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Weak actions do not supply pentagon coherence data

Statement refuted

The pairwise invertible comparisons of a weak action of a group on a category, in the sense of Weak action of a group on a category, automatically satisfy the pentagon (associativity) coherence condition, so that every weak action can be used as a genuine coherent $2$ -action without further argument.

Facts & Assumptions

Given: The group $G = Z / 2 \times Z / 2$ presented as $⟨ a, b ∣ a^{2} = b^{2} = 1, a b = b a ⟩$ , the category $Q$ of complex vector spaces, the constant functor assignment $F_{g} = {Id}_{Q}$ for every $g \in G$ , and the weak-action definition and its pentagon of Weak action of a group on a category.

[L1]

Complex vector spaces and complex-linear maps form a category $Q$ , namely the category of left modules over the field $C$ with module homomorphisms (Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$ , Unital left and right modules over a ring; unqualified module means left module), and functors between categories and natural transformations between them are as in Covariant functor, identity functor, composite functor, and contravariant functor and Natural isomorphism.

[L2]

A weak action of $G$ on $Q$ is a functor assignment $g \mapsto F_{g}$ with $F_{1} = {Id}_{Q}$ and isomorphisms $F_{f g} ≅ F_{f} F_{g}$ for all $f, g$ ; a coherent action additionally requires chosen isomorphisms $μ_{f, g} ⁣ : F_{f} F_{g} \to F_{f g}$ whose two pentagon composites at every triple $(f, g, h)$ agree (Weak action of a group on a category).

[L3]

A natural endomorphism $η ⁣ : {Id}_{Q} \to {Id}_{Q}$ is multiplication by a scalar: evaluating at $C$ gives $λ : = η_{C} (1)$ , and naturality of $η$ with respect to the linear maps $C \to V$ , $1 \mapsto v$ , for $v \in V$ gives $η_{V} (v) = λ v$ for every complex vector space $V$ and every $v \in V$ . Consequently every natural isomorphism ${Id}_{Q} \to {Id}_{Q}$ is multiplication by a nonzero scalar $λ \in C^{\times}$ , and composition of such natural transformations is multiplication of scalars.

Counterexample

technique · direct

1.1L1L2

The data. Since $F_{g} = {Id}_{Q}$ for all $g \in G$ , the composite functors satisfy $F_{f} F_{g} = F_{f g} = {Id}_{Q}$ on the nose for every pair $(f, g) \in G \times G$ , and $F_{1} = {Id}_{Q}$ ; in particular the functor assignment is a weak action of $G$ on $Q$ in the sense of [L2] whenever the pairwise isomorphisms exist.

2.1step 1.1L3

The scalar comparisons. By [L3] a natural isomorphism $F_{f} F_{g} \to F_{f g}$ between identity functors is uniquely a nonzero scalar, so the following choices are well-defined natural isomorphisms: $μ_{a, b} : = (- 1) \cdot id$ , that is, multiplication by $- 1$ on every complex vector space, and $μ_{f, g} : = id$ (multiplication by $1$ ) for every other pair $(f, g) \in G \times G$ , including the pairs $(a, a)$ , $(a, a b)$ and $(1, b)$ .

3.1step 2.1L2L3

The pentagon at $(a, a, b)$ . The two composites of the pentagon for the triple $(a, a, b)$ map $F_{a} F_{a} F_{b} = {Id}_{Q}$ to $F_{a^{2} b} = F_{b} = {Id}_{Q}$ and are, by [L3], multiplication by the scalars $μ_{1, b} μ_{a, a} = 1 \cdot 1 = 1$ and $μ_{a, a b} μ_{a, b} = 1 \cdot (- 1) = - 1$ respectively; the first composite uses $μ_{a, a} ⁣ : F_{a} F_{a} \to F_{a^{2}} = F_{1}$ followed by $μ_{1, b}$ , and the second uses $F_{a} μ_{a, b} ⁣ : F_{a} F_{a} F_{b} \to F_{a} F_{a b}$ followed by $μ_{a, a b}$ , and composition of scalar natural transformations is multiplication. Since $1 \neq - 1$ in $C^{\times}$ , the two composites are distinct natural transformations, so the pentagon diagram does not commute.

4.1step 2.1step 3.1∎

Conclusion. The functor assignment $F_{g} = {Id}_{Q}$ together with the chosen pairwise isomorphisms of step 2.1 satisfies the letter of the weak-action definition of [L2] but not the pentagon of a coherent action, by step 3.1. Hence pairwise invertible comparisons do not automatically supply coherence data, the refuted statement is false, and the weak Khovanov–Seidel action of this page must not be assumed coherent without further argument.

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Definition 2.6 and the discussion after Proposition 2.7 (standard reference, not scraped)
Mikhail Khovanov and Richard Thomas, Braid cobordisms, triangulated categories, and flag varieties, Homology Homotopy Appl. 9 (2007) 19-94 (arXiv:math/0609335v2), Section 1 (weak action versus genuine action) (standard reference, not scraped)