Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Carles Casacuberta[en] After looking over the difficulties to give an adequate ground to the Theory of Categories, we give a presentation of the basic concepts of this theory from a proposal of definitions that will be useful for dealing with the changes of universe rigorously. In this way, once fixed the set-theoretic framework where we will be moving, we will define a κ-category, a bundle, a definable bundle of categories and a change of universe or enlargement. We will reformulate all the concepts that will be necessary for us to finally speak about the locally presentable change of universe and the logical enlargement of the k-locally presentable k-categories and we will see how the locally presentable enlargements, which in general they do not have to coincide
with the logical enlargement, preserve k-colimits
