Group actions on categories and Elagin’s theorem revisited

Abstract

After recalling basic definitions and constructions for a finite group G action on a k-linear category we give a concise proof of the following theorem of Elagin: if $\mathscr {C}= \langle \mathscr {A}, \mathscr {B}\rangle$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of G, and $\mathscr {C}^G$ is triangulated, then there is a semiorthogonal decomposition $\mathscr {C}^G = \langle \mathscr {A}^G, \mathscr {B}^G \rangle$. We also prove that any G-action on $\mathscr {C}$ is weakly equivalent to a strict G-action which is the analog of the Coherence theorem for monoidal categories.

1 Introduction

1.1 The setting of finite groups acting on categories is a well-studied ground, see e.g. [2,3,4,5, 9] and references therein. A useful way to define the action is to require for every $g \in G$ an autoequivalence $\rho _g:\mathscr {C}\rightarrow \mathscr {C}$ together with a choice of isomorphisms $\rho _g \rho _h \simeq \rho _{gh}$ satisfying a cocycle condition, see 2.1. One would then study the category of equivariant objects , see 2.4.

1.2 The main goal of this paper is to give a direct proof of Elagin’s theorem [3, 4] stating that if $\mathscr {C}= \langle \mathscr {A}, \mathscr {B}\rangle$ is a semi-orthogonal decomposition of triangulated categories and G is a finite group acting on $\mathscr {C}$ by triangulated autoequivalences in such a way that the category of equivariant objects $\mathscr {C}^G$ is triangulated and preserving $\mathscr {A}$ and $\mathscr {B}$, then there is a semi-orthogonal decomposition , see Theorem 6.2.

1.3 In our proof we construct the functors and adjoint to the inclusion functors. The key step in the proof is to show that if $\Phi :\mathscr {A}\rightarrow \mathscr {C}$ is a G-equivariant functor which admits a left or right adjoint functor $\Psi$, then $\Psi$ is automatically equivariant: see Proposition 3.9.

1.4 We also prove that every G-action is G-weakly equivalent to a strict G-action, that is to an action satisfying $\rho _g \rho _h = \rho _{gh}$, see Theorem 5.4. This is analogous to the Coherence Theorem for monoidal categories: every monoidal category is equivalent to a strict monoidal category, see e.g. [7, 1.2.15].

1.5 In order to formulate and prove these facts we need to develop the language of G-functors, G-natural transformations and so on. Perhaps relevant definitions and constructions are well known to experts but we include these for completeness as we could not find the reference that fits our purpose.

1.6 All categories, functors, etc are k-linear where . Groups acting on categories are finite and we denote by $1 \in G$ the neutral element of the group.

We use the symbol “

” to denote vertical composition of natural transformations of functors, the other types of compositions are denoted by concatenation.

2 G-categories and equivariant objects

2.1 2.1 Definition

By a G -action on $\mathscr {C}$ we mean the following data [4, Definition 3.1]:

• For each element $g \in G$ an autoequivalence $\rho _g:\mathscr {C}\rightarrow \mathscr {C}$.

• For each pair $g,h \in G$ an isomorphism of functors

  $$\begin{aligned} \phi _{g,h}:\rho _g \rho _h \cong \rho _{g h}. \end{aligned}$$

The data must satisfy the following associativity axiom: for all $g,h,k \in G$ the diagram of functors $\mathscr {C}\rightarrow \mathscr {C}$ is commutative:

2.2 It follows from the definition that there is an isomorphism of functors

$$\begin{aligned} \phi _1:\rho _1 \simeq \mathrm{id} \end{aligned}$$

obtained by post-composing $\phi _{1,1}:\rho _1 \rho _1 \rightarrow \rho _1$ with . That is we have

$$\begin{aligned} \phi _{1,1} = \rho _{1} \phi _{1}. \end{aligned}$$

Furthermore one can show that $\phi _1$ satisfies [5, 2.1.1 (e)]:

$$\begin{aligned} \phi _{g,1} = \rho _g \phi _1:\rho _g \rho _1 \rightarrow \rho _g, \qquad \phi _{1,g} = \phi _1 \rho _g:\rho _1 \rho _g \rightarrow \rho _g \end{aligned}$$

so that the definition 2.1 coincides with that of [5, 2.1].

On the other hand if one asks for $\phi _1$ to be the identity transformation, one gets a slightly stronger definition of a G-descent datum of [8, Definition 1.1].

2.3 Using the language of monoidal functors [7, Definition 1.2.10], one can give a very concise definition of a group acting on a category. For that consider G as a monoidal category: G is discrete as a category and its monoidal structure is defined by

Now a G-action on $\mathscr {C}$ amounts to the same thing as an action of monoidal category G on $\mathscr {C}$ [7, Example 1.2.12], i.e. a weak monoidal functor

where on the right is the category of functors $\mathscr {C}\rightarrow \mathscr {C}$ with monoidal structure given by composing functors.

2.2 2.4 Definition

One defines the category of G equivariant objects $\mathscr {C}^G$ [4, 5] as follows: objects of $\mathscr {C}^G$ are linearized objects, i.e. objects $c \in \mathscr {C}$ equipped with isomorphisms

$$\begin{aligned} \theta _g:c \rightarrow \rho _g (c), \qquad g \in G, \end{aligned}$$

satisfying the condition that the following diagrams are commutative:

Morphisms of equivariant objects consist of those morphisms of the underlying objects in $\mathscr {C}$ which commute with all $\theta _g$, $g \in G$.

3 G-functors and G-natural transformations

3.1 3.1 Definition

Given two categories $\mathscr {C},\mathscr {D}$ with G-actions and a functor , $\Phi$ is called a right lax G-functor if there are given natural transformations

such that the two natural transformations coincide:

This commutative diagram is called the pentagon axiom. Similarly ${\Phi }$ is called a left lax G-functor if there are given natural transformations

satisfying the dual pentagon axiom. A right (or left) lax G-functor ${\Phi }$ is called a weak G-functor if all $\delta _g$ are isomorphisms.

The following lemma is a useful criterion for a weak G-functor.

3.2 3.2 Lemma

Let ${\Phi }$ be a right (or left) lax G-functor. The following conditions are equivalent:

• (i) The natural transformation is an isomorphism.

• (ii) ${\Phi }$ satisfies the identity element axiom:

  

• (iii) ${\Phi }$ is a weak G-functor.

Proof

Implications (iii) $\Rightarrow$ (i), (ii) $\Rightarrow$ (i) are obvious. Let us prove that (i) $\Rightarrow$ (iii). Consider the case of the right lax G-functor. Applying the pentagon axiom to the pair gives

Since the natural transformation on the right-hand side is an isomorphism (note that $\delta _1$ is an isomorphism by the identity element axiom) and $\rho _{g},\rho _{g^{-1}}$ are equivalences, it follows that $\delta _{g^{-1}}$ is left invertible and $\delta _g$ is right invertible. Thus we see that all $\delta _g$ are isomorphisms.

Now we prove (i) $\Rightarrow$ (ii). Consider the natural transformation

We are given that $\varepsilon$ is an isomorphism and we need to prove that $\varepsilon$ is in fact an identity.

We use Lemma 3.3 applied to the trivial group and the composition

$$\begin{aligned} (\mathscr {C}, \mathrm{id}) \xrightarrow {(\mathrm{id},\,\phi _1)} (\mathscr {C}, \rho _1) \xrightarrow {({\Phi },\, \delta _1)} (\mathscr {D}, \rho _1) \xrightarrow {(\mathrm{id},\, \phi _1^{-1})} (\mathscr {D}, \mathrm{id}) \end{aligned}$$

which gives a lax G-functor $(\mathscr {C}, \mathrm{id}) \xrightarrow {({\Phi }, \varepsilon )} (\mathscr {D}, \mathrm{id})$. The pentagon axiom for this functor yields

and we deduce that $\varepsilon = \mathrm{id}$. $\square$

3.3 3.3 Lemma

If $({\Phi }, \delta ^{\Phi }):\mathscr {C}\rightarrow \mathscr {D}$, $({\Psi }, \delta ^{\Psi }):\mathscr {D}\rightarrow \mathscr {E}$ are right/left/weak G-functors, then their composition is a right/left/weak G-functor.

For the proof one needs to check that the composition satisfies the pentagon and/or the identity element axioms; this is a straightforward check.

3.4 3.4 Lemma

A weak G-functor induces a functor on the categories of equivariant objects such that the following diagram is commutative:

Proof

For $(c, \theta ) \in \mathscr {C}^G$ we define linearization on ${\Phi }(c)$ as a composition of isomorphisms

of with $\delta _g$. It is now a standard check that ${\Phi }(c)$ becomes an equivariant object and that ${\Phi }^G$ is a functor. $\square$

3.5 3.5 Definition

A natural transformation between two weak G-functors is called a G-natural transformation if for every $g \in G$ the following diagram commutes:

3.6 3.6 Lemma

A G-natural transformation $\mu$ between two weak G-functors ${\Phi }_1, {\Phi }_2:\mathscr {C} \rightarrow \mathscr {D}$ induces a natural transformation .

Proof

To prove that $\mu$ descends to a natural transformation we check that for every $(c, \theta ) \in \mathscr {C}^G$ the morphism $\mu :{\Phi }_1(c) \rightarrow {\Phi }_2(c)$ commutes with linearizations:

The transformation $\mu ^G$ is natural since the original transformation $\mu$ is natural and the forgetful functor is faithful. $\square$

3.7 3.7 Definition

Two weak G-functors , are called G-adjoint if they are adjoint and the unit and counit of the adjunction are G-natural transformations.

3.8 3.8 Lemma

A G-adjoint pair of functors ${\Phi },{\Psi }$ induces an adjoint pair between the categories of equivariant objects.

Proof

From 3.6 it follows that we have natural transformations , . The condition for ${\Psi }$ and ${\Phi }$ to be adjoint is that two compositions

and

are identities. Since the forgetful functor is faithful, the same holds for . $\square$

3.9 3.9 Proposition

A left or right adjoint ${\Psi }$ to a weak G-functor ${\Phi }$ can be made into a weak G-functor in such a way that ${\Psi }$ and ${\Phi }$ become G-adjoint.

Proof

Let ${\Psi }$ be the left adjoint to . We construct the structure of a left lax G-functor on ${\Psi }$ using the structure of a right lax G-functor on ${\Phi }$.

Let and be the unit and the counit of the adjunction. Given a right lax G-structure on ${\Phi }$, we define the left lax G-structure on ${\Psi }$ as a mate of $\delta _g$ with respect to the adjunction [6, Proposition 2.1], [7, pp. 185–186], i.e.

The pentagon axiom can be expressed as an equality of certain compositions in the double category of [6, p. 86], hence is preserved under taking mates by [6, Proposition 2.2]. Checking the identity axiom for $\delta _1'$ is straightforward.

Now by 3.2, ${\Psi }$ becomes a weak G-functor. The proof for right adjoints is analogous.

We now need to prove that the unit and counit transformations $\varepsilon ,\eta$ are G-natural. We do the proof for the unit $\varepsilon$. We need to check that the following diagram commutes:

Here $\delta _{{\Phi }{\Psi }}$ is defined using 3.3. Unraveling the definitions we are left with checking the diagram (where we use simplified notation for the natural transformations to denote the obvious compositions)

which is easily seen to commute. $\square$

3.10 3.10 Corollary

Let be a weak G-functor. Then the following conditions are equivalent:

• (a) ${\Phi }$ is an equivalence of categories

• (b) There exists a weak G-functor and G-natural isomorphisms , .

In this case we will call ${\Phi }$ a weak G-equivalence.

Proof

We only need to prove (a) $\Rightarrow$ (b) as the opposite implication is trivial. Let be the quasi-inverse functor to ${\Phi }$. In particular ${\Psi }$ and ${\Phi }$ are adjoint (both ways) so that by 3.9 ${\Psi }$ has a structure of a weak G-functor with compositions G-isomorphic to identity functors. $\square$

4 Example: G-actions on the category of vector spaces

4.1 In this section we review a well-known example of how equivalence classes of G-actions on the category of k-vector spaces correspond bijectively to cohomology classes $H^2(G, k^*)$.

4.2 Let $\mathscr {C}= \mathrm{Vect}_k$ be the category of k-vector spaces, and let $\rho$ be the G-action on $\mathrm{Vect}_k$. As every autoequivalence of $\mathscr {C}$ is isomorphic to the identity functor, let us assume $\rho _g = \mathrm{id}$ for every $g \in G$. In this setup the data of the G-action $\rho$ defined in 2.1 is equivalent to specifying a cocycle $\phi \in Z^2(G,k^*)$.

4.3 Consider two G-actions on $\mathrm{Vect}_k$ given by cocycles . For the G-actions to be equivalent there needs to exist a weak G-functor

which is an equivalence of categories. Then the pentagon axiom 3.1 requires existence of an element $\delta = (\delta _g)_{g \in G} \in Z^1(G,k^*)$ such that for all g, h. Thus G-categories $(\mathrm{Vect}_k, \phi )$ and $(\mathrm{Vect}_k, \phi ')$ are equivalent if and only if .

4.4 The category of equivariant objects $(\mathrm{Vect}_k, \phi )^G$ is the category of $\phi$-twisted G-representations with objects given by vector spaces V together with isomorphism $\theta _g:V \rightarrow V$ satisfying $\theta _{gh} =$ and G-equivariant morphisms. In particular, if $\phi$ is the trivial cocycle, so that G-action on $\mathrm{Vect}_k$ is trivial, $\mathrm{Vect}_k^G$ is the category of G-representations.

5 Strictifying G-actions

5.1 Let ${\Omega }(G)$ denote the category with one object for every element $g \in G$ with for $g \ne h$.

5.2 Let $\mathscr {C}$ be a category with a G-action. Consider the category of weak G-functors and G-natural transformations from ${\Omega }(G)$ to $\mathscr {C}$

We endow $\mathscr {C}'$ with the strict G-action induced by the G-action on ${\Omega }(G)$.

5.3 Explicitly the objects of $\mathscr {C}'$ consist of families together with isomorphisms satisfying the cocycle condition that two ways of getting an isomorphism coincide. The morphisms from $(c_g)_{g \in G}$ to $(d_g)_{g \in G}$ are morphisms satisfying the condition that the two natural ways of forming a morphism coincide.

5.1 5.4 Theorem

The functor sending $(c_g)_{g \in G}$ to $c_1$ is a weak G-equivalence. Hence, every G-action is weakly equivalent to a strict G-action.

Proof

We need to check that ${\Phi }$ has a structure of a weak G-functor and that ${\Phi }$ is fully faithful and essentially surjective.

The structure of a weak G-functor on ${\Phi }$ is in fact simply given by the structure maps $\delta _{h,g}$. That is we have functorial isomorphisms

and the pentagon axiom follows from the cocycle condition on $\delta$.

To check that ${\Phi }$ is essentially surjective, one checks that for any $c \in \mathscr {C}$ the family $(\rho _g(c))$ has a structure of an object from $\mathscr {C}$. Furthermore, one can see that any object $(c_g)_{g \in G}$ is isomorphic to $(\rho _g(c_1))_{g \in G}$.

Thus to check that ${\Phi }$ is fully faithful, we may take two objects and $(d_g)_{g \in G} = (\rho _g(d_1))$ and a morphism between them. It is then easy to see that and that conversely for any , the collection $\rho _g(f_1)$ defines a morphism between c and d. $\square$

6 Elagin’s theorem

6.1 If $\mathscr {C}$ is a triangulated category and G acts by triangulated autoequivalences, then $\mathscr {C}^G$ is endowed with a shift functor and a set of distinguished triangles: these are the triangles that are distinguished after applying the forgetful functor . Furthermore under some mild technical assumptions this gives $\mathscr {C}^G$ the structure of a triangulated category [4, Theorem 6.9], for instance existence of a dg-enhancement of $\mathscr {C}$ is a sufficient condition for $\mathscr {C}^G$ to be triangulated [4, Corollary 6.10].

6.1 6.2 Theorem

Let $\mathscr {C}= \langle \mathscr {A}, \mathscr {B}\rangle$ be a semi-orthogonal decomposition of triangulated categories. Let G act on $\mathscr {C}$ by triangulated autoequivalences which preserve $\mathscr {A}$ and $\mathscr {B}$. Assume that the equivariant category $\mathscr {C}^G$ is triangulated with respect to triangles coming from $\mathscr {C}$. Then are triangulated and there is a semi-orthogonal decomposition

Proof

The existence of an adjoint pair between $\mathscr {C}$ and $\mathscr {C}^G$ [4, Lemma 3.7] implies that and . In particular, $\mathscr {A}^G$ and $\mathscr {B}^G$ are triangulated subcategories of .

Now in order to establish the semi-orthogonal decomposition it suffices to show that the embedding has a left adjoint [1, 1.5]. This holds true by 3.9, 3.8: the functor $i:\mathscr {A}\rightarrow \mathscr {C}$ is (strictly) G-equivariant, hence its left adjoint $p:\mathscr {C}\rightarrow \mathscr {A}$ induces an adjoint $p^G$ to the embedding . $\square$