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A higher-dimensional categorical perspective on 2-crossed modules

  • Emre Özel ORCID logo EMAIL logo , Ummahan Ege Arslan ORCID logo and İbrahim İlker Akça ORCID logo
Published/Copyright: December 21, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of a 2-truncated simplicial group. Therefore, the 2-crossed module is an algebraic homotopy model for the homotopy 3-types. Tricategories are a three-dimensional generalization of the bicategory concept. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product. Briefly, a Gray category is a semi-strict 3-category for homotopy 3-types. Naturally, the tricategory perspective is used in homotopy theory. The 2-crossed module is associated with the concept of the Gray category. The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups.

Keywords: crossed module; 2-crossed module; categorification; sestertius category; tricategory
MSC 2010: 18G45; 18N10; 18N20; 18N25

1 Introduction

Whitehead described crossed modules in [1,2] for modeling homotopy 2-types from the algebraic homotopy theory perspective. Roughly speaking, in terms of nonabelian algebraic homotopy 2-types, any crossed module of groups G 1 = ( E → ∂ G ) has a classification space X such that π 2 ( X ) ≅ ker ( ∂ ) and π 1 ( X ) ≅ coker ( ∂ ) , while each of the homotopy groups π i ( X ) for i > 2 is trivial. Loday introduced cat 1 -groups, a similar structure for homotopy 2-types in [3]. From a categorical point of view, Brown and Spencer [4,5] associated crossed modules of groupoids with double groupoids. In terms of structures in categories, a crossed module of groups is equivalent to an internal category in the category of groups Grp  [6,7]. From the point of view of monoidal categories in [8], crossed modules are equivalent to strict 2-groups.

Gordon et al. [9] introduced the tricategory concept, the natural three-dimensional generalization of bicategory [10]. Also, bicategories are called weak 2-categories [8]. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product [9]. Briefly, the Gray category means exactly a strict cubical tricategory. Furthermore, a strict cubical tricategory is a semi-strict 3-category for homotopy 3-types, as a Gray category [11,12].

Conduché [13] defined a 2-crossed module for the algebraic homotopy 3-type models using simplicial homotopy theory. In brief, the 2-crossed module is the Moore complex of a 2-truncated simplicial group. In [14,15], the 2-crossed module is associated with the concept of the Gray category.

The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups. For this, we will first construct a strict 2-group from a crossed module of groups by a categorical point of view. In contrast to crossed modules, the categorical structure must change as one moves towards higher dimensions. Because the 2-crossed module defined on the pre-crossed module does not form a strict 3-category structure. The algebraic homotopy 3-type models related to the concept of semi-strict 3-group are sestertius categories. The word sestertius is derived from the Latin phrase “SEMIS TERTIVS,” which is translated as “2 1/2.” Mathematically, a sestertius category is defined by a three-dimensional generalization of a strict 2-category, called a sesquicategory, for which there is no interchange law (for a different perspective, refer [16]). Therefore, we will obtain a sestertius category with one object in which all 1-, 2-, and 3-morphisms are invertible from a 2-crossed module of groups. Afterwards, we will construct a tricategory from this sestertius category.

2 Preliminaries

Before giving the definition of a tricategory, we will simply give an overview of the two- and three-dimensional categories.

Roughly speaking, a category consists of objects and the morphisms between them, as follows:

2

For these structures, we will use the nomenclature 0- and 1-cells, respectively. If we use the 0-source and 0-target functions to describe the 0-cells, we obtain s 0 ( f ) = a and t 0 ( f ) = b . Naturally, this structure has to satisfy unit and associative composition:

2

Note that the composition of the 1-cells is “ ♯ 0 .” We may write this composition as g ♯ 0 f (meaning first g and f ). Throughout our work, we will describe all compositions in this way. Of course, the composite of g ♯ 0 f of two 1-cells is defined if and only if t 0 ( f ) = s 0 ( g ) . Thus, for composition g ♯ 0 f , s 0 ( g ♯ 0 f ) = s 0 ( f ) , and t 0 ( g ♯ 0 f ) = t 0 ( g ) . In addition, if the morphisms of 1-cells are invertible, this structure is a groupoid.

If we want to add 2-cell to this structure, we have the following:

2

As the diagram shows, the 1-source and 1-target of the 2-cell α are s 1 ( α ) = f and t 1 ( α ) = f ′ , respectively.

Let us have 2-cells with source and target compatibility as follows:

2

Thus, we can define whiskering diagrams as follows:

2
2

In addition, the horizontal composition of 2-cells “ ♯ 0 ” is defined as follows:

2

Further, if we have 2-cells α and α ′ where the 1-target of α is f ′ and the 1-source of α ′ is f ′ , then the vertical composition of these 2-cells, “ ♯ 1 ,” is defined as follows:

2

Additionally, the vertical composition ♯ 1 of 2-cells is compatible with the horizontal composition ♯ 0 of 2-cells. If we show the interchange law diagrammatically, we have

2

We can upgrade this structure as much as we want. If we add 3-cells, we have

2

Note that the 2-source and 2-target of 3-cell π are s 2 ( π ) = α 1 and t 2 ( π ) = α 2 , respectively. Thus, we would expect to have a vertical composition ♯ 2 for 3-cells. Of course, it is necessary to satisfy the interchange law for 3-cells. If the morphisms of these 1-, 2-, and 3-cells are invertible, then this structure is a 3-groupoid.

We will consider the tricategory structure, which is a particular type of the 3-category structure. Tricategory is some kind of algebraic notation for weak 3-categories. We recall the definition of tricategory given in [9,17].

Definition 2.1

A tricategory T consists of following data subject to the following axioms.

DATA:

  1. A set O b T of objects of T .

  2. For ( a , b ) ∈ O b T × O b T , a bicategory (weak 2-category) T ( a , b ) , called the hom-bicategory of T at a and b , exists. The objects of T ( a , b ) will be referred to as the 1-cells of T with source a and target b , the arrows of T ( a , b ) will be referred to as 2-cells of T (with their same source and target), and the 2-cells of T ( a , b ) will be referred to as 3-cells of T (also with their same source and target).

  3. For objects a , b , and c of O b T , a functor ⊗ : T ( b , c ) × T ( a , b ) ⟶ T ( a , c ) exists called composition.

  4. For an object a of O b T , a functor I a : 1 ⟶ T ( a , a ) , exists where 1 denotes the unit bicategory.

  5. For objects a , b , c , and d of O b T , an adjoint equivalence α

in Bicat ( T ( c , d ) × T ( b , c ) × T ( a , b ) , T ( a , d ) ) exists.

  1. For objects a , b of O b T , adjoint equivalences l and r

in Bicat ( T ( a , b ) , T ( a , b ) ) exists.

  1. For a , b , c , d , e ∈ O b T , an isomorphism 2-cell π ( i.e., an invertible modification)

in the bicategory Bicat ( T 4 ( a , b , c , d , e ) , T ( a , e ) ) exists, where T 4 = T 4 ( a , b , c , d , e ) is abbreviation for T ( d , e ) × T ( c , d ) × T ( b , c ) × T ( a , b ) .

  1. For objects a , b , c of T , invertible modifications exist.

AXIOMS:

(1) [Non-abelian 4-cocycle condition] The following equation of 2-cells holds in the bicategory T ( a 1 , a 5 ) , where we will denote the 2-cell elements by a i , 1 ≤ i ≤ 14 for easy reading of the diagrams. For elements k , j , h , g , and f of 2-cells in T

a 1 = ( k ⊗ ( j ⊗ ( h ⊗ g ) ) ) ⊗ f , a 2 = k ⊗ ( ( j ⊗ ( h ⊗ g ) ) ⊗ f ) , a 3 = k ⊗ ( j ⊗ ( ( h ⊗ g ) ⊗ f ) ) , a 4 = k ⊗ ( j ⊗ ( h ⊗ ( g ⊗ f ) ) ) , a 5 = ( k ⊗ j ) ⊗ ( h ⊗ ( g ⊗ f ) ) , a 6 = ( ( k ⊗ j ) ⊗ h ) ⊗ ( g ⊗ f ) , a 7 = ( ( ( k ⊗ j ) ⊗ h ) ⊗ g ) ⊗ f , a 8 = ( ( k ⊗ ( j ⊗ h ) ) ⊗ g ) ⊗ f a 9 = ( k ⊗ ( ( j ⊗ h ) ⊗ g ) ) ⊗ f , a 10 = k ⊗ ( ( ( j ⊗ h ) ⊗ g ) ⊗ f ) , a 11 = ( k ⊗ ( j ⊗ h ) ) ⊗ ( g ⊗ f ) , a 12 = k ⊗ ( ( j ⊗ h ) ⊗ ( g ⊗ f ) ) , a 13 = ( ( k ⊗ j ) ⊗ ( h ⊗ g ) ) ⊗ f , a 14 = ( k ⊗ j ) ⊗ ( ( h ⊗ g ) ⊗ f ) ;

(2) [Left normalization] The following equation of 2-cells holds in the bicategory T ( a 1 , a 4 ) , where the unmarked isomorphisms are either naturality isomorphisms for α or unique coherence isomorphisms from the H o m -bicategory.

a 1 = ( h ⊗ ( l ⊗ g ) ) ⊗ f , a 2 = ( h ⊗ g ) ⊗ f , a 3 = h ⊗ ( g ⊗ f ) , a 4 = ( ( h ⊗ l ) ⊗ g ) ⊗ f , a 5 = ( h ⊗ l ) ⊗ ( g ⊗ f ) , a 6 = h ⊗ ( ( l ⊗ g ) ⊗ f ) , a 7 = h ⊗ ( l ⊗ ( g ⊗ f ) ) ;

(3) [Right normalization] The following equation of 2-cells holds in the bicategory T ( a 1 , a 4 ) .

a 1 = h ⊗ ( ( g ⊗ l ) ⊗ f ) , a 2 = h ⊗ ( g ⊗ ( l ⊗ f ) ) , a 3 = h ⊗ ( g ⊗ f ) , a 4 = ( h ⊗ g ) ⊗ f , a 5 = ( h ⊗ ( g ⊗ l ) ) ⊗ f , a 6 = ( ( h ⊗ g ) ⊗ l ) ⊗ f , a 7 = ( h ⊗ g ) ⊗ ( l ⊗ f ) ;

Example 2.2

  1. A monoidal bicategory is a tricategory with a single object.

  2. BiCat is a tricategory with objects being bicategories; 1-cells are homomorphisms between these bicategories; 2-cells are pseudonatural transformations; and 3-cells are modifications. As seen in [18], BiCat is not triequivalent to Gray, but it can be Gray-categorized in a special way [17].

  3. Fundamental 3-groupoids are tricategories. 3-groupoids are a generalization of 2-groupoids in higher-dimensional category theory. If we look at fundamental groupoids with a higher categorical approach, the fundamental 3-groupoid is a tricategory whose objects are spaces, 1-cells are continuous maps, and higher cells are the appropriate mediator homotopy types. This structure is explored in detail in [19]. In [17], in terms of the fundamental 3-groupoid, the objects of an X space are points of X , and the higher cells are paths, two-dimensional disks, and equivalence classes of three-dimensional disks in X . This structure is also a tricategory.

Remark 2.3

A tricategory is strict if all of its constraints are identities; that is, if the invertible modifications (5)–(8) in Definition 2.1 are identities [9].

Lemma 2.4

Let T be a tricategory; if it satisfies the following conditions, it is a cubical tricategory [17].

  1. Each bicategory T ( a , b ) is a strict 2-category.

  2. Each functor I a : 1 ⟶ T ( a , a ) is a cubical functor.

  3. For objects a , b , and c of O b T , each ⊗ : T ( b , c ) × T ( a , b ) ⟶ T ( a , c ) is a cubical functor.

Definition 2.5

Let T and T ′ be tricategories. A trihomomorphism F from T to T ′ is a map that consists of the following data and axioms:

DATA:

  1. A morphism O b T → O b T ′ .

  2. For a , b ∈ O b T , a functor F a b : T ( a , b ) → T ′ ( F a , F b ) .

  3. For a , b , and c ∈ O b T , an adjoint equivalence ξ : ⊗ ′ ∘ ( F × F ) ⇒ F ∘ ⊗ with left adjoint is illustrated in the following diagram:

  1. For all a ∈ O b T , an adjoint equivalence ι : I F a ′ ⇒ F ∘ I a with left adjoint is illustrated in the following diagram:

  1. For a , b , c , d ∈ O b T , an invertible modification θ is illustrated in the following diagram:

  1. For a , b ∈ O b T , invertible modifications κ and χ is illustrated in the following diagram:

and

AXIOMS:

(1) For all 1-morphisms ( x , y , z , w ) ∈ T ( d , e ) × T ( c , d ) × T ( b , c ) × T ( a , b ) , the following equation of modifications holds:

a 1 = F ( ( ( f g ) j ) k ) , a 2 = F ( ( f ( g j ) ) k ) , a 3 = F ( f ( ( g j ) k ) ) , a 4 = F ( f ( g ( j k ) ) ) , a 5 = F f F ( g ( j k ) ) , a 6 = F f ( F g F ( j k ) ) , a 7 = F f ( F g ( F j F k ) ) , a 8 = ( F f F g ) ( F j F k ) , a 9 = ( ( F f F g ) F j ) F k , a 10 = ( F ( f g ) F j ) F k , a 11 = F ( ( f g ) j ) F k , a 12 = F ( f ( g j ) ) F k , a 13 = ( F f F ( g j ) ) F k , a 14 = ( F f ( F g F j ) ) F k , a 15 = F f F ( ( g j ) ) k , a 16 = F f ( F ( g j ) F k ) , a 17 = F f ( F ( g j ) F k ) , a 18 = F ( f g ) ( F j F k ) , a 19 = ( F f F g ) F ( j k ) , a 20 = F ( f g ) F ( j k ) , a 21 = F ( ( f g ) ( j k ) ) ;

(2) For all 1-cells ( x , y ) ∈ T ( b , c ) × T ( a , b ) , the following equation of modifications holds:

a 1 = F ( ( f I ) g ) , a 2 = F ( f ( I g ) ) , a 3 = F ( f g ) , a 4 = F f F g , a 5 = F f F ( I g ) , a 6 = F f ( F I F g ) , a 7 = ( F f F I ) F g , a 8 = F ( f I ) F g , a 9 = ( F f I ) F g , a 10 = F f ( I F g ) ;

3 Algebraic type models for homotopy theory

3.1 Algebraic 2-types

In this section, we will give information about crossed modules.

Definition 3.1

A crossed module of groups, G 1 = ( E → ∂ G ) , consists of two groups G and E with a left group action of G on E , and group homomorphim ∂ : E → G satisfying the following conditions:

XℳOD -1 ∂ ( e g ) = g ∂ ( e ) g − 1 , ∀ g ∈ G and e ∈ E ,

XℳOD -2 ∂ ( e ) e ′ = e e ′ e − 1 , ∀ e , e ′ ∈ E .

If G 1 = ( E → ∂ G ) only satisfies the XℳOD -1 condition, it is a pre-crossed module.

Example 3.2

  1. Let G be a group and Aut ( G ) be the automorphism group of group G . With the conjugation homomorphism ϕ , the following structure is a crossed module:This structure is called the automorphism crossed module of group G .

  2. Let F → i E → p B be a fibration sequence of pointed spaces. Then, p is a fibration that satisfies F = p − 1 ( b 0 ) , where b 0 is the basepoint of B . Say that the fibre F is pointed at f 0 , and f 0 is also considered to be the basepoint of E . Then, we have an induced map of fundamental groups as follows:Triple ( π 1 ( F ) , π 1 ( E ) , π 1 ( i ) ) is a crossed module. Refer [20] for detailed information.

Definition 3.3

Let G 1 = ( E → ∂ G ) and G 1 ′ = ( E ′ → ∂ ′ G ′ ) be two crossed modules of groups. A f = ( f 1 , f 0 ) morphism of crossed modules from G 1 to G 1 ′ consists of group homomorphisms f 1 : E → E ′ and f 0 : G → G ′ , also the following equations are satisfied:

  1. f 0 ∂ = ∂ ′ f 1 ,

  2. f 1 ( e g ) = f 1 f 0 ( g ) ( e ) ,

for each g ∈ G and e ∈ E .

Thus, we can define the category XℳOD G r p , formed by the crossed modules of groups and their morphisms.

Now let us categorically express the crossed module G 1 = ( E → ∂ G ) . Kamps and Porter [14] barely mentioned that one converts a crossed module to a 2-groupoid, but we will give in detail how to construct a 2-group G ( G 1 ) (strict 2-groupoid with one object) as follows.

{ • } is the 0-cells of the G ( G 1 ) 2-groupoid that we will construct. The bottom group of the crossed module, G , forms the 1-cells (elements of group G ) of this 2-groupoid. The composition “ ♯ 0 ” is defined as follows:

The semi-direct product group E ⋊ G will give the 2-cells of G ( G 1 ) . For the 2-cell ( e , g ) , ( g ∈ G and e ∈ E ), the 1-source and 1-target are s 1 ( ( e , g ) ) = g and t 1 ( ( e , g ) ) = ∂ ( e ) g , respectively. In addition, for each g ∈ G , i 1 ( g ) = ( 1 E , g ) . This is the picture of 2-cell ( e , g ) from the 2-categorical point of view, as follows:

3.1.1 Horizontal composition of 2-cells

Suppose ( e 2 , g 2 ) : g 2 ∂ ( e 2 ) g 2 : and ( e 1 , g 1 ) : g 1 ∂ ( e 1 ) g 1 : ∈ E ⋊ G , then ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) = ( e 1 , g 1 ) ( e 2 , g 2 ) = ( e 1 e 2 , g 1 g 2 ) . Thus, the horizontal composition “ ♯ 0 ” for 2-cells is defined as follows. First, let us define whiskering compositions as follows.

Whiskering as

clearly gives ( e 1 , g 1 g 2 ) and ( e 2 g 1 , g 1 g 2 ) , respectively. The duality of whiskerings is defined similarly. Naturally, the horizontal composition of 2-cells of 2-groupoid G ( G 1 ) is shown from the 2-categorical point of view as follows:

From the definition of the 1-source s 1 and 1-target t 1 , we have

s 1 ( ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) ) = s 1 ( e 1 g 1 e 2 , g 1 g 2 ) = g 1 g 2

and

t 1 ( ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) ) = t 1 ( ( e 1 e 2 g 1 , g 1 g 2 ) ) = ∂ ( e 1 e 2 g 1 ) g 1 g 2 = ∂ ( e 1 ) ∂ ( e 2 g 1 ) g 1 g 2 = ∂ ( e 1 ) g 1 ∂ ( e 2 ) g 1 − 1 g 1 g 2 = ∂ ( e 1 ) g 1 ∂ ( e 2 ) g 2 .

In the last calculations, we used XℳOD -1 in Definition 3.1.

3.1.2 Vertical composition of 2-cells

Suppose ( e ′ , ∂ ( e ) g ) is another element in ( E ⋊ G ) . The vertical composition “ ♯ 1 ” of 2-cells is given by

( e ′ , ∂ ( e ) g ) ♯ 1 ( e , g ) = ( e ′ e , g ) .

The vertical composition of 2-cells is illustrated as follows:

s 1 ( ( e ′ , ∂ ( e ) g ) ♯ 1 ( e , g ) ) = s 1 ( ( e ′ e , g ) ) = g ,

and

t 1 ( ( e ′ , ∂ ( e ) g ) ♯ 1 ( e , g ) ) = t 1 ( ( e ′ e , g ) ) = ∂ ( e ′ e ) g .

3.1.3 Interchange law of 2-cells

The XℳOD -2 condition in the definition of crossed module of groups 3.1 establishes the interchange law between the horizontal and vertical compositions, and as a result, we have a strict 2-groupoid. The interchange law holds. It is explicitly expressed as

( ( e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 1 ( e 1 , g 1 ) ) ♯ 0 ( ( e 2 ′ , ∂ ( e 2 ) g 2 ) ♯ 1 ( e 2 , g 2 ) ) = ( e 1 ′ e 1 , g 1 ) ♯ 0 ( e 2 ′ e 2 , g 2 ) = ( e 1 ′ e 1 g 1 ( e 2 ′ e 2 ) , g 1 g 2 ) = ( e 1 ′ e 1 g 1 e 2 ′ e 1 − 1 e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ′ ∂ ( e 1 ) ( e 2 ′ g 1 ) e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ′ e 2 ′ ( ∂ ( e 1 ) g 1 ) e 2 ′ e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ′ ( ∂ ( e 1 ) g 1 ) e 2 ′ , ∂ ( e 1 ) g 1 ∂ ( e 2 ) g 2 ) ♯ 1 ( e 1 e 2 g 1 , g 1 g 2 ) = ( ( e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 0 ( e 2 ′ , ∂ ( e 2 ) g 2 ) ) ♯ 1 ( ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) ) ,

and it can be drawn as follows:

Consequently, we construct the following strict 2-group structure:

Conversely, any strict 2-group is a crossed module of groups. More specifically, let G be a 2-groupoid with a single object as follows:

To obtain the crossed module of groups from a strict 2-group G , we have { • } = G 0 , G = G 1 , and E = Ker s 1 ⊆ G 2 . Let ∂ : E → G be the restriction of the 1-target map t 1 : G 2 → G 1 , and the left action is as follows:

e g = i 1 ( g ) ♯ 0 e ♯ 0 i 1 ( g ) − 1

for each g ∈ G and e ∈ E .

Naturally, any crossed module is a sesquicategory. If we look at it in more detail as follows, it satisfies the condition of being a sesquicategory becoming a 2-category in [12,21]. For all g ∈ G and e ∈ E , we have:

Therefore, we have

( ( e 1 , g 1 ) ♯ 0 ∂ ( e 2 ) g 2 ) ♯ 1 ( g 1 ♯ 0 ( e 2 , g 2 ) ) = ( e 1 , g 1 ∂ ( e 2 ) g 2 ) ♯ 1 ( e 2 g 1 , g 1 g 2 ) = ( e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 e 2 g 1 e 1 − 1 e 1 , g 1 g 2 ) = ( ∂ ( e 1 ) ( e 2 g 1 ) e 1 , g 1 g 2 ) = ( e 2 ∂ ( e 1 ) g 1 e 1 , g 1 g 2 ) = ( e 2 ∂ ( e 1 ) g 1 , ∂ ( e 1 ) g 1 g 2 ) ♯ 1 ( e 1 , g 1 g 2 ) = ( ∂ ( e 1 ) g 1 ♯ 0 ( e 2 , g 2 ) ) ♯ 1 ( ( e 1 , g 1 ) ♯ 0 g 2 ) .

We can diagrammatically represent this sesquicategory condition as follows:

The 2-group construction that we obtained above can also be constructed from the bicategory perspective that Jean Bénabou gives in [10]. Roughly, in the bicategory B ( G 1 ) , we construct from the crossed module G 1 ,

O b B ( G 1 ) = { • } is the set of objects, and for • ∈ O b B ( G 1 ) , B ( G 1 ) ( • , • ) is a category, as shown below:

  • Objects: Every 1-cell, g : • → • ∈ G , is an object.

  • 1-Morphisms: Every 2-cell, ( e , g ) : g ⇒ ∂ ( e ) g ∈ E ⋊ G , is a 1-morphism.

B ( G 1 ) ( • , • ) is a strict category with the composition ♯ 1 .

3.2 Algebraic 3-types

First of all, we give related knowledge about the 2-crossed modules of groups given in [13,22,23].

Definition 3.4

A 2-crossed module of groups G 2 = ( L → δ E → ∂ G , { − , − } ) is a semi-exact sequence of G-groups, together with left group actions of G on E and L (and on G by conjugation), and a G - equivariant function, called the Peiffer lifting. Here G -equivariance means:

{ e 1 , e 2 } g = { e 1 g , e 2 g } .

For each g ∈ G and e 1 , e 2 ∈ E the following should be satisfied:

  • 2 XℳOD -1    ∂ δ = 1 ,

  • 2 XℳOD -2    δ ( { e 1 , e 2 } ) = ⟨ e 1 , e 2 ⟩ = e 1 e 2 e 1 − 1 e 2 − 1 ∂ ( e 1 ) ,

  • 2 XℳOD -3    { δ ( l 1 ) , δ ( l 2 ) } = [ l 1 , l 2 ] = l 1 l 2 l 1 − 1 l 2 − 1 ,

  • 2 XℳOD -4    { δ ( l ) , e } { e , δ ( l ) } = l l − 1 ∂ ( e ) ,

  • 2 XℳOD -5    { e 1 e 2 , e 3 } = { e 1 , e 2 e 3 e 2 − 1 } { e 2 , e 3 } ∂ ( e 1 ) ,

  • 2 XℳOD -6    { e 1 , e 2 e 3 } = { e 1 , e 2 } { e 1 , e 3 } ( ∂ ( e 1 ) e 2 ) ,

for each g ∈ G , e ∈ E , and l ∈ L .

Remark 3.5

It should be noted that ( L → δ E ) is a crossed module, with

l e = l { δ ( l ) − 1 , e }

for each e ∈ E and l ∈ L . However, ( E → ∂ G ) is just a pre-crossed module.

Lemma 3.6

Let G 2 = ( L → δ E → ∂ G , { − , − } ) be a 2-crossed module of groups. Then, ( l e ) g = e g ( l g ) , for each g ∈ G , e ∈ E , and l ∈ L [22,23].

Example 3.7

  1. From a functorial point of view, any crossed module of groups E → ∂ G is a 2-crossed module 1 → 0 E → ∂ G , with trivial Pfeifer lifting [20].

  2. The quadratic modules that Baues [24] defined are a special version of the 2-crossed modules, satisfying certain additional nilpotency conditions. Refer [25] for its relation to 2-crossed modules.

Definition 3.8

Let G 2 = ( L → δ E → ∂ G , { − , − } ) and G ′ = ( L ′ → δ ′ E ′ → ∂ ′ G ′ , { − , − } ′ ) be two 2-crossed modules of groups. A morphism of 2-crossed modules from G to G ′ is illustrated by the following commutative diagram:

where f 2 : L → L ′ , f 1 : E → E ′ , and f 0 : G → G ′ are group homomorphisms, and also the following equations are satisfied:

f 1 ( e g ) = f 1 f 0 ( g ) ( e ) , f 2 ( l g ) = f 2 f 0 ( g ) ( l ) , f 2 ( { e 1 , e 2 } ) = { f 1 ( e 1 ) , f 1 ( e 2 ) } ,

for each g ∈ G , e ∈ E , and l ∈ L .

We shall let 2 XℳOD G r p , denote the category whose objects are 2-crossed modules of groups and morphisms are morphisms of 2-crossed modules, as defined above.

Now let us construct a 2-crossed module G 2 = ( L → δ E → ∂ G , { − , − } ) as a categorical G ( G 2 ) . We have:

G ( G 2 ) 0 = { • } , G ( G 2 ) 1 = G , G ( G 2 ) 2 = E ⋊ G , G ( G 2 ) 3 = L ⋊ E ⋊ G .

The 1-, 2-, and 3-cells are as follows:

for g ∈ G , e ∈ E , and l ∈ L . For the 3-cell ( l , e , g ) , the 2-source and 2-target are s 2 ( ( l , e , g ) ) = ( e , g ) and t 2 ( ( l , e , g ) ) = ( δ ( l ) e , g ) , respectively. In addition, for each g ∈ G and e ∈ E , i 2 ( ( e , g ) ) = ( 1 L , e , g ) . We can show it 2-categorically diagrammatically as follows:

If we want to picture this structure 3-categorical diagrammatically, we can describe it as follows:

3.2.1 Vertical composition of 3-cells

Suppose ( l ′ , δ ( l ) e , g ) is another element in ( L ⋊ E ⋊ G ) . The vertical composition “ ♯ 2 ” of 3-cells is given by

( l ′ , δ ( l ) e , g ) ♯ 2 ( l , e , g ) = ( l ′ l , e , g ) .

The vertical composition of 3-cells is illustrated as follows:

s 2 ( ( l ′ , δ ( l ) e , g ) ♯ 2 ( l , e , g ) ) = s 2 ( ( l ′ l , e , g ) ) = ( e , g )

and

t 2 ( ( l ′ , δ ( l ) e , g ) ♯ 2 ( l , e , g ) ) = t 2 ( ( l ′ l , e , g ) ) = ( δ ( l ′ l ) e , g ) .

3.2.2 Horizontal composition of 3-cells

Whiskering as

clearly gives ( l ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( e , g ) = ( l ′ , e ′ e , g ) and ( e ′ , ∂ ( e ) g ) ♯ 1 ( l , e , g ) = ( l e ′ , e ′ e , g ) , respectively. Naturally, 2-source and 2-target are defined as follows: s 2 ( ( l ′ , e ′ e , g ) ) = ( e ′ e , g ) = s 2 ( ( l e ′ , e ′ e , g ) ) , t 2 ( ( l ′ , e ′ e , g ) ) = ( δ ( l ′ ) e ′ e , g ) , and t 2 ( ( l e ′ , e ′ e , g ) ) = ( e ′ δ ( l ) e , g ) . The horizontal composition “ ♯ 1 ” of 3-cells is given by

( l ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l , e , g ) = ( l ′ l e ′ , e ′ e , g ) .

s 2 ( ( l ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l , e , g ) ) = s 2 ( ( l ′ e ′ l , e ′ e , g ) ) = ( e ′ e , g )

and

t 2 ( ( l ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l , e , g ) ) = t 2 ( ( l ′ e ′ l , e ′ e , g ) = ( δ ( l ′ e ′ l ) e ′ e , g ) ) = ( δ ( l ′ ) δ ( l l e ′ ) e ′ e , g ) = ( δ ( l ′ ) δ ( l { δ ( l ) − 1 , e ′ } ) e ′ e , g ) = ( δ ( l ′ ) δ ( l ) δ ( { δ ( e ) − 1 , g ′ } ) e ′ e , g ) = ( δ ( l ′ ) δ ( l ) ⟨ δ ( l ) − 1 , e ′ ⟩ e ′ e , g ) = ( δ ( l ′ ) δ ( l ) δ ( l ) − 1 e ′ ( δ ( l ) − 1 ) − 1 ∂ ( δ ( l ) − 1 ) e ′ − 1 e ′ e , g ) = ( δ ( l ′ ) e ′ δ ( l ) e , g ) .

In the last calculations, we used 2 XℳOD -1,-2 in Definition 3.4 and Remark 3.5.

The compositions ♯ 2 and ♯ 1 of 3-cells are compatibles

( ( l 2 ′ , δ ( l 1 ′ ) e ′ , ∂ ( e ) g ) ♯ 1 ( l 2 , δ ( l 1 ) e , g ) ) ♯ 2 ( ( l 1 ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l 1 , e , g ) ) = ( l 2 ′ l 2 ( δ ( l 1 ′ ) e ′ ) , δ ( l 1 ) e ′ δ ( l 1 ) e , g ) ♯ 2 ( l 1 ′ l 1 e ′ , e ′ e , g ) = ( l 2 ′ δ ( l 1 ′ ) ( l 2 e ′ l 2 e ′ ) l 1 ′ l 1 e ′ , e ′ e , g ) = ( l 2 ′ l 1 ′ l 2 e ′ l 1 ′ − 1 l 1 ′ l 1 e ′ , e ′ e , g ) = ( l 2 ′ l 1 ′ l 2 e ′ l 1 e ′ , e ′ e , g ) = ( l 2 ′ l 1 ′ ( l 2 l 1 ) e ′ , e ′ e , g ) = ( l 2 ′ l 1 ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l 2 l 1 , e , g ) = ( ( l 2 ′ , δ ( l 1 ′ ) e ′ , ∂ ( e ) g ) ♯ 2 ( l 1 ′ , e ′ , ∂ ( e ) g ) ) ♯ 1 ( ( l 2 , δ ( l 1 ) e , g ) ♯ 2 ( l 1 , e , g ) ) .

Diagrammatically,

As it can be seen, this equality is satisfied because ( L → δ E ) is a crossed module, see Remark 3.5. Interchange law,

( ( l 2 ′ , δ ( l 1 ′ ) e ′ , ∂ ( e ) g ) ♯ 1 ( l 2 , δ ( l 1 ) e , g ) ) ♯ 2 ( ( l 1 ′ , e ′ , ∂ ( e ) g ) ♯ 1 ( l , e , g ) ) = ( ( l 2 ′ , δ ( l 1 ′ ) e ′ , ∂ ( e ) g ) ♯ 2 ( l 1 ′ , e ′ , ∂ ( e ) g ) ) ♯ 1 ( ( l 2 , δ ( l 1 ) e , g ) ♯ 2 ( l 1 , e , g ) )

is satisfied for vertical ♯ 2 and horizontal ♯ 1 compositions of 3-cells.

3.2.3 0-Horizontal composition of 3-cells

Furthermore, it is necessary to specify the 0-horizontal composition between the H o m -sets of 1-cells whose 0-sources and 0-targets are compatible. The 0-horizontal composition “ ♯ 0 ” of 3-cells is given by

( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 , e 2 , g 2 ) = ( l 1 e 1 ( l 2 g 1 ) , e 1 e 2 g 1 , g 1 g 2 ) .

Whiskering as

gives ( l 1 , e 1 e 2 g 1 , g 1 g 2 ) and ( e 1 ( l 2 g 1 ) , e 1 e 2 g 1 , g 1 g 2 ) , respectively. The composition “ ♯ 0 ” of 3-cells from a 3-categorical perspective is as follows:

s 2 ( ( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 , e 2 , g 2 ) ) = s 2 ( ( l 1 e 1 g 1 l 2 , e 1 e 2 g 1 , g 1 g 2 ) ) = ( e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 )

and

t 2 ( ( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 , e 2 , g 2 ) ) = t 2 ( ( l 1 e 1 ( l 2 g 1 ) , e 1 e 2 g 1 , g 1 g 2 ) ) = ( δ ( l 1 ) e 1 ( l 2 g 1 ) e 1 e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) δ ( ( l 2 g 1 ) e 1 ) e 1 e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) δ ( l 2 g 1 { δ ( l 2 g 1 ) − 1 , e 1 } ) e 1 e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) δ ( l 2 g 1 ) δ ( { δ ( l 2 g 1 ) − 1 , e 1 } ) e 1 e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) δ ( l 2 g 1 ) δ ( l 2 g 1 ) − 1 e 1 ( δ ( l 2 g 1 ) − 1 ) − 1 ∂ ( δ ( l 2 g 1 ) − 1 ) e 1 − 1 e 1 e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) e 1 δ ( l 2 g 1 ) e 2 g 1 , g 1 g 2 ) = ( δ ( l 1 ) e 1 g 1 ( δ ( l 2 ) e 2 ) , g 1 g 2 ) = ( δ ( l 1 ) e 1 , g 1 ) ♯ 0 ( δ ( l 2 ) e 2 , g 2 ) .

In the last calculations, we used 2 XℳOD -1,-2 in Definition 3.4 and Remark 3.5.

Furthermore, for 3-cells in the following diagram:

the composition ♯ 0 is associative as follows:

( l 1 , e 1 , g 1 ) ♯ 0 ( ( l 2 , e 2 , g 2 ) ♯ 0 ( l 3 , e 3 , g 3 ) ) = ( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 e 2 ( l 3 g 2 ) , e 2 e 3 g 2 , g 2 g 3 ) = ( l 1 e 1 ( g 1 ( l 2 e 2 ( l 3 g 2 ) ) , e 1 g 1 ( e 2 e 3 g 2 ) , g 1 g 2 g 3 ) ) = ( l 1 e 1 ( l 2 g 1 ) e 1 ( g 1 e 2 ( l 3 g 2 ) ) , e 1 e 2 g 1 g 1 ( e 3 g 2 ) , g 1 g 2 g 3 ) = ( l 1 e 1 ( l 2 g 1 ) e 1 e 2 g 1 ( l 3 g 1 g 2 ) , e 1 e 2 g 1 e 3 g 1 g 2 , g 1 g 2 g 3 ) = ( l 1 e 1 ( l 2 g 1 ) , e 1 e 2 g 1 , g 1 g 2 ) ♯ 0 ( l 3 , e 3 , g 3 ) = ( ( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 , e 2 , g 2 ) ) ♯ 0 ( l 3 , e 3 , g 3 ) .

We used Lemma 3.6 to show that the composition ♯ 0 is associative.

The interchange law between compositions ♯ 1 and ♯ 0 is not satisfied as follows:

( ( l 1 ′ , e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 0 ( l 2 ′ , e 2 ′ , ∂ ( e 2 ) g 2 ) ) ♯ 1 ( ( l 1 , e 1 , g 1 ) ♯ 0 ( l 2 , e 2 , g 2 ) ) ≠ ( ( l 1 ′ , e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 1 ( l 1 , e 1 , g 1 ) ) ♯ 0 ( ( l 2 ′ , e 2 ′ , ∂ ( e 2 ) g 2 ) ♯ 1 ( l 2 , e 2 , g 2 ) ) .

Because the tail of the 2-crossed module G 2 , ( E → ∂ G ) , is just a pre-crossed module, see Remark 3.5. Therefore, the following structure is not a strict 3-groupoid with a single object. We have sestertius category (sestertius groupoid):

Also, this structure is 3-globular (globularity means s n s n + 1 = s n t n + 1 , t n s n + 1 = t n t n + 1 . )

s 1 s 2 ( ( l , e , g ) ) = s 1 ( ( e , g ) ) = g = s 1 ( ( δ ( l ) e , g ) ) = s 1 t 2 ( ( l , e , g ) )

and

t 1 s 2 ( ( l , e , g ) ) = t 1 ( ( e , g ) ) = ∂ ( e ) g = ∂ δ ( l ) ∂ ( e ) g = ∂ ( δ ( l ) e ) g = t 1 ( ( δ ( l ) e , g ) ) = t 1 t 2 ( ( l , e , g ) ) ,

and reflexive (reflexive means I d = s n i n = t n i n .)

s 2 i 2 ( ( e , g ) ) = s 2 ( ( 1 L , e , g ) ) = ( e , g ) = ( δ ( 1 L ) e , g ) = t 2 ( ( 1 L , e , g ) ) = t 2 i 2 ( ( e , g ) ) = I d ( ( e , g ) )

for each g ∈ G , e ∈ E , and l ∈ L .

It is also possible to proceed the other way for similar construction. Let us have a globular, reflexive, and strict sestertius category with one object as follows:

Then, the following left actions exist for each a ∈ G 1 , b ∈ G 2 , and c ∈ G 3 . With the help of these left actions, the following isomorphisms are defined:

For each b ∈ G 2 ,

b ↦ ( b ♯ 1 i 1 s 1 ( b − 1 ) , s 1 ( b ) ) ↦ b ♯ 1 i 1 s 1 ( b − 1 ) ♯ 1 i 1 s 1 ( b ) = b .

Similarly, for each b ∈ Ker s 1 and a ∈ G 1 ,

( b , a ) ↦ b ♯ 1 i 1 ( a ) ↦ ( b ♯ 1 i 1 ( a ) ) ♯ 1 i 1 s 1 ( b ♯ 1 i 1 ( a ) ) − 1 , s 1 ( b ♯ 1 i 1 ( a ) ) = ( b , a ) .

As a result, G 2 ≅ Ker s 1 ⋊ G 1 . In the same way, for each c ∈ G 3 ,

c ↦ ( c ♯ 2 i 2 s 2 ( c − 1 ) , s 2 ( c ) ♯ 1 i 1 s 1 ( s 2 ( c ) − 1 ) , s 1 s 2 ( c ) ) ↦ c ♯ 2 i 2 s 2 ( c − 1 ) ♯ 2 i 2 ( s 2 ( c ) ♯ 1 i 1 s 1 ( s 2 ( c ) − 1 ) ) ♯ 2 i 2 i 1 ( s 1 s 2 ( c ) ) = c ,

and for c ∈ Ker s 2 , b ∈ Ker s 1 , and a ∈ G 1 ,

( c , b , a ) ↦ c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ↦ ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ♯ 2 i 2 s 2 ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ) ) − 1 , s 2 ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ) ♯ 1 i 1 s 1 ( s 2 ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ) ) − 1 , s 1 s 2 ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ) = ( c ♯ 2 i 2 ( b ) ♯ 2 i 2 i 1 ( a ) ♯ 2 i 2 i 1 ( a ) − 1 ♯ 2 i 2 ( b ) − 1 ♯ 2 i 2 s 2 ( c ) − 1 , s 2 ( c ) ♯ 1 s 2 i 2 ( b ) ♯ 1 i 1 ( a ) ♯ 1 i 1 s 1 s 2 i 2 i 1 ( a ) − 1 ♯ 1 i 1 s 1 s 2 i 2 ( b ) − 1 ♯ 1 i 1 s 1 s 2 ( c ) − 1 , s 1 s 2 ( c ) ♯ 0 s 1 s 2 i 2 ( b ) ♯ 0 s 1 s 2 i 2 i 1 ( a ) ) = ( c , b , a ) .

Consequently, G 3 ≅ Ker s 2 ⋊ Ker s 1 ⋊ G 1 . Therefore, L = Ker s 2 ⊆ G 3 , E = Ker s 1 ⊆ G 2 , G = G 1 , and target transformations have the following restrictions: G 2 ( G ) is a semi-exact sequence of G -groups. ( Ker s 1 → t ¯ 1 G 1 ) is a pre-crossed module of groups with left group action. In more detail, the axiom XℳOD -1 in Definition 3.1 is satisfied:

t ¯ 1 ( b a ) = t ¯ 1 ( i 1 ( a ) ♯ 0 b ♯ 0 i 1 ( a ) − 1 ) = t ¯ 1 i 1 ( a ) ♯ 0 t ¯ 1 ( b ) ♯ 0 t ¯ 1 i 1 ( a ) − 1 = a ♯ 0 t ¯ 1 ( b ) ♯ 0 a − 1 ,

for each a ∈ G 1 and b ∈ Ker s 1 . Note that the reflexivity feature ( t ¯ 1 i 1 = I d ) is used in the above steps. Similarly, ( Ker s 2 → t ¯ 2 Ker s 1 ) is a crossed module of groups with left group action. Because the interchange law is satisfied between the ♯ 2 and ♯ 1 compositions of the sestertius category G , the XℳOD -2 axiom in Definition 3.1 is satisfied. Refer also [6].

There is also the Pfeifer lifting map { − , − } * , defined as follows:

Additionally, for each a ∈ G 1 , b 1 , b 2 ∈ Ker s 1 ;

{ b 1 , b 2 } * a = i 2 a ( b 1 ♯ 0 b 2 ♯ 0 b 1 − 1 ♯ 1 b 2 − 1 t ¯ 1 ( b 1 ) ) = i 2 a ( b 1 ) ♯ 0 a i 2 ( b 2 ) ♯ 0 a i 2 ( b 1 ) − 1 ♯ 1 i 2 a ( b 2 − 1 t ¯ 1 ( b 1 ) ) = i 2 i 1 ( a ) ♯ 0 i 2 ( b 1 ) ♯ 0 i 2 i 1 ( a ) − 1 ♯ 0 i 2 i 1 ( a ) ♯ 0 i 2 ( b 2 ) ♯ 0 i 2 i 1 ( a ) − 1 ♯ 0 i 2 i 1 ( a ) ♯ 0 i 2 ( b 1 − 1 ) ♯ 0 i 2 i 1 ( a ) − 1 ♯ 1 i 2 i 1 ( a ) ♯ 0 i 2 i 1 t ¯ 1 ( b 1 ) ♯ 0 i 2 ( b 2 − 1 ) ♯ 0 i 2 i 1 t ¯ 1 ( b 1 ) − 1 ♯ 0 i 2 i 1 ( a ) − 1 = i 2 ( i 1 ( a ) ♯ 0 b 1 ♯ 0 i 1 ( a ) − 1 ♯ 0 i 1 ( a ) ♯ 0 b 2 ♯ 0 i 1 ( a ) − 1 ♯ 0 i 1 ( a ) ♯ 0 b 1 − 1 ♯ 0 i 1 ( a ) − 1 ♯ 1 i 1 ( a ) ♯ 0 i 1 t ¯ 1 ( b 1 ) ♯ 0 b 2 − 1 ♯ 0 i 1 t ¯ 1 ( b 1 ) − 1 ♯ 0 i 1 ( a ) − 1 ) = i 2 ( b 1 a ♯ 0 b 2 a ♯ 0 b 1 − 1 a ♯ 1 b 2 − 1 a ♯ 0 t ¯ 1 ( b 1 ) ) = i 2 ( b 1 a ♯ 0 b 2 a ♯ 0 b 1 − 1 a ♯ 1 a ♯ 0 t ¯ 1 ( b 1 ) ♯ 0 a − 1 ( b 2 − 1 a ) ) = i 2 ( b 1 a ♯ 0 b 2 a ♯ 0 b 1 − 1 a ♯ 1 t ¯ 1 ( b 1 a ) ( b 2 − 1 a ) ) = { b 1 a , b 2 a } * .

Moreover, there exist the left action equality c ( b , a ) = ( c i 1 ( a ) ) b and

( c b ) a = ( i 2 ( b ) ♯ 0 c ♯ 0 i 2 ( b ) − 1 ) a = i 2 a ( b ) ♯ 0 c a ♯ 0 i 2 a ( b ) − 1 = i 2 i 1 ( a ) ♯ 0 i 2 ( b ) ♯ 0 i 2 i 1 ( a ) − 1 ♯ 0 c a ♯ 0 i 2 i 1 ( a ) ♯ 0 i 2 ( b ) − 1 ♯ 0 i 2 i 1 ( a ) − 1 = i 2 ( i 1 ( a ) ♯ 0 b ♯ 0 i 1 ( a ) − 1 ) ♯ 0 c a ♯ 0 i 2 ( i 1 ( a ) ♯ 0 b − 1 ♯ 0 i 1 ( a ) − 1 ) = i 2 ( b a ) ♯ 0 c a ♯ 0 i 2 ( b a ) − 1 = b a ( c a ) ,

for each a ∈ G 1 , b ∈ Ker s 1 , and c ∈ Ker s 2 . These definitions satisfy the axioms given in Definition 3.4. Now we will only satisfy axioms 2 XℳOD -1 and 2 XℳOD -2. We leave the remaining axioms to the readers. It is pretty fun to prove.

  • 2 XℳOD -1   for each c ∈ Ker s 2 , t ¯ 1 t ¯ 2 ( c ) = t ¯ 1 s 2 ( c ) = 1 , ( ∵ the globularity),

  • 2 XℳOD -2   we have

    t ¯ 2 ( { b 1 , b 2 } * ) = t ¯ 2 i 2 ( b 1 ♯ 0 b 2 ♯ 0 b 1 − 1 ♯ 1 b 2 − 1 t ¯ 1 ( b 1 ) ) = t ¯ 2 ( i 2 ( b 1 ) ♯ 0 i 2 ( b 2 ) ♯ 0 i 2 ( b 1 − 1 ) ♯ 1 i 2 ( b 2 − 1 t ¯ 1 ( b 1 ) ) ) = ( t ¯ 2 i 2 ) ( b 1 ) ♯ 0 ( t ¯ 2 i 2 ) ( b 2 ) ♯ 0 ( t ¯ 2 i 2 ) ( b 1 − 1 ) ♯ 1 ( t ¯ 2 i 2 ) ( b 2 − 1 t ¯ 1 ( b 1 ) ) = b 1 ♯ 0 b 2 ♯ 0 b 1 − 1 ♯ 1 b 2 − 1 t ¯ 1 ( b 1 )

for each b 1 , b 2 ∈ Ker s 1 , ( ∵ the reflexivity t n i n = I d n ).

Along with these definitions and constructions, the following theorem is given.

Theorem 3.9

A 2-crossed module of groups is a globular, reflexive, and strict sestertius groupoid with one object (sestertius group).

Now, we will obtain a tricategory in light of these knowledges.

3.2.4 From 2-crossed modules of groups to tricategories

In this section, we will construct tricategory from a 2-crossed module. Let G 2 = ( L → δ E → ∂ G , { − − } ) be a 2-crossed module of groups. We construct the T ( G 2 ) tricategory as follows:

DATA:

  1. O b T ( G 2 ) = { • }

  2. For • ∈ O b T ( G 2 ) . T ( G 2 ) ( • , • ) is a bicategory as follows:

  • Objects: Every 1-cell, g : • → • ∈ G , is an object.

  • 1-Morphisms: Every 2-cell, ( e , g ) : g ⇒ ∂ ( e ) g ∈ E ⋊ G , is a 1-morphism.

  • 2-Morphisms: Every 3-cell, ( l , e , g ) : ( e , g ) ⇛ ( δ ( l ) e , g ) ∈ L ⋊ E ⋊ G , is a 2-morphism.

The vertical composition of T ( G 2 ) category is ♯ 2 , and the horizontal composition is ♯ 1 . Note that T ( G 2 ) is a strict 2-category (see in Remark 2.3).

  1. The functor

is called composition:

For composable 2-morphisms of T ( G 2 ) , the functor ⊗ is cubical [17, page 38, Definition 3.1]. Therefore, we have either

( ( e 1 , g 1 ) ♯ 0 ( e 2 ′ , ∂ ( e 2 ) g 2 ) ) ♯ 1 ( ( 1 E , g 1 ) ♯ 0 ( e 2 , g 2 ) ) = ( e 1 e 2 ′ g 1 , g 1 ∂ ( e 2 ) g 2 ) ♯ 1 ( e 2 g 1 , g 1 g 2 ) = ( e 1 e 2 ′ g 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ( e 2 ′ e 2 ) g 1 , g 1 g 2 ) = ( e 1 , g 1 ) ♯ 0 ( e 2 ′ e 2 , g 2 ) = ( ( e 1 , g 1 ) ♯ 1 ( 1 E , g 1 ) ) ♯ 0 ( ( e 2 ′ , ∂ ( e 2 ) g 2 ) ♯ 1 ( e 2 , g 2 ) ) ,

diagrammatically,

or

( ( e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 0 ( 1 E , ∂ ( e 2 ) g 2 ) ) ♯ 1 ( ( e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) ) = ( e 1 ′ , ∂ ( e 1 ) g 1 ∂ ( e 2 ) g 2 ) ♯ 1 ( e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ′ e 1 e 2 g 1 , g 1 g 2 ) = ( e 1 ′ e 1 , g 1 ) ♯ 0 ( e 2 , g 2 ) = ( ( e 1 ′ , ∂ ( e 1 ) g 1 ) ♯ 1 ( e 1 , g 1 ) ) ♯ 0 ( ( 1 E , ∂ ( e 2 ) g 2 ) ♯ 1 ( e 2 , g 2 ) ) ,

diagrammatically,

  1. The functor I • : 1 T ( G 2 ) ( • , • ) , where 1 denotes the unit bicategory:

I • is also cubical.

  1. For object • of O b T ( G 2 ) as follows:

an adjoint equivalence α ;

α : ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) ⟶ ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) .

Since every T ( G 2 ) ( • , • ) is a strict 2-category, we have

( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) .

Note that ⊗ is associative

in Bicat ( T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) , T ( G 2 ) ( • , • ) ) .

  1. For each g ∈ G , e ∈ E , and l ∈ L , adjoint equivalences l and r

and

in BiCat ( T ( G ) ( • , • ) , T ( G ) ( • , • ) ) .

  1. For • ∈ O b T ( G ) as follows:

and a 2-cell isomorphism π (Instead of T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) , we use T 4 ( G 2 ) as its abbreviation):

a 1 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ⊗ ( l 4 , e 4 , g 4 ) a 2 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ⊗ ( l 4 , e 4 , g 4 ) a 3 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ⊗ ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) a 4 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ⊗ ( l 4 , e 4 , g 4 ) a 5 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) a 6 = ( l 1 ( l 2 g 1 ) e 1 e 1 e 2 g 1 ( l 3 g 1 g 2 ) , e 1 e 2 g 1 e 3 ( g 1 g 2 ) , g 1 g 2 g 3 ) ⊗ ( l 4 , e 4 , g 4 ) a 7 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 e 2 e 3 g 2 ( l 4 g 3 ) , e 2 e 3 g 2 e 4 ( g 2 g 3 ) , g 2 g 3 g 4 ) a 8 = ( l 1 ( l 2 g 1 ) e 1 e 1 e 2 g 1 ( l 3 g 1 g 2 ) e 1 e 2 g 1 g 3 g 1 g 2 ( l 4 g 1 g 2 g 3 ) , e 1 e 2 g 1 g 3 ( g 1 g 2 ) e 4 ( g 1 g 2 g 3 ) , g 1 g 2 g 3 g 4 ) = ( l 1 e 1 ( g 1 ( l 2 ( l 3 g 2 ) e 2 e 2 e 3 g 2 ( l 4 g 3 ) ) ) , e 1 g 1 ( e 2 e 3 g 2 e 4 ( g 2 g 3 ) ) , g 1 g 2 g 3 g 4 ) .

in the bicategory Bicat ( T 4 ( G 2 ) , T ( G 2 ) ( • , • ) )

  1. For invertible modifications,

a 1 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) , a 2 = ( l 1 , e 1 , g 1 ) ⊗ ( 1 L , 1 E , 1 G ) ⊗ ( l 2 , e 2 , g 2 ) , a 3 = ( l 1 e 1 ( l 2 g 1 ) , e 1 e 2 g 1 , g 1 g 2 ) ;

Note that the tricategory T ( G 2 ) we have defined provides Remark 2.3; that is why it is strict.

AXIOMS:

(1) [Non-abelian 4-cocycle condition] For any strict 2-category T ( G 2 ) ( − , − ) , the following 2-cells equation diagram preserves the natural isomorphism of α . We have 2-cells elements as follows:

We will use the following elements as abbreviations for the objects of T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) ;

a 1 = ( ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) ) ) ⊗ ( l 5 , e 5 , g 5 ) a 2 = ( l 1 , e 1 , g 1 ) ⊗ ( ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) ) ⊗ ( l 5 , e 5 , g 5 ) ) a 3 = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) ⊗ ( l 5 , e 5 , g 5 ) ) ) a 4 = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( ( l 3 , e 3 , g 3 ) ⊗ ( l 4 ( l 5 g 4 ) e 4 , e 4 e 5 g 4 , g 4 g 5 ) ) ) a 5 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ⊗ ( l 4 ( l 5 g 4 ) e 4 , e 4 e 5 g 4 , g 4 g 5 ) ) a 6 = ( ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) ⊗ ( l 4 ( l 5 g 4 ) e 4 , e 4 e 5 g 4 , g 4 g 5 ) a 7 = ( ( ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) ⊗ ( l 4 , e 4 , g 4 ) ) ⊗ ( l 5 , e 5 , g 5 ) a 8 = ( ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ) ⊗ ( l 4 , e 4 , g 4 ) ) ⊗ ( l 5 , e 5 , g 5 ) a 9 = ( ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ⊗ ( l 4 , e 4 , g 4 ) ) ) ⊗ ( l 5 , e 5 , g 5 ) a 10 = ( l 1 , e 1 , g 1 ) ⊗ ( ( ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ⊗ ( l 4 , e 4 , g 4 ) ) ( l 5 , e 5 , g 5 ) ) a 11 = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ) ⊗ ( l 4 ( l 5 g 4 ) e 4 , e 4 e 5 g 4 , g 4 g 5 ) a 12 = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ⊗ ( l 4 ( l 5 g 4 ) e 4 , e 4 e 5 g 4 , g 4 g 5 ) ) a 13 = ( ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) ) ⊗ ( l 5 , e 5 , g 5 ) a 14 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( ( l 3 ( l 4 g 3 ) e 3 , e 3 e 4 g 3 , g 3 g 4 ) ⊗ ( l 5 , e 5 , g 5 ) ) .

Thus, we have the following commutative diagrams:

(2) [Left normalization] The following equation of 2-cells in any strict 2-category T ( G 2 ) ( − , − ) , where the unmarked isomorphisms are either naturality isomorphisms for α or unique coherence isomorphisms

from the hom-strict 2-category. The 2-cell elements are as follows:

We will use the following elements as abbreviations for the objects of T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) ;

a 1 = ( ( l 1 , e 1 , g 1 ) ⊗ ( ( 1 L , 1 E , 1 G ) ⊗ ( l 2 , e 2 , g 2 ) ) ) ⊗ ( l 3 , e 3 , g 3 ) = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) a 2 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) a 3 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) a 4 = ( ( l 1 , e 1 , g 1 ) ⊗ ( ( 1 L , 1 E , 1 G ) ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) a 5 = ( ( l 1 , e 1 , g 1 ) ⊗ ( 1 L , 1 E , 1 G ) ) ⊗ ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) a 6 = ( l 1 , e 1 , g 1 ) ⊗ ( ( ( 1 L , 1 E , 1 G ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) ) = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) a 7 = ( l 1 , e 1 , g 1 ) ⊗ ( ( 1 L , 1 E , 1 G ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) ) .

Thus, we have the following commutative diagrams:

(3) [Right normalization] Similarly, the following equation of 2-cells holds in any strict 2-category T ( G 2 ) ( − , − ) . The 2-cell elements are as follows:

We will use the following elements as abbreviations for the objects of T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) × T ( G 2 ) ( • , • ) ;

a 1 = ( l 1 , e 1 , g 1 ) ⊗ ( ( ( l 2 , e 2 , g 2 ) ⊗ ( 1 L , 1 E , 1 G ) ) ⊗ ( l 3 , e 3 , g 3 ) ) = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) a 2 = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( ( 1 L , 1 E , 1 G ) ⊗ ( l 3 , e 3 , g 3 ) ) ) = ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( l 3 , e 3 , g 3 ) ) a 3 = ( l 1 , e 1 , g 1 ) ⊗ ( l 2 ( l 3 g 2 ) e 2 , e 2 e 3 g 2 , g 2 g 3 ) a 4 = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) a 5 = ( ( l 1 , e 1 , g 1 ) ⊗ ( ( l 2 , e 2 , g 2 ) ⊗ ( 1 L , 1 E , 1 G ) ) ) ⊗ ( l 3 , e 3 , g 3 ) = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( l 3 , e 3 , g 3 ) a 6 = ( ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( 1 L , 1 E , 1 G ) ) ⊗ ( e 3 , g 3 , h 3 ) a 7 = ( ( l 1 , e 1 , g 1 ) ⊗ ( l 2 , e 2 , g 2 ) ) ⊗ ( ( 1 L , 1 E , 1 G ) ⊗ ( l 3 , e 3 , g 3 ) ) = ( l 1 ( l 2 g 1 ) e 1 , e 1 e 2 g 1 , g 1 g 2 ) ⊗ ( l 3 , e 3 , g 3 ) .

Thus, we have the following commutative diagrams:

Note that this structure is a strict cubical tricategory with a single object, as it satisfies Remark 2.3. and Lemma 2.4. We can also obtain reverse construction; we can derive a 2-crossed module from any strict cubical tricategory with a single object (sestertius groupoid with one object). See how to obtain a 2-crossed module of groups from a sestertius category with one object in Section 3.2.

In this way, we can construct the morphism of 2-crossed modules in the form of trihomomorphism. Let f be a 2-crossed module morphism between G 2 and G 2 ′ as shown below:

Roughly speaking, trihomomorphism consists of the following data and provides the axioms in Definition 2.5:

  2. For • ∈ O b T ( G 2 ) , the following diagram represents

For each g ∈ G , e ∈ E , and l ∈ L ;

F 0 ( g ) = f 0 ( g ) , F 0 ( ∂ ( e ) g ) = f 0 ( ∂ ( e ) g ) , F 1 ( ( e , g ) ) = ( f 1 ( e ) , f 0 ( g ) ) , F 1 ( ( δ ( l ) e , g ) ) = ( f 1 ( δ ( l ) e ) , f 0 ( g ) ) , F 2 ( ( l , e , g ) ) = ( f 2 ( l ) , f 1 ( e ) , f 0 ( g ) ) .

The morphisms F n are compatible with n -source and n -target morphisms for 0 ≤ n ≤ 2 .

s 0 ′ ( F 0 ( g ) ) = t 0 ′ ( F 0 ( g ) ) = s 0 ′ ( F 0 ( ∂ ( e ) g ) ) = t 0 ′ ( F 0 ( ∂ ( e ) g ) ) = F O b ( • ) = • ′ ,

s 1 ′ ( F 1 ( ( e , g ) ) ) = s 1 ′ ( ( f 1 ( e ) , f 0 ( g ) ) ) = f 0 ( g ) ,

t 1 ′ ( F 1 ( ( e , g ) ) ) = t 1 ′ ( ( f 1 ( e ) , f 0 ( g ) ) ) = ∂ ′ ( f 1 ( e ) ) f 0 ( g ) = ∂ ′ f 1 ( e ) f 0 ( g ) = f 0 ∂ ( e ) f 0 ( g ) ,

s 1 ′ ( F 1 ( ( δ ( l ) e , g ) ) ) = s 1 ′ ( ( f 1 ( δ ( l ) e ) , f 0 ( g ) ) ) = f 0 ( g ) ,

t 1 ′ ( F 1 ( ( δ ( l ) e , g ) ) ) = t 1 ′ ( ( f 1 ( δ ( l ) e ) , f 0 ( g ) ) ) = ∂ ′ ( f 1 ( δ ( l ) e ) ) f 0 ( g ) = ∂ ′ ( f 1 ( δ ( l ) ) ) ∂ ′ ( f 1 ( e ) ) f 0 ( g ) = ∂ ′ δ ′ ( f 2 ( l ) ) ∂ ′ f 1 ( e ) f 0 ( g ) = ∂ ′ f 1 ( e ) f 0 ( g ) = f 0 ∂ ( e ) f 0 ( g ) ,

and

s 2 ′ ( F 2 ( ( l , e , g ) ) ) = s 2 ′ ( ( f 2 ( l ) , f 1 ( e ) , f 0 ( g ) ) ) = ( f 1 ( e ) , f 0 ( g ) ) ,

t 2 ′ ( F 2 ( ( l , e , g ) ) ) = t 2 ′ ( ( f 2 ( l ) , f 1 ( e ) , f 0 ( g ) ) ) = ( δ ′ ( f 2 ( l ) ) f 1 ( e ) , f 0 ( g ) ) = ( f 1 δ ( l ) f 1 ( e ) , f 0 ( g ) ) = ( f 1 ( δ ( l ) e ) , f 0 ( g ) ) .

  1. For • ∈ O b T ( G 2 ) , an adjoint equivalence ξ : ⊗ ′ ∘ ( F × F ) ⇒ F ∘ ⊗ with left adjoint is illustrated in the following diagram:

  1. For • ∈ O b T ( G 2 ) , an adjoint equivalence ι : I F O b ( • ) ′ ⇒ F ∘ I • with left adjoint illustrated in the following diagram:

Since f n s are group homomorphisms, unit elements are preserved.

Other data (invertible modifications) and axioms in Definition 2.5 are easily provided, because ⊗ is associative and I • is preserved under F .

4 Conclusion

In this study, we obtained a strict cubical tricategory with a single object by looking at the 2-crossed module from a higher-dimensional perspective, and we also defined the reverse construction. With this type of definition, non-abelian objects and other algebraic 3-type models can be defined. The important point here is the difference in understanding between the crossed module and the 2-crossed module.

Acknowledgement

The authors would like to thank the handling editor and the referees for their helpful and useful comments and suggestions.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-29
Revised: 2024-04-29
Accepted: 2024-07-30
Published Online: 2024-12-21

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0061
Keywords for this article
crossed module; 2-crossed module; categorification; sestertius category; tricategory
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