ABSTRACT
A category C is given by a collection of objects (denoted by upper case letters, e.g. X) and arrows (also called morphisms or maps) between objects (denoted by lower case letters f : X Ñ Y ) satisfying certain conditions. We will assume that the collection of objects of C, denoted ObpCq, forms a class, and that for every pair of objects X and Y of C, the collection of arrows from X to Y , denoted HomCpX,Y q, forms a set, called a homset of C. Moreover, for every triple of objects X, Y , and Z of C, there is a binary operation
˝ X,Y,Z
: HomCpX,Y q ˆ HomCpY, Zq Ñ HomCpZ,Zq pf, gq ÞÑ g ˝
X,Y,Z f
To not overload the notation, we simply write ˝X,Y,Z as ˝. The operation ˝ is associative on every triple of composable arrows f , g, h, i.e., ph ˝ gq ˝ f “
h ˝ pg ˝ fq. The last condition in the definition of a category is that for every object X there exists a unique morphism idX , called the identity on X, such that f ˝ idX “ f and idX ˝ g “ g, for every arrow f : X Ñ Y and g : Z Ñ X.
