Title:

Representations of crossed squares and cat2groups

The concept of crossed modules was introduced by J.H.C. Whitehead in the late 1940s and then Loday [27] reformulated it as cat1groups. Crossed modules and cat1groups are twodimensional generalisations of a group. Loday showed in [9] that crossed modules can be understood also as 2groups. In much the same way, a higher dimensional analogue of crossed modules, the concept of crossed squares was introduced by Loday and GuinValery [27] and then Arvasi [2] linked it to the concept of higher categorical groups, namely cat2groups. From the same point of view, crossed squares and cat2groups are analogues of a threedimensional generalisation of a group namely 3groups. 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 2categorical analogue of the category of vector spaces over a _eld K has been described by ForresterBarker [17] as the concept of a 2category of length 1 chain complexes. Here, we describe a 3groupoid of length 2 chain complexes as a 3categorical analogue of the category of vector spaces over a _eld K. In this thesis, we _rst construct a 3groupoid 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 cat2groups and crossed squares, which will be de_ned as 3functors. This extends the previous work by ForresterBarker [17] where he de_ned the representation theory of cat1groups and crossed modules, which are given by 2functors 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.
