Theory and Applications of Categories, Vol. 30, No. 43, 2015, pp. 1429–1468.

TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS

JEFFREY C. MORTON AND ROGER PICKEN

Abstract. Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in set-theoretic terms. This paper examines the analogous situation for structures described in categorytheoretic terms, where symmetry is expressed as the action of a 2-group G (equivalently, a categorical group) on a category C. It describes the construction of a transformation groupoid in diagrammatic terms, and considers this construction internal to Cat, the category of categories. The result is a double category C//G which describes the local symmetries of C. We define this and describe some of its structure, with the adjoint action of G on itself as a guiding example.

1. Introduction

Symmetry is a fundamental notion both in mathematics and in physics. In this paper, we consider the symmetries of categories: in particular, if a symmetry of a category C is represented as an endofunctor of C, one also has ’symmetries of symmetries’, because there are natural transformations between these functors. These two levels of symmetry can be represented by using a 2-group. Thus we consider actions of 2-groups on categories, and develop the analog of the “transformation groupoid” construction, which captures the local view of the symmetries for a group action on a set.

1.1. Symmetries, group actions and 2-group actions. The usual mathematical representation of symmetry is as a group action on a set, or perhaps on a manifold, vector space, or whatever sort of structure is of interest. Not only the objects on which the groups act, but also the groups themselves may have various sorts of structure - one may be interested in algebraic groups, or Lie groups, for example. In particular, symmetry in this sense is described by the action of a group object in an appropriate category: an object in the category equipped with structure maps, such as the multiplication map m : G × G → G, satisfying the axioms which define a group.

The authors would like to acknowledge Dany Majard for many helpful discussions in the writing of this paper. This work was supported in part by the Graduiertenkolleg 1670 “Mathematics Inspired by String Theory and QFT” of the Mathematics Department of the University of Hamburg, and by the Fundação para a Ciência e a Tecnologia, Portugal, through projects PTDC/MAT/101503/2008, EXCL/MAT-GEO/0222/2012 and PEst-OE/EEI/LA0009/2013.

Received by the editors 2014-10-03 and, in revised form, 2015-10-07. Transmitted by James Stasheff. Published on 2015-10-26. 2010 Mathematics Subject Classification: 18B40, 18D10, 20L99. Key words and phrases: 2-group, categorical group, crossed module, action, double category, adjoint

action.

\begin{matrix}c \\ ◯ \end{matrix} Jeffrey C. Morton and Roger Picken, 2015. Permission to copy for private use granted.

Even superficially very different constructions, such as quantum groups, fit this pattern to some extent. A quantum group is a rather broad concept with many variations, but the starting point is the idea of a Hopf algebra, a bialgebra with a certain structure. Quantum groups play a role in describing symmetry in noncommutative geometry and other algebraic settings. Hopf algebras may be seen as group objects in Alg^{op}, the opposite category of some category of algebras. (Which category of algebras depends on the context within noncommutative geometry).

The notion of a group action, also in the broader sense above, describes what are known as global symmetries: global operations on the object which supports the symmetry, leaving it in some sense unchanged. The exact sense in which it is to be unchanged, and what a transformation of the object can consist of, is the guiding criterion in finding its group of symmetries. On the other hand, there is also a local concept of symmetry. Following Weinstein [28], local symmetry may be described in terms of groupoids. Recall that groupoids are categories in which all morphisms are invertible. Now the entire set which supports the symmetry, such as the set of points of a space, forms the objects of the groupoid, and the morphisms of the groupoid represent symmetry relations which relate one point to another, making them “indistinguishable” under the relation in question. Such symmetry relations may or may not arise from a global group action on the set, and thus the local “groupoid approach” to symmetry is more fine-grained than the global “group action approach”.

Of course, given a global group action, it can also be viewed from an equivalent local perspective, in terms of the transformation groupoid associated to the global group action. Note that not every groupoid representing local symmetry is a transformation groupoid^{1}. In our generalization of group actions to 2-group actions, we will develop both global and local perspectives, and show them to be equivalent.

TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS 1431

The construction of a transformation groupoid, coming from a group action on a set, makes sense not only in the category of sets, but also in other contexts. Typical examples would include a Lie group acting on a manifold, a topological group acting on a topological space, or an algebraic group acting on a variety. In reasonable situations (which always apply in a topos, such as the category of sets, or of topological spaces), one can construct a transformation groupoid which captures the local picture implied by such global symmetry groups. These will be, respectively, Lie groupoids, topological groupoids, or algebraic groupoids in the cases mentioned above. We can say that each of these situations is an instance of a general process, which can be described diagrammatically, and independently of the type of objects playing the role of “sets”. In other words, there is a general construction of a transformation groupoid, which makes sense in a category (of a suitable kind). Each of the cases we mentioned are examples of this construction “internal to” a chosen ambient category, such as topological spaces, manifolds, varieties, and so forth.

1Some easy counter-examples appear as sub-groupoids of transformation groupoids. For instance, suppose we have an action of a Lie group G on a manifold M. We can consider a small neighborhood U in M, and take the sub-groupoid of the transformation groupoid associated to the action of G with only those objects corresponding to points in U. This has all the morphisms taking points in U to other points in U, even if U does not contain any entire orbit of the G action. That is, it retains all the ’local’ information about M’s symmetries that can be detected in U.

The situation we describe in this paper generalizes these ideas in a different direction. We are interested in describing symmetry of categories by taking the notion of a group action on a set to a higher categorical level, both for the group and for the set it acts on. Thus we will be studying the action of a 2-group on a category.

In order to understand this step, it is useful to recall that a group is a particular type of category, with just a single object and all morphisms invertible. The product on the group is the composition of morphisms. Now, given an object X in a category, X, there is a sub-category End(X) of X with one object X and morphisms consisting of all the endomorphisms of X. Restricting to the invertible such morphisms, one gets Aut(X). This is a subcategory of X, but it is also a group in the sense just mentioned. It is, indeed, the full symmetry group of the object X. In these terms, the action of a group G on X is described by a functor from the category G to the category of automorphisms of X, say ϕ : G → Aut(X).

Now, just as a group may be regarded as a special sort of category, so too a 2-group can be seen as a special kind of 2-category, namely one with just a single object and invertible 1- and 2-morphisms. Given a category C, i.e. an object in the 2-category Cat, one has endofunctors from C to itself, but also natural transformations between such endofunctors. So End(C) is a 2-category with a single object C, whose 1-morphisms are endofunctors of C and whose 2-morphisms are natural transformations between endofunctors. If we restrict to only those endofunctors and natural transformations which are invertible, we have Aut(C), which is a 2-group. It is, indeed, the full symmetry 2-group of C. Then, in complete analogy with the action of a group G on a set X, the action of a 2-group G on a category C is described by a 2-functor from the 2-category G to the 2-category of automorphisms of C:

Φ : G → Aut(C).

Intuitively, following Baez and Lauda in [1], we are considering not only symmetries of C (the 1-morphism level of Φ), but also “symmetries between symmetries” (the 2-morphism level of Φ).

As an aside, we note that there are other approaches to generalizing the action of G on X to higher settings. Some recent work has been done on categories equipped with actions of a group G, (note, not a 2-group) and their “equivariantizations” (see, for example, [15, 24, 9] among others). These equivariantizations generalize the notion of the fixed points of a group action on a (possibly structured) set. From our perspective this is a rather special case of the most general sense of symmetry for categories since we expect to have not only morphisms between categories (namely, functors), but also morphisms between these morphisms (namely, natural transformations). This means that, for us, the notion of “symmetry” of a category is inherently somewhat more subtle than that for a set, or even for any set with an additional structure, such as a topological space. Another generalization, going beyond ours, is to consider the action of 2-groupoids, or yet more general categorical structures, on a category [6]. A 2-groupoid is a 2-category with invertible 1- and 2-morphisms, but it may have more than one object. However, we wish to focus on the 2-group case, because it naturally generalizes the important and ubiquitous notion of actions of groups, and because the application we have in mind (in higher gauge theory, to be described shortly) motivated us to study 2-group actions, and not the action of anything more general. Finally, we note the recent work by J. Elgueta [13], where a weak 2-group acts on a groupoid by equivalences, rather than by endofunctors. Here the self-equivalences of an object in a 2-category naturally constitute a weak 2-group. This is not directly comparable to our approach, since weak 2-groups are used throughout in Elgueta’s paper.

Returning to our main discussion, the essential point in this paper is that 2-group actions, too, can be understood in terms of an internal construction like those described above, i.e. transformation groupoids internal to some category. The ambient category is now Cat, the category whose objects are (small) categories and whose morphisms are functors. Then we can use the fact that an alternative way to see 2-groups is as categorical groups, namely group objects in Cat. Starting with an ordinary group action on a set, the corresponding transformation groupoid is obtained from a function, ϕ̂: G × X → X, satisfying the usual properties for an action. Here G is treated as a set, although the properties involve the group operation on G. In the same way, a functor:

Φ̂ : G × C → C

satisfying analogous properties to ϕ̂, and where G is treated as a category as opposed to a 2-category, is equivalent to an action of the 2-group G on the category C, as described above (Theorem 3.4). From Φ̂ one obtains the description of the action as a transformation groupoid internal to Cat. In this way we get a local symmetry picture for the global action of a 2-group on a category, just like the global and local pictures that we had for the action of an ordinary group.

We have been using two different understandings of a 2-group action on a category C. The view where the 2-group is taken to be a group object in Cat is generally called “internal”. By analogy, we may call the other “external”, i.e. in terms of a 2-functor from the 2-category G to the 2-category Aut(C), whose sole object is C . The “internal” description makes sense in any category where a group object can be defined, but describes a 2-group action only in Cat. The “external” description always describes the action of a 2-group, by its very definition. The existence of both internal and “external” views of symmetry for a category relies on the fact that Cat is a closed category, that is, a category with an “internal Hom”, so that the morphisms from A to B form an object Hom(A, B) within the category. Thus Aut(C) can be seen as a category, equipped with certain structure maps which make it into a 2-group.

Now, the internal, local symmetry picture associated with Φ̂ above gives rise to a transformation groupoid in the same ambient category as our group object, and the object it acts on. So for categories, we find that this local symmetry structure is most naturally encoded by a category (indeed, it will be a groupoid) internal to Cat itself. Such a structure is called Cat-category, or often a double category (though to avoid ambiguity, we will reserve this name for a distinct but equivalent structure). In particular, this structure is not generally just a 2-category. An external view of it shows that it has two different kinds of morphism, denoted horizontal and vertical, and higher morphisms which naturally have the shape of squares.^{2} We will show in Section 4 that there are several complementary perspectives for viewing this structure, all highlighting particular features of the 2-group action. The groupoid internal to Cat perspective has a dual description as a category internal to Gpd (the category of groupoids), and a third, symmetric perspective views the structure as a double category (becoming a double groupoid when the acted-on category C is a groupoid). Thus our treatment here demonstrates that there is extra subtlety, through these “variations in shape”, when we try to extend the concept of symmetry to higher structures.

1.2. Motivation from higher gauge theory. Our motivation for the present study comes from geometry, although this will play no direct role in this paper. We will, however, describe a little of this motivation here, and in a subsequent paper we will give more details about the example which prompted us to study this question.

The example we had in mind comes from higher gauge theory (HGT) [3]. This is a general program of developing analogs to concepts in gauge theory, for higher-categorical groups. It provides generalizations of, among other things, the theory of connections on bundles over manifolds. These analogs include the study of connections on (non-abelian) gerbes [14, 23, 20, 19, 27]. These entities have been the study of considerable interest in recent years, particularly since they can be used to describe theories in which one can naturally describe the parallel transport of extended objects such as (one-dimensional) strings or (higher-dimensional) “membranes”.

Behind the current paper is a desire to better understand the symmetries of the “moduli spaces” for such theories. That is, in ordinary gauge theory, given a manifold M and a Lie group G, there is a space of all connections on a given principal G-bundle over M. Furthermore, there is a group which acts on this space, the group of all gauge transformations. There is a local picture of this symmetry, in which both connections and gauge transformations are described in terms of certain forms and functions on M. The relation between the global symmetries and this local picture is well-understood. The local picture describes a groupoid, whose objects are the connections, coming from the global group action.

In the case of connections on gerbes, there is also a well-understood local picture in terms of forms and functions. However, these naturally form something more complex than a groupoid. The relationship between this local picture and a notion of global symmetries is less well-established. In our subsequent paper, we will show a simple example of how the local picture arises from a global symmetry 2-group acting on a category which plays the role of the moduli space of connections on gerbes over M. For even higher-categorical structures, known as n-gerbes, one can describe the transport of higher-dimensional objects, or n-branes. This naturally has connections to string theory [26, 2], and to other areas in theoretical physics.

2The shape of higher morphisms is a typical issue in higher-categorical constructions: see e.g. [10, 18] for discussion of variants of higher categories.

Our main specific motivation within HGT relates to work of Yetter [29], and Martins and Porter [21], extending a certain field theory (the Dijkgraaf-Witten model) to 2-groups. One of the ultimate goals in our project is to reproduce for these theories a groupoid-based approach to such a model [22], using the double groupoids we find here, relating these field theories to the representations of the double groupoids.

While the motivation from HGT is of interest to us, it is not necessary to understand the present paper. For the moment, we are simply interested in understanding better how the relationship between local and global symmetries applies when the entity whose symmetries are involved happens to be a category. Since the notion of symmetry is one of the most fundamental, and categories have proven to be a vital and important part of modern mathematics, the present study should be of interest well beyond the context of our original example, and applicable in a variety of areas.

1.3. Results and outline. The main results of this work are structural theorems relating to the action of a strict 2-group G on a category C, and the different ways in which this action may be described.

We begin in Section 2 with a discussion of (strict) 2-groups, and various equivalent definitions which are useful in talking about them. This includes the correspondence between 2-groups and crossed modules and the definition of the double groupoid associated to the 2-group G, which is both an introduction to the notion of double category and a useful 2-dimensional diagrammatic tool for 2-group calculations.

This is followed by Section 3 which defines both the global and local notions of a 2-group G acting on a category C (see the first part of the introduction), i.e. Definitions 3.2 and 3.3, and shows the equivalence between the two definitions in Theorem 3.4. This is followed in subsection 3.6 by a basic example, namely the adjoint action of a 2-group on itself, analogous to the adjoint action of a group on itself by conjugation. (In the forthcoming paper on higher gauge theory, we show that the adjoint action of G on itself describes higher gauge theory for the simple case when the manifold is the circle, S^{1}.)

In our main Section 4, we start by describing the construction of the transformation groupoid associated to a group action on a set, in a diagrammatic form, which captures the description of this groupoid in terms of elements. In subsection 4.2, repeating this construction in Cat gives a groupoid internal to Cat, as shown in Theorem 4.4. We develop a few different points of view on this internal groupoid in subsection 4.5, first taking a transposed internal view, giving rise to a category internal to Gpd, the category of groupoids (Theorem 4.8), and then giving a symmetrical definition in terms of a double category (Theorem 4.10). The transposed viewpoint turns out to have close connections to ordinary transformation groupoids for certain group actions. Further aspects of these viewpoints are developed in subsection 4.12, and in particular we show that there are two 2-categories associated to the transformation double category, namely the horizontal 2-category and the vertical 2-category, Theorems 4.16 and 4.17. Our standard example, that of the adjoint action of a 2-group on itself, gives a concrete case throughout, which we develop in detail in 4.18.

2. 2-Groups and 2-Group Actions on a Category

In this paper, we will be describing certain actions of 2-groups. The structures known as 2-groups are also sometimes called categorical groups, gr-categories or groupal groupoids. Moreover, all of these can be shown to be equivalent to crossed modules (of groups). As the abundance of terminology suggests, there are several conceptually different, but logically equivalent, definitions which can be given for these structures. We will primarily use the term “2-group”, but in fact it will be useful for our discussion to be able to move back and forth between the different definitions. In Section 2.1 we will recall those definitions which will be used, and note how they are equivalent.

This prepares the ground for section 3, where we will consider from two of these points of view how 2-groups can act on categories, and the “transformation” structure which results. This will be the analog of the transformation groupoid associated to a group action on a set.

Note that in the following, to avoid cumbersome notation, we use the notation X^{(0)} and X^{(1)} to denote, respectively, the objects and morphisms of a category X, and similar notation for the object and morphism maps of a functor. For a 2-category X, when appropriate, we denote the 2-morphisms by X^{(2)}.

2.1. 2-Groups, Categorical Groups, and Crossed Modules. Here we lay out three definitions of equivalent structures: 2-groups, categorical groups, and crossed modules. In this section, we will be careful to distinguish the three, but in the rest of the paper we will generally use the term “2-group”. The main point of this section is to highlight the well-known result that there is an equivalence between the three definitions.

2.1.1. 2-Groups as 2-Categories. One very useful definition of a 2-group is motivated by analogy to the definition of a group as a (small) one-object category whose morphisms are all invertible. In this case, the elements of the group are the set of morphisms of the category, and the group multiplication is the composition.

This highlights the way the definition of a category generalized the notion of “composition” which begins with algebraic gadgets such as groups. One might equally well define a (small) category in algebraic language as a unital semigroupoid. Taking the definition of category as more basic gives a group as a special case. While of course not the standard definition of a group, it is at least straightforward to generalize to “higher” groups in the sense of higher category theory.

(We will assume some familiarity with 2-categories in what follows, though for readers who are not so familiar we suggest the succinct note by Leinster [17] as a starting point, and the more comprehensive survey by Lack [16] and Chapter 7 of Borceux [4] for more detail.)

2.2. Definition. A 2-group G is a 2-category with one object (G^{(0)} = {⋆}), for which all 1-morphisms γ ∈ G^{(1)} and 2-morphisms χ ∈ G^{(2)} are invertible.

Note that in the case of χ ∈ G^{(2)}, we require invertibility with respect to both horizontal and vertical composition.

Let us unpack this definition more explicitly.

Suppose G is a 2-group. There are various 1-morphisms from ⋆ to itself, which have a composition operation, since all such morphisms have matching source and target:

\xymatrix{⋆ & ⋆ \ar[l]^{γ_{1}} & ⋆ \ar[l]^{γ_{2}}} =  \xymatrix{⋆ & ⋆ \ar[l]^{γ_{1}∘γ_{2}}}  (1)

Composition of morphisms is the multiplication, which is therefore associative, and every morphism must have an inverse by the definition of a 2-group, so (G^{(1)}, ∘) forms a group.

Moreover, there are 2-morphisms between the 1-morphisms. They have both a horizontal and a vertical composition. The horizontal composition we denote ∘, since this is the same direction as the composition of 1-morphisms. The vertical composition we denote ·. The two compositions must be compatible, in the sense that the following composite is well-defined:

[figure: ⋆; ⋆; γ_{1}; γ_{2}; γ_{3}; ⋆; γ′_{1}; γ′_{2}; γ′_{3}; χ_{2}; χ_{1}; χ′_{2}; χ′_{1}]  (2)

This is expressed as the “interchange law”

(χ_{1} · χ_{2}) ∘ (χ′_{1} · χ′_{2}) = (χ_{1} ∘ χ′_{1}) · (χ_{2} ∘ χ′_{2})  (3)

Again, these 2-morphisms are assumed to be invertible, so it follows that (G^{(2)}, ∘) forms a group. Note that we do not say that (G^{(2)}, ·) forms a group, since · may not be defined for all pairs of 2-morphisms, unless they have compatible source and target 1-morphisms.

However, one can find a group structure using vertical composition of 2-morphisms, upon choosing a given 1-morphism γ, and considering Hom(γ, γ), the collection of 2morphisms from γ to itself. This is a group with operation ·, since 2-morphisms are invertible under ·.

(Note that Hom(γ, γ) does not necessarily close under ∘, unless γ = 1_{G}. In that case, Hom(1_{G}, 1_{G}) is an abelian group by the so-called Eckmann-Hilton argument, which uses the interchange property (3) to show that horizontal and vertical composition agree, and define an abelian group.)

We have noted the composition operations for both types of morphisms. It is also significant that there is also a straightforward interaction between the two types, namely whiskering. That is, a 1-morphism can act on a 2-morphism, either on the left or the right, where it acts by composition with an identity 2-cell. That is, by convention we say:

[figure: ⋆; ⋆; γ; ⋆; γ_{2}; γ_{1}; χ] =  [figure: ⋆; ⋆; γ∘γ_{1}; γ∘γ_{2}; Id_{γ}∘χ]  (4)

and similarly for whiskering on the right.

Again, all these remarks are only unpacking what is implied by the definition of a 2-category with one object and invertible 1- and 2-morphisms. We will want to use this definition occasionally, particularly when describing 2-group actions. However, for the most part is will be useful to think of 2-groups in a way which is more amenable to explicit calculations. The first step toward this is the definition of a categorical group, and then even more explicit is the presentation in terms of a crossed module. We recall these next.

2.2.1. Categorical Groups. Another natural approach to defining a “higher” analog of a group is based on the view that a group is a “group object in Set”. A group object in a category C with finite products (in particular, C = Set or C = Cat with Cartesian products) is an object G ∈ C, equipped with structure maps such as the multiplication map m : G × G → G satisfying the same axioms as those for a group. These axioms can be expressed as commuting diagrams which make sense in any category with finite products and specialize to the usual axioms in (Set, ×).

2.3. Definition. A (strict) categorical group is a (strict) group object in the monoidal category (Cat, ×).

We remark here that there is a more general concept of (weak) categorical groups, but we will restrict our attention to the strict case in the present paper. See Baez and Lauda [1] for an exposition of the weak case.

That is, categorical groups are (small) categories G, equipped with functors ⊗ : G × G → G, and inv : G → G satisfying the group axioms.

Since Cat is, in fact, a monoidal 2-category (in which the monoidal product on objects is the cartesian product of categories), our definition specifies that the functors ⊗ and inv satisfy the axioms for a group strictly. That is, the usual equations such as associativity of multiplication are still equations, rather than 2-isomorphisms. The corresponding weak notion, in which equations are replaced by such 2-isomorphisms, necessarily has additional coherence conditions they must satisfy.

Notice that the “group multiplication” functor, which we have written as ⊗, satisfies the properties for a monoidal product, as the notation suggests. Indeed, a (strict) monoidal category is simply a monoid object in Cat. A (strict) categorical group is therefore a strict monoidal category with inverses. That is, every object and morphism has an inverse with respect to ⊗.

It is standard, and easy to see, that any 2-group as defined in Section 2.1.1 gives rise to a categorical group, and vice versa. We describe the correspondence here to settle notation.

2.4. Definition. Given a 2-group G, the categorical group (C(G), ⊗) associated to G is defined as follows:

• Objects: C(G)^{(0)} = G^{(1)} (objects are morphisms of G)

• Morphisms: C(G)^{(1)} = G^{(2)} (morphisms are 2-morphisms of G)

• The composition of morphisms of C(G) is vertical composition of 2-morphisms in G

• The monoidal product ⊗ of C(G) is:

– On C(G)^{(0)}, same as composition in G^{(1)}

– On C(G)^{(1)}, same as horizontal composition in G^{(2)}

• The inverse functor inv : C(G) → C(G) is given, for each object or morphism, by the inverse or vertical inverse of the corresponding 1- or 2- morphism in G

Given a categorical group (G, ⊗), the 2-group ⋆//G has:

• Object: just one, ⋆

• Morphisms: (⋆//G)^{(1)} = G^{(0)}, the objects of G

• 2-Morphisms: (⋆//G)^{(2)} = G^{(1)}, the morphisms of G

• Composition for (⋆//G)^{(1)} and horizontal composition for (⋆//G)^{(2)} are ⊗^{(0)} and ⊗^{(1)} from G respectively

• Vertical composition for (⋆//G)^{(2)} is composition for G^{(1)}

It is well known, and the unfamiliar reader may easily check, that these two correspondences give an equivalence of the two definitions. For example, the properties of invertibility for ⋆//G follow from the existence of inv in (G, ⊗), and the fact that it satisfies group axioms. It is similarly easy to check that monoidal functors between categorical groups correspond to 2-functors between 2-groups. For example, the interchange law for 2-groups is expressed, in categorical groups, as the fact that the monoidal product is functorial. The convention for whiskering in a 2-group is easily translated to the usual convention for the monoidal product of an object with a morphism.

Another equivalent presentation is a result of the fact that there is a correspondence between groups internal to categories, and categories internal to groups (that is, categories whose sets of objects and sets of morphisms each have the structure of a group, and where source, target, and composition operations are all group homomorphisms). This equivalence follows naturally, since given a categorical group, the structure maps, such as ⊗, are functors which satisfy the group axioms. The object and morphism maps for these functors therefore each separately satisfy the same axioms. Thus, the objects of a categorical group form a group, as do the morphisms of the categorical group. Thinking of these separate group structures naturally leads to the third of the definitions we will use in this paper, namely crossed modules.

2.4.1. Crossed Modules. A well-known theorem due to Brown and Spencer [8] says that (strict) 2-groups are classified by crossed modules. See also [5] for further background.

2.5. Definition. A crossed module consists of (G, H, ▷, ∂), where G and H are groups, G ▷ H is an action of G on H by automorphisms and ∂ : H → G is a homomorphism, satisfying the equations:

∂(g ▷ η) = g∂(η)g^{−1}  (5)

and

∂(η) ▷ ζ = ηζη^{−1}  (6)

We describe the correspondence with categorical groups, which will be the form we make the most use of, though of course the correspondence with 2-groups follows immediately as well:

2.6. Definition. The categorical group (G, ⊗) given by (G, H, ▷, ∂) has:

• Objects: G^{(0)} = G

• Morphisms: G^{(1)} = G × H, with source and target maps

s(g, η) = g  (7)

and

t(g, η) = ∂(η)g  (8)

• Identities:

Id_{g} = (g, 1_{H})  (9)

• Composition:

(∂(η)g, ζ) ∘ (g, η) = (g, ζη).  (10)

• Monoidal product: given on objects by g_{1} ⊗ g_{2} = g_{1}g_{2} and on morphisms by

(g_{1}, η) ⊗ (g_{2}, ζ) = (g_{1}g_{2}, η(g_{1} ▷ ζ))  (11)

which corresponds to the horizontal composition of 2-morphisms in G.

The theorem alluded to above asserts that any strict 2-group is equivalent to one of this form. We will usually assume that any 2-group we use from now on is presented by a crossed module, and use the corresponding notation for explicit calculations.

The structure of a category which lies behind the definition of a crossed module is somewhat disguised. Much of it is included in the fact that G and H are groups. The explicit action of G on H captures the notion of whiskering, remarked upon above, and the projection onto the first factor, together with the “boundary” map ∂, determines the source and target structure maps for the categorical group. However, for example, the vertical composition of 2-morphisms is not explicitly mentioned. Full details of how it and other categorical group structures can be reconstructed from a crossed module are well recorded elsewhere.

2.7. Double Groupoid of a 2-Group. A useful construction for calculations later on is the double groupoid of G.

Double categories were introduced by Ehresmann [12, 11], and can be defined in several equivalent ways, three of which are noted by Brown and Spencer [7]. The most pertinent later in this paper is that they are categories internal to Cat, the category of all categories. This terse definition is the most common in current use, but is somewhat opaque, and obscures the basically symmetric nature of these structures.

One can also describe a (small) double category in a more manifestly symmetric way. Although the two definitions are equivalent, they are superficially rather different. In Section 4.5 we will use both points of view to describe the structure which it is the goal of this paper to construct. Therefore, to avoid confusion, we will reserve “double category” for the symmetric view, and use “Cat-category” to mean the equivalent structure seen as a category in Cat.

We will not give a full definition of “double category” here, though, since it is equivalent to and can be deduced from the terse definition, but it is intuitively useful to see a double category D as consisting of:

• a set of objects O

• two sets H and V of morphisms (we use calligraphic letters here to avoid confusion with the group H), denoted horizontal and vertical, together with structure maps making them into horizontal and vertical categories with objects O

• a set of squares S which form the morphisms of two category structures whose objects are H and V respectively

The category structures on the squares S satisfy some extra properties making them compatible with the category structures on H and V. Intuitively, they say that “horizontal” structure maps commute with “vertical” structure maps. For example, one can take a horizontal composite of two squares σ_{1} and σ_{2}, and then take the morphism s_{v}(σ_{1} ∘_{h} σ_{2}) ∈ H which is its vertical source. This is the same as s_{v}(σ_{1}) ∘_{h} s_{v}(σ_{2}), the horizontal composite of the vertical sources of each square. There are many other conditions of this form, such as the “interchange law” for composition of squares, which has the same form as (3).

A double groupoid is a double category in which all morphisms and squares are invertible. Further discussion of double groupoids is found in Brown and Spencer [7], while an interesting summary of general folklore about this symmetric definition and variations on it can be found online [25].

The above intuition essentially justifies a graphical notation which makes 2-group calculations more straightforward. This notation has been extensively used elsewhere, see e.g. [5], and we now describe it for reference.

2.8. Definition. Given a 2-group G, the double groupoid of G, denoted D(G), is the double groupoid with:

• A unique object of D(G): O = G^{(0)} = {⋆}

• Horizontal and vertical morphisms of D(G) are H = V = G^{(∞)}

• Squares of D(G) are given by all squares of the form

g_{4} g^{3} η g_{2} g_{1}

where g_{i} ∈ G, i = 1, …, 4, η ∈ H and ∂(η) = g_{1}g_{2}g_{3}^{−1}g_{4}^{−1}

• horizontal and vertical composition of squares is given by

g^{3} g_{4} η_{1}  g^{7} g_{2} η_{2} g_{6} g_{1}  g_{5}  = g_{4}  g^{3}g^{7} η_{1}(g_{4}g_{3}g_{2}^{−1}) ▷ η_{2} g_{6} g_{1}g_{5}  = g_{4}  g^{3}g^{7} (g_{1} ▷ η_{2})η_{1} g_{6} g_{1}g_{5}

where the two expressions for the H element in the horizontal compositions are the same, using ∂(η_{1}^{−1}) ▷ η_{2} = η_{1}^{−1}η_{2}η_{1}, and

g_{4} g^{3} η_{1} g_{2} g_{1} g_{5} η_{2} g_{7} g_{6}  = g_{5}g_{4}  g^{3} η_{2}(g_{5} ▷ η_{1}) g_{7}g_{2} g_{6}

and these operations satisfy the interchange law, i.e. the equality of evaluating a 2 by 2 array of squares in two different ways (first horizontal and then vertical composition, or vice-versa).

In Ehresmann’s terminology, this is the double category of quintets of the bicategory G: the term refers to the fact that squares are determined by five pieces of data: the four 1-morphisms which are its edges, and the 2-morphism which fills the square.

2.9. Remark. Due to the interchange law there is a consistent evaluation of any rectangular array of squares, which is independent of the order in which the horizontal and vertical multiplications are performed. Also squares in D(G) have horizontal and vertical inverses, which are respectively given by (for the square in the definition):

g_{2} g_{3}^{−1} η^{−h} g_{4} g_{1}^{−1}  = g_{2}  g_{3}^{−1} g_{1}^{−1} ▷ η^{−1} g_{4} g_{1}^{−1}  g_{4}^{−1} g^{1} η^{−v} g_{2}^{−1} g_{3}  = g_{4}^{−1}  g^{1} g_{4}^{−1} ▷ η^{−1} g_{2}^{−1} g_{3}

In particular, the double groupoid D(G) contains a copy of G by considering only the horizontal morphisms, and the squares for which the vertical source and target are identities. Thus, a typical square of this sort is of the following form:

\begin{matrix}g_{1} \\ η \\ g_{2} \end{matrix}

where here, and henceforth, any unlabelled edge or square is taken to be labelled with the identity of the corresponding group. This can be identified with a 2-morphism in the 2-group G, namely:

[figure: ⋆; ⋆; g_{2}; g_{1}; η]  (12)

or equivalently, with the morphism (g_{1}, η) in the categorical group.

3. Actions of 2-Groups on Categories

Our aim is to describe actions of a 2-group G on a category C. There are two equivalent ways of describing these, depending on whether one views G as a 2-group or a categorical group. In this section, we will outline these two views and see the relation between them. Then we will consider a natural example, namely the adjoint action of a categorical group on its own underlying category.

There are two different, but equivalent, definitions of group actions on a set, and both will be relevant for us. To begin with, consider a group G, seen as a one-object category whose morphisms are all invertible. Then a G action ϕ on a set X may be described as a functor

ϕ : G → Set  (13)

where X = ϕ(⋆) is the image of the unique object of G, i.e. the image of ϕ is End(X), the full subcategory of Set with the single object X (in fact, since G is a group, the image is Aut(X), consisting of only the invertible endomorphisms of X).

Next we show how to construct the transformation groupoid in a way that naturally generalizes to the 2-group case, by regarding the group G as a set equipped with a multiplication map m : G × G → G and an inverse inv : G → G, satisfying the group axioms. Thus we have the familiar definition of an action on a set X as a function

ϕ̂: G × X → X  (14)

This function is related to ϕ by ϕ̂(γ, x) = ϕ_{γ}(x). Functoriality of ϕ means, in particular, that ϕ̂ satisfies a compatibility condition with the multiplication map m : G × G → G, namely that the following commutes:

\xymatrix{G × G × X \ar[r]^{m×Id_{X}} \ar[d]_{Id_{G}×ϕ̂} & G × X \ar[d]^{ϕ̂} \\ G × X \ar[r]_{ϕ̂} & X}  (15)

ϕ̂ also satisfies a unit condition:

ϕ̂(1, x) = x, ∀x ∈ X  (16)

This definition is, of course, equivalent to the point of view of an action as a functor. First it determines ϕ(⋆) = X. The two definitions are then related by turning a function G → Hom(X, X) into an X-valued function of G × X, taking (γ, x) to ϕ_{γ}(x). (The term for this in logic is “uncurrying”, while “currying” denotes the process which turns a function of n variables into a chain of n one-variable functions, each one returning the next function in the chain).

3.1. Actions of 2-Groups on Categories. Next we extend the two viewpoints of the previous introduction for the action of a group on a set to the action of a 2-group on a category. First, by analogy with (13), regarding a 2-group as a 2-category, an action will be a 2-functor into the 2-category Cat.

3.2. Definition. A 2-group G acts (strictly) on a category C if there is a (strict) 2functor:

Φ : G → Cat  (17)

whose image lies in End(C), the full sub-2-category of Cat with the single object C.

Thus Φ(∗) = C, and on 1- and 2-morphisms Φ is given by assignments:

• for each γ ∈ G^{(1)} we have the endofunctor Φ_{γ} : C → C, acting as

(x →^{f} y) ↦ (Φ_{γ}(x) ⟶^{Φ_{γ}(f)} Φ_{γ}(y))

• for each 2-morphism (γ_{1}, χ) ∈ G^{(2)}, we have the natural transformation Φ_{(γ_{1},χ)} : Φ_{γ_{1}} → Φ_{γ_{2}}, where γ_{2} = ∂(χ)γ_{1}, given by assignments (C^{(0)} ∋ x) ↦ (Φ_{(γ_{1},χ)}(x) ∈ C^{(1)}) satisfying the naturality condition

\xymatrix{Φ_{γ_{1}}(x) \ar[r]^{Φ_{γ_{1}}(f)} \ar[d]_{Φ_{(γ_{1},χ)}(x)} & Φ_{γ_{1}}(y) \ar[d]^{Φ_{(γ_{1},χ)}(y)} \\ Φ_{γ_{2}}(x) \ar[r]_{Φ_{γ_{2}}(f)} & Φ_{γ_{2}}(y)}  (18)

These assignments must satisfy the conditions to be a strict 2-functor, i.e. they preserve all vertical and horizontal compositions and identities, which here means:

Ver 1) Φ_{(γ_{2},χ_{2})}(x) ∘_{v} Φ_{(γ_{1},χ_{1})}(x) = Φ_{(γ_{1},χ_{2}χ_{1})}(x) where γ_{2} = ∂(χ_{1})γ_{1} 2) Φ_{(γ,1)}(x) = id_{Φ_{γ}(x)}

Hor 1) Φ_{γ_{1}} ∘ Φ_{γ_{3}} = Φ_{γ_{1}γ_{3}}, Φ_{1} = id_{C}

2) Φ_{(γ_{1},χ_{1})} ∘_{h} Φ_{(γ_{3},χ_{2})} = Φ_{(γ_{1}γ_{3},χ_{1}(γ_{1}▷χ_{2}))}. Writing out explicitly the horizontal composition of natural transformations on the l.h.s., this final condition becomes:

Φ_{(γ_{1},χ_{1})}(Φ_{γ_{4}}(x)) ∘ Φ_{γ_{1}}(Φ_{(γ_{3},χ_{2})}(x)) = Φ_{(γ_{1}γ_{3},χ_{1}(γ_{1}▷χ_{2}))}(x)

where γ_{4} = ∂(χ_{2})γ_{3}. Here the underlying array of squares in D(G) is

γ^{1} χ_{1}  γ^{3} χ_{2} γ_{2}  γ_{4}  (19)

The second viewpoint for a 2-group to act on a category is to regard the 2-group G as a categorical group, i.e. a monoidal category, and proceed by analogy with (14), (15).

3.3. Definition. A strict action of a categorical group G on a category C is a functor Φ̂ : G × C → C satisfying the action square diagram in Cat (strictly):

\xymatrix{G × G × C \ar[r]^{⊗×Id_{C}} \ar[d]_{Id_{G}×Φ̂} & G × C \ar[d]^{Φ̂} \\ G × C \ar[r]_{Φ̂} & C}  (20)

and the unit condition

Φ̂(1, x) = x, ∀x ∈ C^{(0)}  (21)

Thus, from functoriality, we have the conditions:

Φ̂((γ_{2}, χ_{2}), g) ∘ Φ̂((γ_{1}, χ_{1}), f) = Φ̂((γ_{1}, χ_{2}χ_{1}), g ∘ f)  (22)
Φ̂((γ, 1_{H}), id_{x}) = id_{Φ̂(γ,x)}  (23)

where, in (22), γ_{2} = ∂(χ_{1})γ_{1}. The action square diagram corresponds to the equations (on objects and morphisms respectively):

Φ̂(γ_{1}γ_{3}, x) = Φ̂(γ_{1}, Φ̂(γ_{3}, x))  (24)

Φ̂((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f) = Φ̂((γ_{1}, χ_{1}), Φ̂((γ_{3}, χ_{2}), f)).  (25)

Here the underlying array of squares in D(G) is (19).

The following theorem shows that these two viewpoints are equivalent.

3.4. Theorem. A strict 2-functor Φ : G → End(C) is equivalent to a strict action functor Φ̂ : G × C → C.

Proof. Given Φ, set: Φ̂(γ, x) := Φ_{γ}(x) and

( [figure: Φ̂(γ_{1}, x); Φ̂((γ_{1}, χ), f); Φ̂(γ_{2}, y)] ) := ( [figure: Φ_{γ_{1}}(x); Φ_{(γ_{1},χ)}(y) ∘ Φ_{γ_{1}}(f); Φ_{γ_{2}}(y)] )  (26)

By the naturality condition (18), this is also equal to

( [figure: Φ_{γ_{1}}(x); Φ_{γ_{2}}(f) ∘ Φ_{(γ_{1},χ)}(x); Φ_{γ_{2}}(y)] )  (27)

Now functoriality for Φ̂ follows from putting together four naturality squares (18) in the obvious way, and then using functoriality of Φ_{γ} horizontally and the first Ver condition vertically.

The first action square equation (24) and the unit condition (3.3) are immediate. The second action square equation (25) follows from:

Φ̂((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f) = Φ_{(γ_{1}γ_{3},χ_{1}(γ_{1}▷χ_{2}))}(y) ∘ Φ_{γ_{1}γ_{3}}(f)
= Φ_{(γ_{1},χ_{1})}(Φ_{γ_{4}}(y)) ∘ (Φ_{γ_{1}}(Φ_{(γ_{3},χ_{2})}(y)) ∘ Φ_{γ_{1}}(Φ_{γ_{3}}(f)))
= Φ_{(γ_{1},χ_{1})}(Φ_{γ_{4}}(y)) ∘ (Φ_{γ_{1}}(Φ_{(γ_{3},χ_{2})}(y)) ∘ Φ_{γ_{3}}(f)))
= Φ̂((γ_{1}, χ_{1}), Φ_{(γ_{3},χ_{2})}(y) ∘ Φ_{γ_{3}}(f))
= Φ̂((γ_{1}, χ_{1}), Φ̂((γ_{3}, χ_{2}), f))

using the first Hor condition for Φ and associativity in the second equality.

Conversely, given Φ̂, set Φ_{γ}(x) := Φ̂(γ, x) (on objects),

( [figure: Φ_{γ}(x); Φ_{γ}(f); Φ_{γ}(y)] ) := ( [figure: Φ̂(γ, x); Φ̂((γ, 1_{H}), f); Φ(γ, y)] )

(on morphisms), and on 2-morphisms the natural transformation Φ_{(γ_{1},χ)} : Φ_{γ_{1}} → Φ_{γ_{2}}, where γ_{2} = ∂(χ)γ_{1}, is given by:

( [figure: Φ_{γ_{1}}(x); Φ_{(γ_{1},χ)}(x); Φ_{γ_{2}}(x)] ) := ( [figure: Φ̂(γ_{1}, x); Φ̂((γ_{1}, χ), id_{x}); Φ̂(γ_{2}, x)] )  (28)

Now, the Hor properties for Φ follow in a straightforward manner using the functoriality conditions (22), (23) for Φ̂. The first Ver property for Φ is immediate, and the second Ver property follows from:

Φ_{(γ_{1},χ_{1})}(Φ_{γ_{4}}(x)) ∘ Φ_{γ_{1}}(Φ_{(γ_{3},χ_{2})}(x)) = Φ̂((γ_{1}, χ_{1}), id_{Φ̂(γ_{4},x)}) ∘ Φ̂((γ_{1}, 1_{H}), Φ̂((γ_{3}, χ_{2}), id_{x}))
= Φ̂((γ_{1}, χ_{1}), Φ̂((γ_{3}, χ_{2}), id_{x}))
= Φ̂((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), id_{x}))
= Φ_{(γ_{1}γ_{3},χ_{1}(γ_{1}▷χ_{2}))}(x)

where we use functoriality of Φ̂ (22) in the second equality and the second action square condition (25) applied to f = id_{x} in the penultimate equality.

It is convenient to introduce a succinct notation for a 2-group action analogous to the usual notation g ▷ x = ϕ_{g}(x) for a group action. There are actually three possible maps.

Since Φ_{γ} is a functor with both object and morphism maps, the objects of G act on both the set of objects and the set of morphisms of C. Moreover, the action functor Φ̂ also has both object and morphism maps. The object map Φ̂(γ, −) is the same as the object map for Φ_{γ}, by the above argument. However, the morphism map determines an action of the morphisms of G on the morphisms of C, which is a different action again.

We will return to the relation between these three actions again in Corollary 4.14. For the moment, is convenient to use the symbol ▶ to denote all three, as follows.

3.5. Definition. If G is a 2-group classified by the crossed module (G, H, ▷, ∂), let the notation ▶ denote the following.

• Given γ ∈ G^{(0)} = G and x ∈ C^{(0)}, let

γ ▶ x = Φ_{γ}(x) = Φ̂(γ, x)  (29)

• Given γ ∈ G^{(0)} = G and f ∈ C^{(1)}, let

γ ▶ f = Φ_{γ}(f) = Φ̂((γ, 1_{H}), f)  (30)

• Given (γ, χ) ∈ G^{(1)} = G ⋉ H and (f : x → y) ∈ C^{(1)}, let

\begin{aligned}(γ, χ) ▶ f & = & Φ̂((γ, χ), f) \\ & = & Φ_{(γ,χ)}(y) ∘ (γ ▶ f) \\ & = & (∂(χ)γ ▶ f) ∘ Φ_{(γ,χ)}(x) \end{aligned}  (31)

The last two expressions are the same as those given in the proof of Theorem 3.4, written in our new notation. It will be revisited in (66).

3.6. Example: Adjoint Action of 2-Groups. Here we want to describe the adjoint action of a 2-group G on itself. This is the analog of the usual adjoint action of a group G on itself by conjugation. The adjoint action is a functor

Φ : G → End(G)  (32)

given by the property that

Φ_{γ}(g) = γgγ^{−1}  (33)

We want a 2-functor given “by conjugation”, insofar as this makes sense. In fact, as we shall see, this is easy to do in the language of crossed modules, since the axioms (5) and (6) for a crossed module (G, H, ▷, ∂) imply that the action ▷ of G on H resembles conjugation as much as possible. This will be made even clearer by using the square calculus in the double groupoid of G, D(G).

In accordance with Definition 3.2, we take the action of G on itself to be given by a 2-functor from G to Cat with image in End(G), i.e. the “acting” G is regarded as a 2category, whilst the “acted on” G is regarded as a (monoidal) category. (This is analogous to the situation for the adjoint action of a group G, regarded as a category, acting on G, regarded as a set).

3.7. Definition. Suppose G is the 2-group given by a crossed module (G, H, ▷, ∂). Then we define a strict 2-functor:

Φ : G → Cat  (34)

with image in End(G) in the following way. At the object level, Φ(∗) = G, where G on the right is taken to be a category. For each morphism γ ∈ G^{(1)}

Φ_{γ} : G → G  (35)

is a morphism of End(G), namely an endofunctor of G. Its object map is given by:

Φ_{γ}(g) = γgγ^{−1}  (36)

and its morphism map is given by

Φ_{γ}(g, η) = (γgγ^{−1}, γ ▷ η)  (37)

For each 2-morphism (γ, χ) ∈ G^{(2)} there is a natural transformation from Φ_{γ} to Φ_{∂(χ)γ}, given by:

Φ_{(γ,χ)}(g) = (γgγ^{−1}, χ(γgγ^{−1}) ▷ χ^{−1}))  (38)

In the action notation of Definition 3.5, the first two simply say that γ ▶ g = γgγ^{−1} and γ ▶ (g, η) = (γ ▶ g, γ▷η). The third part of that definition will define (γ, χ) ▶ (g, η), which is not directly given by Φ. However, we can understand it using the calculus of squares introduced in Section 2.7 for the double groupoid of quintets D(G) associated to G.

Represent morphisms (g_{1}, η) in the category G as squares, so that the action of Φ_{γ} on such a square is given by:

\begin{matrix}g_{1} \\ η \\ g_{2} \end{matrix}  ↦^{Φ_{γ}}  γ  g^{1} η  γ^{−1} γ  g_{2} γ^{−1} = \begin{matrix}γg_{1}γ^{−1} \\ γ ▷ η \\ γg_{2}γ^{−1} \end{matrix} = \begin{matrix}γ ▶ g_{1} \\ γ ▷ η \\ γ ▶ g_{2} \end{matrix}  (39)

where g_{2} = ∂(η)g_{1}. Likewise we can represent the morphism Φ_{(γ_{1},χ)}(g) (which has no direct analog in the action notation) as

Φ_{(γ_{1},χ)}(g) = γ^{1} χ  g  γ_{1}^{−1} χ^{−h} γ_{2}  g  γ_{2}^{−1} = \begin{matrix}γ_{1} ▶ g \\ χ(γ_{1} ▶ g) ▷ χ^{−1} \\ γ_{2} ▶ g \end{matrix}  (40)

where γ_{2} = ∂(χ)γ_{1}.

It follows from (39) that Φ_{γ}(g_{1}, η) has the correct source and target. Φ_{γ} is a functor, since it preserves identities, Φ_{γ}(g, 1_{H}) = (γgγ^{−1}, 1_{H}) (immediate), and composition, i.e. Φ_{γ}(g_{2}, η_{2}) ∘ Φ_{γ}(g_{1}, η_{1}) = Φ_{γ}(g_{1}, η_{2}η_{1}), where g_{2} = ∂(η_{1})g_{1}, because of:

γ  g^{1} η_{1} γ^{−1} γ  g_{2} η_{2} γ^{−1} γ  g_{3} γ^{−1} = γ  g^{1} η_{2}η_{1} γ^{−1} γ  g_{3} γ^{−1}

It then follows immediately from (40) that Φ_{(γ_{1},χ)}(g) has the correct source and target. The naturality condition (18), i.e. Φ_{γ_{2}}(g_{1}, η) ∘ Φ_{(γ_{1},χ)}(g_{1}) = Φ_{(γ_{1},χ)}(g_{2}) ∘ Φ_{γ_{1}}(g_{1}, η), is the equality:

γ^{1} χ  g^{1} γ_{1}^{−1} χ^{−h} γ_{2}  g_{1} η  γ_{2}^{−1} γ_{2}  g_{2} γ_{2}^{−1} = γ^{1}  g^{1} η  γ_{1}^{−1} γ_{1} χ  g_{2} γ_{1}^{−1} χ^{−h} γ_{2}  g_{2} γ_{2}^{−1}

which follows from the interchange law by evaluating the vertical compositions on both sides first.

The first Ver condition, i.e. Φ_{(γ_{2},χ_{2})}(g) ∘ Φ_{(γ_{1},χ_{1})}(g) = Φ_{(γ_{1},χ_{2}χ_{1})}(g) where γ_{2} = ∂(χ_{1})γ_{1}, follows from the equality:

γ^{1} χ_{1}  g  γ_{1}^{−1} χ_{1}^{−h} γ_{2} χ_{2}  g  γ_{2}^{−1} χ_{2}^{−h} γ_{3}  g  γ_{3}^{−1} =  γ^{1} χ_{2}χ_{1}  g  γ_{1}^{−1} (χ_{2}χ_{1})^{−h} γ_{3}  g  γ_{3}^{−1}

The second Ver condition, i.e. Φ_{(γ,1_{H})}(g) = Id_{Φ_{γ}(g)}, and the first Hor condition, i.e. Φ_{γ_{1}}(Φ_{γ_{3}}(g)) = Φ_{γ_{1}γ_{3}}(g) and Φ_{1}(g) = g, are both immediate. Finally the second Hor condition, i.e. Φ_{(γ_{,}χ_{1})}(Φ_{γ_{4}}(g)) ∘ Φ_{γ_{1}}(Φ_{(γ_{3},χ_{2})}(g)) = Φ_{(γ_{1}γ_{3},χ_{1}(γ_{1}▷χ_{2}))}(g), corresponds to the equality:

γ^{1} γ^{3} χ_{2}  g γ_{3}^{−1} χ_{2}^{−h} γ_{1}^{−1} γ_{1} χ_{1} γ_{4}  g γ_{4}^{−1} γ_{1}^{−1} χ_{1}^{−h} γ_{2} γ_{4}  g γ_{4}^{−1} γ_{2}^{−1} =  γ^{1}γ^{3} χ_{1}(γ_{1} ▷ χ_{2})  g  (γ_{1}γ_{3})^{−1} (χ_{1}(γ_{1} ▷ χ_{2}))^{−h} γ_{2}γ_{4}  g  (γ_{2}γ_{4})^{−1}

This follows from evaluating the two 2 by 2 arrays of squares without g on the left hand side. Thus we have proved:

3.8. Lemma. The 2-group adjoint action of Definition 3.7 is a well-defined action in the sense of Definition 3.2.

3.9. Remark. We can now display the functor Φ̂ of Definition 3.3, in terms of squares, and therefore finish describing this action in the notation of Definition 3.5. Namely (γ_{1}, χ) ▶ (g_{1}, η) = Φ̂((γ_{1}, χ), (g_{1}, η)) is given by:

γ^{1} χ  g^{1} η  γ^{−1} χ^{−h} γ_{2}  g_{2} γ_{2}^{−1} = χ  γ^{1}  g^{1} η  γ^{−1} χ^{−1} ∂(χ) γ_{1}  g_{2} γ_{1}^{−1} ∂(χ)^{−1}

The 5-square array on the right makes it clear that the 2-group adjoint action of (γ_{1}, χ) can be regarded as the ordinary adjoint action of γ_{1} on edges, extended to squares labelled by γ_{1} acting on squares, followed by the adjoint action of χ, in the sense of conjugation with the square labelled χ. Evaluating the array and dropping the indices gives an algebraic formula for this action:

(γ, χ) ▶ (g, η) = (γgγ^{−1}, χ(γ ▷ η)(γgγ^{−1}) ▷ χ^{−1})  (41)

4. Transformation Double Category for a 2-Group Action

In this section, we recall the construction of the transformation groupoid for a group action on a set, and consider the analogous construction for a 2-group action on a category.

If we are given an action of a group on a set, there is a groupoid which corresponds to it.

4.1. Definition. Given a group action ϕ : G → End(X), the transformation groupoid X//G is the groupoid with:

• Objects: x ∈ (X//G)^{(0)} = X

• Morphisms: (γ, x) ∈ (X//G)^{(1)} = G×X, with source and target maps s(γ, x) = x, and t(γ, x) = ϕ_{γ}(x)

• Composition: (γ′, ϕ_{γ}(x)) ∘ (γ, x) = (γ′γ, x)

It is clear that this is a groupoid, whose morphisms are invertible since G is a group. The composition then encodes both the group multiplication (in the first component) and the action (in the second component). Formally, the set P = (X//G)^{(1)} ×_{(X//G)^{(0)}} (X//G)^{(1)} of composable pairs of morphisms in X//G is then given by the pullback square:

\xymatrix{P \ar[r] \ar[d] & G × X \ar[d]^{ϕ̂} \\ G × X \ar[r]_{π_{X}} & X}  (42)

There is an obvious commuting diagram with G×G×X replacing P in the above diagram, and hence, by the universal property of this pullback, there is a unique map from G×G×X to P, given by:

(γ′, γ, x) ↦ ((γ′, ϕ_{γ}(x)), (γ, x))  (43)

which is clearly an isomorphism. In this way, the composition map from P to G × X agrees with the map m × Id_{X} in (15).

In the next subsections, we will develop a similar construction in Cat, which will give a transformation double category for a categorical group action. As discussed in Section 2.7, we reserve the term “double category” for a certain symmetric point of view of this structure. To construct it, however, we use an equivalent definition as a Cat-category.

4.2. Construction of the Transformation Cat-Category for a 2-Group Action. The most obvious way to define an analog of the transformation groupoid in the situation of a 2-group action comes by simply following the same constructions from an ordinary group action, replacing G with G and X with C. Thus, one has action diagrams in Cat.

Next we will construct a transformation groupoid as we did for group actions on sets, but it will be internal to Cat, so we call it a transformation Cat-groupoid.

Thus, now one has a category (C//G)^{(0)} of objects, and a category (C//G)^{(1)} of morphisms. These, of course, have objects and morphisms of their own, but this fact is invisible to the construction and becomes important only when we want to describe the resulting structure concretely.

4.3. Definition. Given a 2-group G, a category C, and an action of G on C as in Definitions 3.2 and 3.3, the transformation Cat-groupoid C//G is the groupoid internal to Cat with:

• Category of objects: (C//G)^{(0)} = C.

• Category of morphisms: (C//G)^{(1)} = G × C, with

– Source functor s = π_{C} : G × C → C

– Target functor t = Φ̂ : G × C → C

– Identity inclusion functor e = 1_{G} × Id_{C} : C → G × C

– Inverse functor inv = (inv_{G}, t) : G × C → G × C

• Category of composable pairs: P, given by the pullback diagram

\xymatrix{P \ar[r] \ar[d] & G × C \ar[d]^{Φ̂} \\ G × C \ar[r]_{π_{C}} & C}  (44)

• Composition functor: given by the action square (20)

The situation is entirely analogous to the transformation groupoid for an ordinary group G, as is seen easily by considering the effect on objects. (Thus the inverse functor, on objects, takes (γ, x) ↦ (γ^{−1}, γ ▶ x), just as with an ordinary transformation groupoid.) Analogously with the situation for groups, we can think of (C//G)^{(1)} as a semidirect product 2-group; and there is a canonical isomorphism between G × G × C and P. We should check that this definition makes sense.

4.4. Theorem. The transformation Cat-groupoid C//G is a well-defined groupoid internal to Cat.

Proof. To confirm this, we must check that the source, target, composition, identity inclusion and inverse functors are well defined, and satisfy the usual properties for a groupoid, primarily associativity and the left and right unit laws.

The fact that the source and target maps are functors is obvious, since they are just the projection from a product, and Φ̂ respectively. The first is necessarily a functor, while the second is a functor by definition.

Recall our notation:

γ ▶ x = Φ_{γ}(x) = Φ̂(γ, x) (γ, χ) ▶ f = Φ̂((γ, χ), f)  (45)

and make explicit the source, target and composition functors at the morphism level as follows:

s((γ, χ), f) = f  t((γ, χ), f) = (γ, χ) ▶ f  (46)
((γ_{1}, χ_{1}), (γ_{3}, χ_{2}) ▶ f) ∘ ((γ_{3}, χ_{2}), f) = ((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f)  (47)

(see (25)). The target of the composition is well-defined due to the commutativity of (20):

(γ_{1}, χ_{1}) ▶ ((γ_{3}, χ_{2}) ▶ f) = (γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})) ▶ f  (48)

To show the functoriality of composition, we refer to the following array of squares in D(G), underlying the calculation:

γ^{1} χ_{1}  γ^{3} χ_{2} γ_{2} χ_{3}  γ_{4} χ_{4} γ_{5}  γ_{6}  (49)

We denote composition in G × C by ¯∘ and composition in C by ∘_{C}, to distinguish them from the composition functor ∘. Then we have:

((γ_{4}, χ_{4}), g) ¯∘ ((γ_{3}, χ_{2}), f) = ((γ_{3}, χ_{4}χ_{2}), g ∘_{C} f)
((γ_{2}, χ_{3}), (γ_{4}, χ_{4}) ▶ g) ¯∘ ((γ_{1}, χ_{1}), (γ_{3}, χ_{2}) ▶ f) = ((γ_{1}, χ_{3}χ_{1}), ((γ_{4}, χ_{4}) ▶ g) ∘_{C}
((γ_{3}, χ_{2}) ▶ f))
= ((γ_{1}, χ_{3}χ_{1}), (γ_{3}, χ_{4}χ_{2}) ▶ (g ∘_{C} f))

where we use (22) in the final equation. We also have:

((γ_{2}, χ_{3}), (γ_{4}, χ_{4}) ▶ g) ∘ ((γ_{4}, χ_{4}), g) = ((γ_{2}γ_{4}, χ_{3}(γ_{2} ▷ χ_{4})), g)
((γ_{1}, χ_{1}), (γ_{3}, χ_{2}) ▶ f) ∘ ((γ_{3}, χ_{2}), f) = ((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f)

Thus it remains to show the equality of two expressions:

((γ_{2}γ_{4}, χ_{3}(γ_{2} ▷ χ_{4})), g) ¯∘ ((γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f) = ((γ_{1}γ_{3}, χ_{3}(γ_{2} ▷ χ_{4})χ_{1}(γ_{1} ▷ χ_{2})), g ∘_{C} f)

and

((γ_{1}, χ_{3}χ_{1}), (γ_{3}, χ_{4}χ_{2}) ▶ (g ∘_{C} f)) ∘ ((γ_{3}, χ_{4}χ_{2}), g ∘_{C} f) = ((γ_{1}γ_{3}, χ_{3}χ_{1}γ_{1} ▷ (χ_{4}χ_{2})), g ∘_{C} f).

This equality is immediate from the interchange law in D(G) applied to the array (49).

Associativity of the composition functor follows from the fact that the horizontal composition of three squares in D(G) has a unique evaluation:

γ^{1} χ_{1}  γ^{3} χ_{2}  γ^{5} χ_{3} γ_{2}  γ_{4}  γ_{6}  (50)

The identity inclusion e is clearly a functor, and satisfies:

((γ, χ), f) ∘ ((1_{G}, 1_{H}), f) = ((γ, χ), f)
((1_{G}, 1_{H}), (γ, χ) ▶ f) ∘ ((γ, χ), f) = ((γ, χ), f)

where in the first equation we have a composable pair, since (if the target of f is y):

t((1_{G}, 1_{H}), f) = (1_{G}, 1_{H}) ▶ f = Φ_{(1_{G},1_{H})}(y) ∘ Φ_{1_{G}}(f) = Φ_{(1_{G},1_{H})}(y) ∘ f = f

using (26) in the second equality, functoriality of Φ in the third equality and property F2 (see Def. 3.2 ) in the final equality.

Finally, the inverse functor assigns, to every ((γ, χ), f) in the category of morphisms, its inverse ((γ^{−1}, χ^{−h}), (γ, χ) ▶ f), where we note that (γ^{−1}, χ^{−h}) = (γ^{−1}, (∂(χ)γ)^{−1} ▷ χ^{−1}) = (γ^{−1}, γ^{−1} ▷ χ^{−1}). This follows immediately from (47). Functoriality of the inverse functor follows easily from (22) and functoriality of the horizontal inverse in D(G).

This construction of a Cat-category has a slight drawback, which is the apparent asymmetry of its definition. In the next section we will address this issue.

4.5. The Transformation Double Category. The construction we have given in the previous section naturally produces a groupoid internal to Cat. However, the view of a double category as an internal category in Cat obscures the underlying symmetry of this structure, and the structure in the “transverse” direction to the categories of objects and morphisms.

As we remarked in Section 2.7, Cat-categories are equivalent to a structure defined in a more symmetric way, having two types of morphisms (horizontal and vertical), and squares. The properties can be deduced from the equivalence: for example, the “interchange law”, which says that mixed horizontal and vertical composites can be taken in any order, amounts to the functoriality of the composition ∘.

Now, the definition of double categories is symmetric under exchanging the roles of “horizontal” and “vertical” morphisms in our diagrams of squares. One can reflect all squares in a diagonal, and obtain another double category, which we call the transpose of the first double category. The resulting structure, naturally, still has an interpretation as a category internal to Cat.

More technically, we have the following:

4.6. Definition. If D is an internal category in Cat, then let the transpose of D, which we denote D̃, be the internal category in Cat defined by the following:

• The category of objects has

– Objects: the objects of the category D^{(0)}

– Morphisms: the objects of the category D^{(1)}

• The category of morphisms has

– Objects: the morphisms of the category D^{(0)}

– Morphisms: the morphisms of the category D^{(1)}

• The identity inclusion, source and target, and composition functors (ẽ, s̃, t̃, ˜∘) have as object maps the corresponding structure maps from the category D^{(0)}, and as morphism maps the corresponding structure maps from the category D^{(1)}

This is a double-category analog of the operation of taking the “opposite” of a category: another category in which morphisms are taken to be oriented in the opposite direction. Indeed, together with such opposite operations in both horizontal and vertical directions, the transpose generates a whole group of operations which take one double category to another: it is plainly isomorphic to the dihedral group D_{4}, the symmetries of a generic square, since each is determined by the source vertex of a generic square, together with the sense of horizontal and vertical. The opposites do not illustrate much new structure, however, so we will restrict our attention to the transpose.

The fact that the transpose is again an internal category in Cat is a standard consequence of basic facts about internalization. We do not want to assume all readers are accustomed to such internal constructions, so will sketch a proof to convey the essential idea.

4.7. Lemma. If D is a category internal in Cat, so is D̃.

Proof. First, one must check that the structure maps (ẽ, s̃, t̃, ˜∘) are functors. This follows from the compatibility conditions for D. We give the example of the source functor s̃ here: the others follow similar lines.

We claim that the source functor in D̃

s̃: D̃^{(1)} → D̃^{(0)}  (51)

preserves composition, that is, for two composable morphisms χ_{1} and χ_{2} in D̃^{(1)}

s̃(χ_{1} ∘ χ_{2}) = s̃(χ_{1}) ∘ s̃(χ_{2})  (52)

where the compositions are in D̃^{(1)} and D̃^{(0)} respectively. Interpreted in D, this is the statement that the composition functor ∘ : D^{(1)} ×_{D^{(0)}} D^{(1)} → D^{(1)} is compatible with the source maps in D^{(1)} and D^{(0)}, which is just part of the fact that ∘ is a functor. Likewise s̃ is compatible with the source, target, and identity inclusion maps in D^{(1)} and D^{(0)}. Similar arguments prove the functoriality and compatibility of t̃, ẽ, and ˜∘. So all the structure maps (ẽ, s̃, t̃, ˜∘) are well-defined morphisms in Cat.

Furthermore, since the object and morphism maps of (ẽ, s̃, t̃, ˜∘) satisfy the axioms for a category separately by definition, the functors satisfy them as well. Since its structure maps are well defined and satisfy all the axioms for a category, D̃ is indeed an internal category in Cat.

In the above sketch, we deliberately chose to compare the interaction of different maps in the two directions, namely source and composition, to emphasize the differences. It is worth remarking that some conditions are symmetrical: for example, the fact that ∘ is functorial, and therefore preserves composition, corresponds in the transpose to exactly the same fact about ˜∘. This is a form of the interchange law.

The above is a special case of a more general duality which can occur with internalization. Suppose we have any two types of objects, X and Y, which can be defined as collections of objects and morphisms satisfying certain diagrammatic axioms. Then it is equivalent to define X-objects internal to the category of Y -objects, and Y -objects internal to the category of X-objects.

Next, we want use the transpose to better understand the structure of the transformation Cat-groupoid C//G associated to a 2-group action. We state the structure of the transpose of C//G as a theorem. The essential new observation is that the transpose { \atop C//G} is built from transformation groupoids associated to group actions in the usual sense.

4.8. Theorem. Given a 2-group G, a category C, and an action of G on C given by Φ̂ : G ×C → C in Cat, the transpose of C//G is the category { \atop C//G} internal to Gpd, with:

• Groupoid of Objects: ({ \atop C//G})^{(0)}, equal to the transformation groupoid C^{(0)}//G^{(0)} associated to the action given by Φ̂^{(0)}.

• Groupoid of Morphisms: ({ \atop C//G})^{(1)}, equal to the transformation groupoid C^{(1)}//G^{(1)} associated to the action given by Φ̂^{(1)}.

• Identity inclusion functor

ẽ: C^{(0)}//G^{(0)} → C^{(1)}//G^{(1)}  (53)

source and target functors

s̃, t̃: C^{(1)}//G^{(1)} → C^{(0)}//G^{(0)}  (54)

and composition functor

˜∘: (C^{(1)}//G^{(1)}) ×_{C^{(0)}//G^{(0)}} (C^{(1)}//G^{(1)}) → C^{(1)}//G^{(1)}  (55)

are the functors whose object maps are the corresponding (e, s, t, ∘) for C and whose morphism maps are those for G × C.

Proof. Since this is a special case of the transpose construction of Lemma 4.7, we just need to show the equalities for ({ \atop C//G})^{(0)} and ({ \atop C//G})^{(1)}. First, consider the action as internal to Cat, by the inclusion of Gpd in Cat. Then the case of the groupoid of objects is clear: its objects are the objects of x ∈ C, and its morphisms are objects of (C//G)^{(1)} = G × C, i.e. they are labeled by pairs (γ, x) ∈ G^{(0)} × C^{(0)}, and (γ, x), as a morphism in ({ \atop C//G})^{(0)}, has source x and target γ ▶ x. Composition is determined by the object map of the composition functor in C//G (47) , i.e. by the monoidal product ⊗ of G^{(0)}. But this is exactly the transformation groupoid C^{(0)}//G^{(0)}, where G^{(0)} is a group with product ⊗, acting on C^{(0)} by Φ̂^{(0)}.

The argument for the groupoid of morphisms ({ \atop C//G})^{(1)} is substantially the same. Its objects are the morphisms of C, i.e. labelled by f ∈ C^{(1)}, and its morphisms are pairs ((γ, χ), f) with source f and target (γ, χ) ▶ f. Composition in the groupoid of morphisms is determined by the morphism part of the composition functor in C//G (47).

Since ({ \atop C//G})^{(0)} and ({ \atop C//G})^{(1)} are in fact groupoids, and all structure maps, being functors, are also morphisms of Gpd, we have that { \atop C//G} is a category internal to Gpd.

We would like to emphasize again, that in the transposed perspective of { \atop C//G}, both the category of objects and the category of morphisms are themselves transformation groupoids in the usual sense, associated to group actions.

We are now ready to combine the perspectives given by C//G and its transpose to express the whole local symmetry structure implied in the action of G on C in a balanced way. This balanced viewpoint is expressed in our definition below of the transformation double category C//G (abusing notation somewhat by keeping the same name as for the internal category). We recall from subsection 2.7 the notation O for the set of objects, H and V for the sets of horizontal and vertical morphisms, and S for the set of squares. By the preceding discussion, the horizontal category is (C//G)^{(0)} = C (Def. 4.3), the vertical category is ({ \atop C//G})^{(0)} = C^{(0)}//G^{(0)} (Thm. 4.8), and for squares, the horizontal composition of squares in S is the morphism map of ∘, while vertical composition of squares in S is the morphism map of ˜∘.

4.9. Definition. Given an action of a 2-group G on a category C, in terms of Φ or Φ̂ of definitions 3.2 and 3.3, the transformation double category C//G is given by:

• Objects: objects x ∈ O = C^{(0)}

• Horizontal Category: (C//G)^{(0)} = C, with morphisms x →^{f} y ∈ H = C^{(1)} composed as usual

• Vertical Category: ({ \atop C//G})^{(0)} = C^{(0)}//G^{(0)}, with morphisms displayed as x ⟶^{(γ,x)} γ ▶ x ∈ V = (C^{(0)}//G^{(0)})^{(1)} with source and target as shown and composed by (γ′, γ ▶ x) ∘ (γ, x) = (γ′γ, x)

• Squares: S = (C//G)^{(1)} = ({ \atop C//G})^{(1)} = G^{(1)} ×C^{(1)}, consisting of pairs of morphisms from G and C, with elements of S denoted by (γ, χ), f , displayed as

[figure: x; f; (γ,x); (γ,χ),f; y; (∂(χ)γ,y); γ ▶ x; (γ,χ)▶f; (∂(χ)γ) ▶ y]  (56)

with horizontal and vertical source and target as shown in the display. Horizontal and vertical composition (equivalent to pasting of diagrams of the form (56)) is given by:

(γ_{2}, χ_{2}), g ∘_{h}  (γ_{1}, χ_{1}), f = (γ_{1}, χ_{2}χ_{1}), g ∘ f  (57)

where γ_{2} = ∂(χ_{1})γ_{1}, and:

(γ_{1}, χ_{1}), (γ_{3}, χ_{2}) ▶ f ∘_{v}  (γ_{3}, χ_{2}), f = (γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), f  (58)

where the underlying squares in D(G) are given in (19).

4.10. Theorem. C//G is a well-defined double category.

Proof. This follows from general considerations, but can also be checked by using the properties of the internal categories C//G and { \atop C//G}, e.g. the vertical target of the vertical composition of squares is well-defined due to (48). We note that the vertical target of the horizontal composition of squares, equals the composition of the vertical targets of the two squares, i.e.

(γ_{1}, χ_{2}χ_{1}) ▶ (g ∘ f) = ((γ_{2}, χ_{2}) ▶ g) ∘ ((γ_{1}, χ_{1}) ▶ f)  (59)

by the functoriality of Φ̂ in (22). Likewise the horizontal target of the vertical composition of squares equals the composition of the horizontal targets of the two squares, i.e.

(∂(χ_{1})γ_{1}∂(χ_{2})γ_{3}, y) = (∂(χ_{1}(γ_{1} ▷ χ_{2}))γ_{1}γ_{3}, y).  (60)

This follows in a straightforward fashion from crossed module properties.

Associativity of the horizontal and vertical composition of squares is an immediate consequence of the associativity of the composition functors in C//G and { \atop C//G}. Finally the interchange law for horizontal and vertical composition of squares corresponds to the proof of functoriality of the composition functor in C//G (Theorem 4.4) and reduces to stating the interchange law for four squares (49) in D(G).

4.11. Remark. If we display the squares of the double category as follows:

[figure: x; f; (γ,x); ((γ,χ),f); y; (∂(χ)γ,y); γ ▶ x; (γ,χ)▶f; (∂(χ)γ) ▶ y]  (61)

with horizontal arrows dashed and vertical arrows dotted, this can be used to illuminate the two internal category perspectives for the double category.

From the perspective of the internal category C//G of Definition 4.3, these squares of the double category display the action of the source and target functors s, t on objects (γ, x), (∂(χ)γ, y), and morphisms ((γ, χ), f). The dashed arrows denote the image of the square under source and target functors, while the dotted arrows denote its source and target internal to the category of morphisms.

Similarly, from the perspective of the internal category { \atop C//G} of Theorem 4.8, these squares display the action of the source and target functors s̃, t̃ on objects f, (γ, χ) ▶ f, and morphisms ((γ, χ), f). Now the dotted arrows denote the image of the square under source and target functors, while the dashed arrows denote its source and target internal to the category of morphisms.

The squares of the double category superimpose the two perspectives giving a balanced view, which makes both internal structures transparent at the same time.

4.12. Structure of the Transformation Double Category. Next, we will consider some consequences of the construction of C//G. As constructed, the main information which can be read directly from C//G, viewed as an internal category in Cat, is precisely the target map, which is just the action Φ̂ itself. Some less obvious consequences are more easily read from the transpose { \atop C//G}, or from C//G seen as a double category.

4.12.1. Relation to Ordinary Transformation Groupoids. We have seen, in Theorem 4.8, that the groupoid of objects and the groupoid of morphisms of { \atop C//G} are both transformation groupoids. The structure maps of { \atop C//G} relate these groupoids to each other. Being functors, these maps consist of two parts, which we can consider separately. Considering the identity-inclusion functor tells us how several transformation groupoids are nested inside one another, and will justify the notation ▶ we previously introduced for our 2-group action. First, note what these groupoids are in crossed module notation.

4.13. Corollary. If G is given by a crossed module (G, H, ∂, ▷), then ({ \atop C//G})^{(0)} =

C^{(0)}//G, and C^{(1)}//G ⊂ ({ \atop C//G})^{(1)} = C^{(1)}//(G ⋉ H).

Proof. In the construction of a categorical group from a crossed module, the group G ⋉ H is the group of morphisms, and the group G of objects occurs as the subgroup of elements of the form (γ, 1_{H}). The groupoid inclusion then follows immediately from the identity-inclusion functor in Theorem 4.8.

The fact that the structure maps are functors means that, from an “external” point of view (that is, if we look at underlying sets), C//G combines three closely related group actions in the standard sense. These three group actions have all been denoted by the symbol ▶ introduced in Definition 3.5. This notation can now be justified because they are all restrictions of the action of G^{(1)} on C^{(1)}, along the identity inclusion maps for G and C. In particular, the associated action groupoids are also related by these restrictions. More precisely, we have:

4.14. Corollary. The identity-inclusion functor ẽ factors into two inclusions:

({ \atop C//G})^{(0)} ⊂ C^{(1)}//G^{(0)} ⊂ ({ \atop C//G})^{(1)}  (62)

where the first inclusion is given by the identity inclusion of C, and the second by the identity inclusion of G.

Proof. The fact that objects of G give endofunctors of C means that the morphism maps Φ_{γ}^{(1)} determine an action of G^{(0)} on C^{(1)}. Thus, objects of G act on both C^{(0)} and C^{(1)}. By functoriality, the inclusion of objects of C as identity morphisms in C^{(1)} is compatible with the action. So the subset C^{(0)} ⊂ C^{(1)} is closed under the action of G. The associated transformation groupoid has as its objects the morphisms f ∈ C, and as morphisms pairs (γ, f) which compose in the usual way. These correspond to the special squares (γ, 1_{H}), f . There is a full inclusion of transformation groupoids induced by the inclusion of their sets of objects C^{(0)} ⊂ C^{(1)}.

The group G^{(1)} ≅ G ⋉ H also acts on C^{(1)}. This action is given by Φ̂, and is related to the action of G on C^{(1)} by the identity inclusion map for G, namely G ⊂ G ⋉ H. In particular, the action of G^{(1)} on C^{(1)} combines both natural transformations acting on objects, and functors acting on morphisms. To see this, note that the squares in the double category C//G occur by construction within a commuting cube which shows the naturality square associated to f, or equivalently the two ways of expressing (γ, χ) ▶ f in terms of Φ - see (26):

[figure: x; (γ,x); f; y; (γ,y); x; (∂(χ)γ,x); f; y; (∂(χ)γ,y); γ ▶ x; Φ_{(γ,χ)}(x); γ▶f; ((γ,χ),f); γ ▶ y; Φ_{(γ,χ)}(y); ∂(χ)γ ▶ x; ∂(χ)γ▶f; ∂(χ)γ ▶ y]  (63)

So the action of morphisms of G on morphisms of C involves natural transformations at the source and target objects, as well as functors acting on the morphisms themselves. The outside faces of the cube (63) are themselves special cases of squares in which one of these two parts is an identity. The case χ = 1_{H}, so that (γ, χ) = Id_{γ} (for example, the front or rear faces of that cube) is precisely the second inclusion of the theorem.

Restricting further to the image of the unit inclusion for C, we have the case f = Id_{x}. In particular, γ ▶ Id_{x} = Id_{γ▶x} is just a consequence of the fact that the action is functorial. (For this reason, it is immediate if we think of the action in terms of Φ̂.)

The action by the subgroup G gives a transformation groupoid with fewer morphisms than the action by the big group G ⋉ H - but the former is strictly a sub-groupoid (by a non-full inclusion) of the latter.

Notice that this factorization of ẽ is given by restricting to the image of the unit inclusions of C and then G, and not the other way around. There is no well-defined action of G^{(1)} on C^{(0)}. This can easily be seen, again by taking a special case of a square of the form (63) with f = Id_{x}. Since γ ▶ x and ∂(χ)γ ▶ x need not be the same object, and even in the case they are, Φ_{(γ,χ)}(x) need not be an identity, the image of e : C^{(0)} → C^{(1)} is not fixed by the action of G^{(1)} on C^{(1)}.

4.14.1. Image of the Composition Functor. Next, consider the composition ˜∘. It is somewhat more complicated than ∘, and derives from the 2-group structure of G, just as composition does in transformation groupoids. A little example calculation with this composite will show how 2-group structure and an action must cohere.

Indeed, composition in ({ \atop C//G})^{(0)} is exactly the composition in the transformation groupoid C^{(0)}//G, by Corollary 4.13. Similarly, the composition in ({ \atop C//G})^{(1)} is the composition in C^{(1)}//G^{(1)}, which is determined by the group multiplication in G^{(1)} ≅ G ⋉ H (in the crossed-module representation), together with the action on C^{(1)}. So we can write this as:

(γ′, χ′), (γ, χ) ▶ f ˜∘ (γ, χ), f = (γ′γ, χ′(γ′ ▷ χ)), f  (64)

We know by construction and the above theorems that this makes a well-defined double category. It is, however, not given directly by the action Φ seen as a 2-functor. In that point of view, the action of morphisms of G on morphisms of C is a consequence of the naturality of Φ_{(γ,χ)} for all (γ, χ) ∈ G^{(1)}.

Writing the target of this composite concretely, it is:

(γ′γ, χ′(γ′ ▷ χ)) ▶ f  (65)

However, we have concrete expressions for (γ, χ) ▶ f, shown in the cube (63), namely:

\begin{aligned}(γ, χ) ▶ f & = & (∂(χ)γ ▶ f) & ∘ & Φ_{(γ,χ)}(x) \\ & = & Φ_{(γ,χ)}(y) & ∘ & (γ ▶ f) \end{aligned}  (66)

where composition is taken in C. Recall that this is a consequence of the naturality of the bottom square of (63) when we apply Φ_{(γ,χ)} at the morphism f.

Now, if we then take (γ, χ) ▶ f as the source of the new square (γ′, χ′), (γ, χ) ▶ f to find its target, we are applying Φ_{(γ′,χ′)} at the morphism (γ, χ) ▶ f. To compute this explicitly, note that each morphism in the commuting square at the bottom of (63) gives rise to another commuting square, and these form four faces of a cube whose other two faces are the image of the original commuting square under the functors Φ_{γ′} and Φ_{∂χ′γ′} respectively:

\xymatrix{∂(χ′)γ′γ ▶ x \ar[dd]_{∂(χ′)γ′▶Φ_{(γ,χ)}(x)} \ar[rr]^{∂(χ′)γ′γ▶f} &  & ∂(χ′γ′γ) ▶ y \ar[dd]^{∂(χ′)γ′▶Φ_{(γ,χ)}(y)} \\  & γ′γ ▶ x \ar[dd]_{γ′▶Φ_{(γ,χ)}(x)} \ar[rr]_{γ′γ▶f} \ar[ul]^{Φ_{(γ′,χ′)}(γ▶x)} \ar@{-->}[dr] &  & γ′γ ▶ y \ar@{-->}[ul]_{Φ_{(γ′,χ′)}(γ▶y)} \ar[dd]^{γ′▶Φ_{(γ,χ)}(y)} \\ ∂(χ′)γ′∂(χ)γ ▶ x \ar[rr]^{∂(χ′)γ′∂(χ)γ▶f} &  & ∂(χ′)γ′∂(χ)γ ▶ y \\  & γ′∂(χ)γ ▶ x \ar[ul]^{Φ_{(γ′,χ′)}(∂(χ)γ▶x)} \ar[rr]_{γ′∂(χ)γ▶f} &  & γ′∂(χ)γ ▶ y \ar@{-->}[ul]_{Φ_{(γ′,χ′)}(∂(χ)γ▶y)}}

(67)

We have omitted the images under Φ_{γ′} and Φ_{(∂χ′)γ′} of the original diagonal (γ, χ) ▶ f for clarity. However, they are the diagonals on the front and back face of this cube. Together with the corresponding edges between these faces, they form a square whose diagonal is the target we want, just as before.

The target morphism of the square (γ′, χ′), (γ, χ) ▶ f , hence also of our composite square, is therefore the dotted arrow across the diagonal of this cube.

The source object of this morphism is γ′γ ▶ x, and its target object is ∂(χ′)γ′∂(χ)γ ▶ y.

However, we already found that the composite square was (γ′γ, χ′(γ′ ▷ χ)), f . Computing the target of this square directly from (63) shows that its target morphism begins at γ′γ ▶ x and ends at (∂(χ′(γ′ ▷ χ))γ′γ ▶ x. A straightforward application of the crossed module axioms confirms that indeed:

∂(χ′)γ′∂(χ)γ = ∂(χ′(γ′ ▷ χ))γ′γ  (68)

as it must be.

Knowing that we are in a double category where composition is well-defined tells us more. Just as the naturality square at the bottom of (63) gave us two expressions for (γ, χ) ▶ f, the cube of naturality squares (67) gives us a total of six expressions which equal the morphism

(γ′γ, ∂(χ′(γ′ ▷ χ))) ▶ f  (69)

As before, each is a composite of morphisms in C, most directly written as:

Φ_{(γ′,χ′)}(∂(χ)γ ▶ y) ∘ (γ′ ▶ Φ_{(γ,χ)}(y)) ∘  (γ′γ ▶ f)
= (∂(χ′)γ′) ▶ Φ_{(γ,χ)}(y)) ∘  Φ_{(γ′,χ′)}(γ ▶ y)  ∘  (γ′γ ▶ f)
= Φ_{(γ′,χ′)}(∂(χ)γ ▶ y) ∘  (γ′∂(χ)γ ▶ f)  ∘ (γ′ ▶ Φ_{(γ,χ)}(x))
= (∂(χ′)γ′∂(χ)γ ▶ f) ∘ Φ_{(γ′,χ′)}(∂(χ)γ ▶ x) ∘ (γ′ ▶ Φ_{(γ,χ)}(x))
= (∂(χ′)γ′ ▶ Φ_{(γ,χ)}(y)) ∘ (∂(χ′)γ′γ ▶ f) ∘ Φ_{(γ′,χ′)}(γ ▶ x)
= (∂(χ′)γ′∂(χ)γ ▶ f) ∘ (∂(χ′)γ′ ▶ Φ_{(γ,χ)}(x)) ∘ Φ_{(γ′,χ′)}(γ ▶ x)

(70)

The fact that these are all equal is a consequence of the fact that Φ is an action, hence every square in the cube is either a naturality square or the image of a naturality square under a functor, and therefore every square commutes. Some of the terms may also be written differently in light of (68) and similar applications of crossed-module axioms.

In general, a composite of n squares will give rise to many different equivalent expressions for the target morphism, in the same way. In particular, one can easily see by induction that one obtains an (n + 1) cube by repeatedly applying Φ at each step. Counting paths around such a cube, one finds (n + 1)! possible composites in C which amount to the same morphism.

Clearly, using the 2-group composition as in (64) to find the target (65) is more direct, and since we know the transpose of C//G is a double category, we can use the 2-group action to find the target directly.

4.14.2. Horizontal and Vertical 2-Categories. Finally, we turn to the double category point of view on C//G. When speaking of double categories, one often refers to various smaller structures which arise from specialization. We already know that the horizontal and vertical categories are, respectively, C itself, and C^{(0)}//G^{(0)}. Moreover, there are also two categories which share the same collection of morphisms (the squares of the double category), but have different sets of objects (namely, the horizontal and vertical morphisms). Again, we have already seen that these are, respectively, G × C and C^{(1)}//G^{(1)}.

However, one also speaks of the horizontal and vertical 2-categories of a double category. These are the last aspect of the structure of C//G we will consider.

4.15. Definition. The horizontal 2-category H(D) of a double category D has the same objects and 1-morphisms as its horizontal category. The 2-morphisms consist only of the squares of D for which the vertical morphisms on the boundary are identities. The vertical 2-category V (D) is defined similarly (i.e. it is the horizontal 2-category of D̃).

It is straightforward to check that these form 2-categories, but since it is a standard fact, we will not verify all the details here. However, we will consider the horizontal and vertical 2-categories for C//G. It should be clear from what we have just said that these will amount to C and C^{(0)}//G^{(0)}, extended by adding certain 2-morphisms.

4.16. Theorem. The horizontal 2-category H(C//G) of the transformation double category associated to a 2-group action has the same objects and 1-morphisms as C. Given objects x, y ∈ C, and a pair of morphisms f, f′ : x → y, the 2-morphisms in Hom(f, f′) are labeled by (1_{G}, χ) : 1_{G} ⇒ 1_{G} such that

f′ = f ∘ Φ_{(1_{G},χ)}(x)  (71)

These 2-morphisms compose horizontally and vertically just as the endomorphisms of 1_{G} in G.

Proof. The objects and 1-morphisms are those of the horizontal category, which is just C by definition.

The 2-morphisms correspond to squares of the form (56) for which the vertical morphisms are identities. These must be of the form (1_{G}, χ), f , since the horizontal source of (γ, χ), f is (γ, x) = Id_{x}, and thus γ = 1_{G}. Moreover, since the target (∂(χ)γ, y) = (∂(χ), y) = Id_{y}, we must also have that ∂(χ) = 1_{G}. So only 2-endomorphisms of the identity of G enter as 2-morphisms.

The vertical target is then the horizontal morphism

(1_{G}, χ) ▶ f = (∂(χ) ▶ f) ∘ Φ_{(1_{G},χ)}(x)  (72)
= f ∘ Φ_{(1_{G},χ)}(x)

When γ_{1} and γ_{2} in (57) and (58) are both equal to 1_{G}, these horizontal and vertical composition rules just amount to multiplication in H, that is, the composition of 2endomorphisms of the identity.

Intuitively, this says that the 2-morphisms with source f in H(C//G) correspond exactly to the endomorphisms (1_{G}, χ) of 1_{G} in G. In the language of crossed modules, they correspond to χ ∈ ker(∂). The target of such a 2-morphism is found by precomposing f with the endomorphisms of x given by the natural transformations Φ_{(1_{G},χ)}. Clearly, these labels χ for 2-morphisms sourced at f will be the same for all f. Depending on the precise details of the action Φ, all such 2-morphisms might be endomorphisms of f itself, or the targets might all be different from f and from each other. This is the type of information about the action Φ which is captured by H(C//G).

4.17. Theorem. The vertical 2-category V (C//G) of the transformation double category associated to a 2-group action has the same objects and 1-morphisms as C^{(0)}//G^{(0)}. Given an object x ∈ C and γ, γ′ ∈ G such that γ ▶ x = γ′ ▶ x, there will be a 2-morphism from (γ, x) to (γ′, x) for each χ such that Φ_{(γ,χ)}(x) = Id_{γ▶x} and γ′ = ∂(χ)γ.

Proof. The objects and 1-morphisms are those of the vertical category, which is C^{(0)}//G^{(0)} by Theorem 4.8.

The 2-morphisms correspond to squares of the form (56) in which x = y and the horizontal morphisms on the boundary are both identities. Since the vertical source is f = Id_{x}, they are squares of the form (γ, χ), Id_{x} .

Moreover, the vertical target must also be the identity. Now, to begin with, this must mean that γ ▶ x = γ′ ▶ x (taking γ′ = ∂(χ)γ), as stated in the theorem. This vertical target is the horizontal morphism

(γ, χ) ▶ f = (γ′ ▶ Id_{x}) ∘ Φ_{(γ,χ)}(x)  (73)
= Id_{γ′▶x} ∘ Φ_{(γ,χ)}(x)

The second equality holds because γ′ acts on f = Id_{x} by the functor Φ_{γ′}, and functors respect identities. Thus, this condition implies that not only are γ ▶ x and γ′ ▶ x equal, but indeed the natural transformation Φ_{(γ,χ)} intertwines them by the identity on γ ▶ x.

The compositions again follow (57) and (58).

Intuitively, the transformation groupoid C^{(0)}//G^{(0)} displays the object part of the action Φ̂. The morphism part can potentially appear here also, relating local symmetry transformations which carry x to the same place. However, as we have seen, such a relation need not exist if γ ≠ γ′, since even though γ ▶ x = γ′ ▶ x, there might be no natural transformation intertwining them, or none intertwining them with identities. Conversely, there may be more than one such natural transformation. This is the information about the full action Φ captured by V (C//G) which the mere groupoid C^{(0)}//G^{(0)} cannot see.

We would next like to explicitly describe the above, by examining the structure of the transformation double groupoid G//G, associated to our example of the adjoint action of a 2-group G on itself.

4.18. The Transformation Double Category for the Adjoint Action. As in all transformation double groupoids, the horizontal category of G//G is just the same as G, seen as a category (which happens to be monoidal). The vertical morphisms and squares will always be labelled by pairs of, respectively, objects and morphisms from G.

Summing up the above, we have:

4.19. Corollary. The transformation double groupoid G//G for the adjoint action of a 2-group G, classified by the crossed module (G, H, ▷, ∂), has:

• Objects: labelled by g ∈ G

• Horizontal Category: the underlying category of G

• Vertical Category: same as G//G, the transformation groupoid for the adjoint action of G

• Squares: labelled by pairs ((γ_{1}, χ_{1}), (g_{1}, η_{1})) ∈ (G × H) × (G × H), denoted by

(γ_{1}, χ_{1}), (g_{1}, η_{1}) . As in (56), the horizontal and vertical source and target of this square are displayed as follows:

[figure: g_{1}; η_{1}; Φ_{γ_{1}}; (γ_{1},χ_{1}),(g_{1},η_{1}); g_{2}; Φ_{γ_{2}}; g_{3}; η_{2}; g_{4}]  (74)

where γ_{2} = ∂(χ_{1})γ_{1}. Here η_{2} is determined by:

\begin{matrix}g_{3} \\ η_{2} \\ g_{4} \end{matrix} = γ^{1} χ_{1}  g^{1} η_{1} γ_{1}^{−1} χ_{1}^{−h} γ_{2}  g_{2} γ_{2}^{−1} .  (75)

• Horizontal composition of squares is given by multiplication in H:

(γ_{2}, χ_{2}), (g_{2}, η_{2}) ∘_{h}  (γ_{1}, χ_{1}), (g_{1}, η_{1}) = (γ_{1}, χ_{2}χ_{1}), (g_{1}, η_{2}η_{1}) (76)

where g_{2} = ∂(η_{1})g_{1}, and vertical composition of squares is given by:

(γ_{1}, χ_{1}), (γ_{3}, χ_{2}) ▶ (g_{1}, η_{2})  ∘_{v}  (γ_{3}, χ_{2}), (g_{1}, η_{2}) =
(γ_{1}γ_{3}, χ_{1}(γ_{1} ▷ χ_{2})), (g_{1}, η_{2})  (77)

which makes them into the transformation groupoid (G ⋉ H)//(G ⋉ H),

Proof. This is a direct consequence of the structure of C//G laid out in Definition 4.9 and Theorem 4.10, with the structure of G as a categorical group given explicitly for C.

Now we will find the horizontal and vertical 2-categories, H(G//G) and V (G//G), of the double category associated to the adjoint action of G

4.20. Corollary. If G is a 2-group classified by the crossed module (G, H, ▷, ∂), then the horizontal 2-category H(G//G) associated to the adjoint action of G has:

• Objects: g ∈ G

• Morphisms: (g, η) ∈ G ⋉ H with source, target, and composition as in G

• 2-Morphisms: ((g, η), χ) ∈ (G ⋉ H) × ker(∂), whose source is (g, η), and whose

target is

(g, η′) = (g, η · (χg ▷ χ^{−1}))  (78)

Composition is given by multiplication in the subgroup ker(∂) ⊂ H.

Proof. Taking the expression (71) for the target of a 2-morphism, in the case of the adjoint action we use the fact that Φ_{(1_{G},χ)}(g) is defined in (38). Substituting this gives the target specified in (78). Composition as multiplication in H also follows from Theorem 4.16.

This tells us how to extend G to a 2-category to reflect part of the symmetry that arises from its action on itself. The vertical 2-category, by contrast, can be better seen as an extension of G//G, the transformation groupoid for the actions of just the group of objects on itself. In particular, we have the following.

4.21. Corollary. If G is a 2-group classified by the crossed module (G, H, ▷, ∂), then the vertical 2-category V (G//G) associated to the adjoint action of G has:

• Objects: g ∈ G

• Morphisms: (γ, g) ∈ G × G with source, target, and composition maps as in the transformation groupoid for the adjoint action of G on itself

• 2-Morphisms: There will be a 2-morphism from (γ, g) to (γ′, g), for each χ ∈ H

such that

(γgγ^{−1}) ▷ χ^{−1} = χ^{−1}  (79)

(That is, χ^{−1} is a fixed point for γgγ^{−1} under the action ▷).

Proof. In Theorem 4.17, we established when there is a 2-morphism in V (C//G). In our case, taking x = g, we find there is such a 2-morphism for any χ such that Φ_{(γ,χ)}(g) = Id_{γgγ^{−1}}.

But we have the expression (38) for the maps defining the natural transformation Φ_{(γ,χ)}. So this condition says that

χ(γgγ^{−1}) ▷ χ^{−1} = 1_{H}  (80)

or in other words, that

(γgγ^{−1}) ▷ χ^{−1} = χ^{−1}  (81)

This is exactly the statement above.

As usual, composition follows (57) and (58).

Intuitively, this tells us that the information about the action Φ captured by the 2-morphisms relating two different symmetry relations γ and γ′ taking g to g is fundamentally information about the crossed module structure, and the action ▷ of G on H.

We have included these calculations of the horizontal and vertical 2-categories, partly because these particular slices of the double category G//G (or C//G generally) are interesting in their own right. But we also include them to illustrate that many squares in the double category do not appear in either the horizontal or vertical 2-categories. Each is missing a great deal of information about the action Φ, and we need to go to the double category G//G (or C//G in general) for the full information.

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1468  JEFFREY C. MORTON AND ROGER PICKEN

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Department of Mathematics and Statistics, The University of Toledo, 2801 W. Bancroft St. Toledo, OH 43606-3390, USA

Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Email: jeffrey.c.morton@theoreticalatlas.net
     roger.picken@tecnico.ulisboa.pt
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