A Natural Transformation that vanishes on generator objects

$\gdef\cal{\mathcal }$ $\gdef\ccal{\mathcal C}$ $\gdef\dcal{\mathcal D}$

Let $\cal C, \dcal$ be pre-triangulated $A_\infty$ (or dg) categories, $F, G: \ccal \to \dcal$ exact $A_\infty$ functors. We assume $\cal C_{gen}$ is a full subcategory of $\cal C$ that generate $\cal C$ in the sense that $Tw(\cal C_{gen}) \cong \cal C$.

It is true that

• Any $A_\infty$ exact functor $F: \ccal \to \dcal$ is determined by its restriction $F_{gen}: \ccal_{gen} \to \dcal$.
• An $A_\infty$ natural transformation $\eta: F \to G$ is determined by its restriction $\eta_{gen}: F_{gen} \to G_{gen}$.

However, it is not true that $\eta_{gen}$ is determined by $\eta_T: F(T) \to G(T)$ for all $T \in \ccal_{gen}$ that satisfies compatibility conditions with morphisms, i.e. for all $f: T_1 \to T_2$, we have $G(f) \circ \eta_{T_1} = \eta_{T_2} \circ F(f)$ in $Hom(F(T_1), G(T_2))$.

In this note, we give an example of a non-trivial natural transformation $\eta$ of degree $0$, such that $\eta_T = 0$ for all generator $T$.

We work throughout with cohomological grading conventions.

An $A_\infty$ category $\mathcal{C}$ consists of a collection of objects together with graded morphism spaces and higher compositions

$$ m_k : \mathrm{Hom}(X_{k-1},X_k)\otimes \cdots \otimes \mathrm{Hom}(X_0,X_1) \to \mathrm{Hom}(X_0,X_k)[2-k], $$

satisfying the usual $A_\infty$ relations. A dg category is the special case where $m_k=0$ for $k\ge 3$.

An $A_\infty$ functor $F:\mathcal{C}\to\mathcal{D}$ consists of maps

$$ F^k : \mathrm{Hom}(X_{k-1},X_k)\otimes \cdots \otimes \mathrm{Hom}(X_0,X_1) \to \mathrm{Hom}(F(X_0),F(X_k))[1-k], $$

satisfying compatibility equations with the $m_k$'s. When $F^k=0$ for $k\ge 2$, this is just an honest dg functor.

Given two $A_\infty$ functors $F,G:\mathcal{C}\to\mathcal{D}$, an $A_\infty$ natural transformation $\eta:F\Rightarrow G$ of degree $d$ consists of multilinear maps

$$ \eta^k : \mathrm{Hom}(X_{k-1},X_k)\otimes \cdots \otimes \mathrm{Hom}(X_0,X_1) \to \mathrm{Hom}(F(X_0),G(X_k))[d-k], $$

satisfying a hierarchy of equations generalizing the usual naturality condition. In particular:

• $\eta^0$ gives the objectwise components;
• $\eta^1$ measures failure of strict naturality;
• higher $\eta^k$ correct higher-order failures.

Let $\ccal_{\mathrm{gen}}$ be a $A_\infty$ category. Its twisted-complex envelope $\mathcal{C} = Tw(C_{\mathrm{gen}})$ is the $A_\infty$ category with objects the twisted complexes $(\oplus_i X_i[d_i], \delta)$.

A basic (and standard) fact is that: any $A_\infty$ functor $F_{gen}:\ccal_{\mathrm{gen}}\to \mathcal{D}$ into a pretriangulated $A_\infty$ category $\mathcal{D}$ extends canonically to an $A_\infty$ functor $${F}:\mathcal{C} \to Tw(\mathcal{D}) \cong \mathcal{D}.$$

$$ (\oplus_i X_i[d_i], \delta) \mapsto (\oplus_i F^0(X_i[d_i]), F^1(\delta) + F^2(\delta,\delta) + \cdots ) $$

The category $C_{\mathrm{gen}}$ has three objects

$$ T_1,\quad T_2,\quad T_3. $$

For each pair $(i,j)$ there is a unique degree $0$ morphism $a_{ij}:T_i\to T_j$, with $a_{ii}=e_i$ the identity. In addition, each object carries a distinguished endomorphism $$t_i:T_i\to T_i.$$

Composition are given by

$$ a_{ij}a_{jk}= \begin{cases} a_{ik} & \text{if } i\le j\le k \text{ or } i\ge j\ge k \cr 0 & \text{ otherwise} \end{cases} $$

$$ \forall i\neq j \quad a_{ji} t_i = t_i a_{ij} = 0, \quad t_i t_i=0. $$

There is no differential on morphisms.

We now define a functor

$$ B : \ccal \to \ccal, $$

which should be thought of as a toy “braiding.”

$$ B(T_1) = T_1,\qquad B(T_3) = T_3, $$

and

$$ B(T_2) = \big[ T_2 \xrightarrow{(a_{21},\,a_{23})} T_1 \oplus T_3 \big], $$

where $T_2$ sits in degree $-1$ and $T_1 \oplus T_3$ in degree $0$.

• $B(e_i)$ is the identity
• $B(t_i)$ acts termwise by $t_i$.
• $B(a_{12}) : B(T_1) \to B(T_2)$, send $T_1$ to the $T_1$ summand.
• $B(a_{32}) : B(T_3) \to B(T_2)$, sned $T_3$ to the $T_3$ summand.
• $B(a_{21}) : B(T_2) \to B(T_1)$, sends $T_3$ summand to $T_1$.
• $B(a_{23}) : B(T_2) \to B(T_3)$, sends $T_3$ summand to to $T_3$.

The composites

$$ T_2 \xrightarrow{a_{21}} T_1 \xrightarrow{a_{12}} T_2,\qquad T_2 \xrightarrow{a_{23}} T_3 \xrightarrow{a_{32}} T_2 $$

are zero in $C_{\mathrm{gen}}$, but their images under $B^1$ are not zero endomorphisms of $B(T_2)$. Therefore, $B$ cannot be a strict dg functor.

We fix this by declaring $B$ to be an $A_\infty$ functor with nonzero $B^2$ on exactly these zig--zag pairs:

• $B^2(T_2 \to T_1 \to T_2)$ is the degree $-1$ endomorphism of $B(T_2)$ sending $T_3 \longrightarrow T_2$.
• $B^2(T_2 \to T_3 \to T_2)$ is the degree $-1$ endomorphism of $B(T_2)$ sending $T_1 \longrightarrow T_2$.

A quick way to summarize the needed $A_\infty$ constraint on $A_\infty$ functors between dg cats $B: C \to D$ is the $k=2$ equation:

$$ \mu_D^1(B^2(f_2,f_1)) + \mu_D^2(B^1(f_2), B^1(f_1)) = B^1(\mu_C^2(f_2,f_1)) + B^2(\mu_C^1(f_2),f_1) + (-1)^{|f_2|} B^2(f_2,\mu_C^1(f_1)). $$

A direct computation shows that the differentials of these homotopies cancel exactly the unwanted composites of $B^1$. No $B^k$ for $k \ge 3$ is needed.

We now define a $A_\infty$ natural transformation of degree $0$

$$ \eta: \mathrm{Id} \to B. $$

We set

$$ \eta^0(T_1) = \eta^0(T_2) = \eta^0(T_3) = 0. $$

So objectwise, $\eta$ vanishes completely.

The only nonzero higher component is

$$ \eta^1(t_2) \in \mathrm{Hom}^{-1}(T_2, B(T_2)), $$

defined by the map

$$ T_2 \xrightarrow{t_2} T_2 $$

landing in the degree $-1$ term of $B(T_2)$. All other $\eta^1(f)$ vanish, and we set $\eta^k = 0$ for $k \ge 2$.

• The $n=1$ naturality equation holds because $(a_{21},a_{23}) \circ t_2 = 0$.
• The $n=2$ equations hold because $t_2$ kills all non-identity compositions, and the maps $B(a_{21})$, $B(a_{23})$ do not see the degree $-1$ term.
• Turning on $B^2$ adds new terms to the naturality equations, but all of them vanish automatically since $\eta^0 = 0$ and $\eta^1$ is supported only on $t_2$.

Finally, $\eta_{\mathrm{gen}}$ is not exact. Any degree $-1$ pre-natural transformation $h: Id \to B$ has only a possible objectwise component

$$ h^0_{T_2} = \alpha e_2 + \beta t_2, $$

and a direct computation shows that

$$ (dh)^1(t_2) = 0 $$

for all choices of $\alpha,\beta$. Hence $\eta_{\mathrm{gen}}$ represents a nontrivial cohomology class.

Moral: natural transformations can be real sneaky — even if nothing happens on objects, genuinely new information can live entirely in higher components.