Peng Zhou

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====== A Natural Transformation that vanishes on generator objects======

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


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 $T$. 

===== $A_\infty$ functors and natural transformations (brief recall) =====

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.

==== From generators to twisted complexes ====

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 quiver-like category $C_{\mathrm{gen}}$ =====

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.

===== The braiding functor $B$ (with homotopies) =====

We now define a functor

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

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

==== On objects ====

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

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 $0$.

==== On morphisms (the $B^1$ part) ====

  * $B(e_i)$ 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 $T_3$.

==== Higher homotopy $B^2$ ====

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.

===== The natural transformation $\eta$ =====

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

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

==== Objectwise component ====

We set

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

So objectwise, $\eta$ vanishes completely.

==== First higher component ====

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 $k \ge 2$.

==== Consistency ====

  * 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 $t_2$.

==== Non-exactness ====

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.//