Representations of Crossed Squares and Cat2-Groups

The concept of crossed modules was introduced by J.H.C. Whitehead in the late 1940s and then Loday [27] reformulated it as cat1-groups. Crossed modules and cat1-groups are two-dimensional generalisations of a group. Loday showed in [9] that crossed modules can be understood also as 2-groups. In much the same way, a higher dimensional analogue of crossed modules, the concept of crossed squares was introduced by Loday and Guin-Valery [27] and then Arvasi [2] linked it to the concept of higher categorical groups, namely cat2-groups. From the same point of view, crossed squares and cat2-groups are analogues of a threedimensional generalisation of a group namely 3-groups. A group can be seen as a category with one object and morphisms given by the elements and with composition being the group multiplications. In classical representation theory the elements of a group can be realised as automorphisms of some object in some category, particularly in the category of vector spaces over a _eld K (see [13]). A 2-categorical analogue of the category of vector spaces over a _eld K has been described by Forrester-Barker [17] as the concept of a 2-category of length 1 chain complexes. Here, we describe a 3-groupoid of length 2 chain complexes as a 3-categorical analogue of the category of vector spaces over a _eld K. In this thesis, we _rst construct a 3-groupoid of length 2 chain complexes and describe it in a matrix language respecting the chain complex conditions. Also, imitating representations of a group G and homomorphisms of the group G into the general linear group of a vector space, we discuss representations of a category, which is a functor into a category of vector spaces over a _eld K. Here we develop a notion of representation of cat2-groups and crossed squares, which will be de_ned as 3-functors. This extends the previous work by Forrester-Barker [17] where he de_ned the representation theory of cat1-groups and crossed modules, which are given by 2-functors from the categorical dimension two to the categorical dimension three. The main objective in this thesis is to construct the general form of the automorphism Aut() after we introduce the path between matrices, which represents length 2 chain complexes and automorphisms of them.