two ways:

so $T$ is uniquely determined by a pair of numbers $(l_1,l_2)$ and a tree $T_r \in \mathsf{Catp}(n_1-l_1,n_2-l_2,d-1)$ , and also by a pair of numbers $(r_1,r_2)$ and a tree $T_l \in \mathsf{Catp}(n_1-r_1,n_2-r_2,d-1)$ . Hence we have bijections

\begin{align*} \begin{aligned} \mathsf{Catp}(n_1,n_2) &\cong \{(l_1,l_2,T_r) \mid l_1+l_2\gt 0,\ T_r \in \mathsf{Catp}(n_1-l_1,n_2-l_2)\} \\[5pt] &\cong \{(T_l,r_1,r_2) \mid r_1+r_2\gt 0,\ T_l \in \mathsf{Catp}(n_1-r_1,n_2-r_2)\}. \end{aligned} \end{align*}

Given $\unicode {x03C1}\in \hom _{A-A}(M,N)$ and $\unicode {x03C4}_{0,0} \in \hom _{{\ell ^e}}^{{\rm cont}}(N,M)$ we then define, for every $T\in \mathsf{Catp}(n_1,n_2)$ , a bimodule pre-morphism as follows. For $n_1=n_2=0$ we set $\unicode {x03C4}_T = \unicode {x03C4}_{0,0}$ , and otherwise we define $\unicode {x03C4}_T$ via the following equivalent recursive formulas:

(14) \begin{equation} \begin{aligned} \unicode {x03C4}_T &= - \unicode {x03C4}_{0,0} \circ \unicode {x03C1}_{l_1,l_2} \circ ({\rm id}^{\otimes l_1} \otimes \underline {\unicode {x03C4}_{T_r}}\otimes {\rm id}^{\otimes l_2})\\[5pt] &= - \unicode {x03C4}_{T_l} \circ ({\rm id}^{\otimes (n_1-r_1)} \otimes \underline {(\unicode {x03C1}_{r_1,r_2}\circ \unicode {x03C4}_{0,0})} \otimes {\rm id}^{\otimes (n_2-r_2)}). \end{aligned} \end{equation}

See again Figure 1 for an example of the map $\unicode {x03C4}_T$ . If $\unicode {x03C4}_{0,0}$ is an inverse for $\unicode {x03C1}_{0,0}$ then the sum of these maps determines an inverse bimodule map.

Lemma 3.17. Let $\unicode {x03C1} \in \hom _{A-A}(M,N)$ be a pre-morphism such that $\unicode {x03C1}_{0,0}$ admits a left-/right-inverse $\unicode {x03C4}_{0,0}$ . Then $\unicode {x03C1}$ admits a left-/right-inverse $\unicode {x03C4} \in \hom _{A-A}(N,M)$ with components

\begin{align*} \unicode {x03C4}_{i,j} = \sum _{T\in \mathsf{Catp}(i,j)} \unicode {x03C4}_T, \end{align*}

where the maps $\unicode {x03C4}_T$ are defined as above.

Proof. Suppose $\unicode {x03C4}_{0,0}$ is a right-inverse for $\unicode {x03C1}_{0,0}$ , then we claim that $\unicode {x03C4}$ is a right-inverse for $\unicode {x03C1}$ . The base case $i=j=0$ is trivial as

\begin{align*} (\unicode {x03C1}\diamond \unicode {x03C4})_{0,0} = \unicode {x03C1}_{0,0} \circ \sum _{T\in \mathsf{Catp}(0,0)} \unicode {x03C4}_T = \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{0,0} = {\rm id}_{M[1]}. \end{align*}

For $i+j\gt 0$ it follows from the recursive definition of the maps $\unicode {x03C4}_T$ that

\begin{align*} \begin{aligned} (\unicode {x03C1}\diamond \unicode {x03C4})_{i,j} &= \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{i,j} + \sum _{l_1+l_2\gt 0}\sum _{T_r\in \mathsf{Catp}(i-l_1,j-l_2)} \unicode {x03C1}_{l_1,l_2} \circ ({\rm id}^{\otimes l_1} \otimes \unicode {x03C1}_{T_r} \otimes {\rm id}^{\otimes l_2}) \\[5pt] &= \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{i,j} + \sum _{l_1+l_2\gt 0}\sum _{T_r\in \mathsf{Catp}(i-l_1,j-l_2)} \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{0,0} \circ \unicode {x03C1}_{l_1,l_2} \circ ({\rm id}^{\otimes l_1} \otimes \unicode {x03C1}_{T_r} \otimes {\rm id}^{\otimes l_2}) \\[5pt] &= \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{i,j} - \sum _{T \in \mathsf{Catp}(i,j)} \unicode {x03C1}_{0,0} \circ \unicode {x03C4}_{T} = 0, \end{aligned} \end{align*}

where the final line uses the bijection between trees in $\mathsf{Catp}(i,j)$ and triples $(l_1,l_2,T_r)$ discussed above. It follows that $\unicode {x03C1}\diamond \unicode {x03C4} = {\rm id}$ , so $\unicode {x03C4}$ is a right-inverse. The case where $\unicode {x03C4}_{0,0}$ is a left-inverse is similar, using the other decomposition into triples $(T_l,r_1,r_2)$ . In particular, $\unicode {x03C4}$ is a two-sided inverse if $\unicode {x03C4}_{0,0}$ is.

We remark that if $\unicode {x03C1}$ is a morphism and $\unicode {x03C4}=\unicode {x03C1}^{-1}$ is a two-sided inverse, then the latter is again a morphism by the equality

\begin{align*} 0 = \unicode {x03C4} \diamond \unicode {x03B4}({\rm id}) = \unicode {x03C4} \diamond \unicode {x03B4}(\unicode {x03C1})\diamond \unicode {x03C4} + \unicode {x03B4}(\unicode {x03C4}) = \unicode {x03B4}(\unicode {x03C4}). \end{align*}

We claim that the map $\unicode {x03C4}$ is, moreover, analytic when $\unicode {x03C1}$ is analytic.

Proof. Because $\unicode {x03C1}$ is analytic and $\unicode {x03C4}_{0,0}$ is continuous, there exists $C\gt 0$ such that ${\left \|\unicode {x03C4}_{0,0}\right \|} \lt C$ and ${\left \|\unicode {x03C1}_{i,j}\right \|} \lt C^{i+j+1}$ for all $i,j\in \mathbb{N}$ . We claim that for $T\in \mathsf{Catp}(n_1,n_2,d)$ the map $\unicode {x03C4}_T$ satisfies ${\left \|\unicode {x03C4}_T\right \|} \lt C^{n_1+n_2+2d+1}$ . For $n_1=n_2=0$ and $d=0$ one has

\begin{align*} {\left \|\unicode {x03C4}_T\right \|} = {\left \|\unicode {x03C4}_{0,0}\right \|} \lt C=C^{0+0+0+1}. \end{align*}

Now let $n_1,n_2$ and $d\gt 0$ be arbitrary, then assuming the bound holds for all $d' \lt d$ it follows by the recursive formula that there exists $(l_1,l_2)$ and a tree $T_r\in \mathsf{Catp}(n_1-l_1,n_2-l_2,d-1)$ such that

\begin{align*} {\left \|\unicode {x03C4}_T\right \|} = {\left \|\unicode {x03C4}_{0,0}\right \|} {\left \|\unicode {x03C1}_{l_1,l_2}\right \|} {\left \|\unicode {x03C4}_{T_r}\right \|} \lt C \cdot C^{l_1+l_2+1} \cdot C^{(n_1-l_1)+(n_1-l_2)+2(d-1)+1} = C^{n_1+n_2+2d+2}. \end{align*}

As the number of internal nodes satisfies $d\leqslant i+j$ , it follows that ${\left \|\unicode {x03C4}_T\right \|} \lt C^{3(i+j)+3}$ for $i+j\gt 0$ . Hence, taking the sum over all trees in $\mathsf{Catp}(i,j)$ one finds that

\begin{align*} {\left \|\unicode {x03C4}_{i,j}\right \|} \leqslant \sum _{T\in \mathsf{Catp}(i,j)} {\left \|\unicode {x03C4}_T\right \|} \lt \sum _{T\in \mathsf{Catp}(i,j)} C^{3(i+j)+3} = |\mathsf{Catp}(i,j)| (C^3)^{i+j+1}. \end{align*}

Each $T\in \mathsf{Catp}(i,j)$ corresponds to a unique planar tree with less than $3(i+j)+3$ total nodes, so the cardinality of the set $\mathsf{Catp}(i,j)$ is clearly bounded by the Catalan number $\mathcal{C}_{3(i+j)+3}$ . Taking again a constant $q\gt 1$ such that $\mathcal{C}_n \lt q^n$ for $n\gg 0$ , it follows that ${\left \|\unicode {x03C4}_{i,j}\right \|} \lt (qC)^{i+j+1}$ for $i+j\gg 0$ . Hence $\unicode {x03C4}$ is analytic.

4.1 Strong homotopy inner products

If $\unicode {x03C3}$ is a cyclic structure on an $A_\infty$ -algebra $B$ then the pullback $f^*\unicode {x03C3}$ along a quasi-morphism $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ will, in general, not be a cyclic structure on $A$ because both the condition

\begin{align*} (f^*\unicode {x03C3})_{i,j} = 0 \;\textrm{if}\; i+j\gt 0, \end{align*}

and the requirement that $(f^*\unicode {x03C3})_{0,0}$ is an isomorphism may be violated; one says that cyclic structures are strict. For this reason it is better to use a homotopy-invariant alternative which relaxes the cyclic conditions. One such alternative was define by Cho [Reference ChoCho08], and we recall it here following the setup of [Reference Amorim and TuAT25].

Any cyclic structure defines a SHIP, but the latter class is much better behaved under $A_\infty$ -morphisms: given $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}(A,B)$ the pullback $f^*$ on bimodule morphisms restricts to a chain map

\begin{align*} f^* \colon \Omega ^{2,{\rm cl}}(B) \to \Omega ^{2,{\rm cl}}(A), \end{align*}

which maps SHIPs to SHIPs when $f$ is a quasi-isomorphism. Following the philosophy of [Reference Kontsevich and SoibelmanKS09], a SHIP can be viewed as a type of noncommutative shifted-symplectic structure: one can view the Hochschild cohomologies $\mathsf{C}^\bullet (A)$ and $\mathsf{C}^\bullet (A,A^\vee )$ as vector fields and differential 1-forms on a noncommutative space, and any cocycle $\unicode {x03C1} \in \Omega ^{2,{\rm cl}}(A)$ defines a contraction map

\begin{align*} \iota _-\unicode {x03C1}\colon \mathsf{C}^\bullet (A) = \mathsf{C}^\bullet (A,A) \to \mathsf{C}^\bullet (A,A_\Delta ^\vee ),\quad \unicode {x03B1} \mapsto \unicode {x03C1} \circ \overline {\unicode {x03B1}}, \end{align*}

which is a quasi-isomorphism if $\unicode {x03C1}$ is a SHIP. We now generalise the notion of strong homotopy inner products to analytic $A_\infty$ -algebras, as follows.

For an analytic $A_\infty$ -algebra $A\in {\mathsf{Alg}^{\infty ,{\rm an}}}$ , we define the complex of analytic closed 2-forms

\begin{align*} \Omega ^{2,{\rm cl},{\rm an}}(A) := \Omega ^{2,{\rm cl}}(A) \cap {\mathcal{A}}(A_\Delta ,A_\Delta ')[-2], \end{align*}

where we view ${\mathcal{A}}(A_\Delta ,A_\Delta ')$ as a subspace of $\mathcal{M}(A_\Delta ,A_\Delta ^\vee )$ via the inclusion $A_\Delta ' \hookrightarrow A_\Delta ^\vee$ . The assignment $A\mapsto \Omega ^{2,{\rm cl},{\rm an}}(A)$ is again functorial, as Lemma3.12 guarantees that there is a chain map

\begin{align*} f^* \colon \Omega ^{2,{\rm cl},{\rm an}}(B) \to \Omega ^{2,{\rm cl},{\rm an}}(A) \end{align*}

for every analytic $A_\infty$ -morphism $f\in {{\rm Hom}}_{{\mathsf{Alg}^\infty }}^{{\rm an}}(A,B)$ . We define the following analogue of a SHIP for an analytic $A_\infty$ -algebra admitting an analytic minimal model.

Definition 4.6. Let $A$ be an analytic $A_\infty$ -algebra $A$ with minimal model ${\rm H}(A)$ . Then an analytic SHIP is a cocycle $\unicode {x03C1}\in \Omega ^{2,{\rm cl},{\rm an}}(A)$ such that the map

\begin{align*} {\rm H}(A)[1] \xrightarrow {\ I\ } A[1] \xrightarrow {\unicode {x03C1}_{0,0}} A^\vee [1] \xrightarrow {\ I^\vee \ } {\rm H}(A)^\vee [1] \end{align*}

admits a continuous inverse.

As before, an analytic SHIP can be thought of as a noncommutative shifted-symplectic structure with contraction map