Category defintions & Examples

Dec 15, 2015

Extract from : CRing Project, Chapter 0 source : https://math.berkeley.edu/~amathew/chcategories.pdf

1. Category definition

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

2. Examples and notations

$1.6 Preorder 2 Examples of categories 2.1 Baby examples 2.2 The category of sets 2.3 Mathematical structures as categories 2.3.1 Preorders 2.3.2 Groups 2.3.3 Matrices 2.4 Categories of sets with structure 3 Properties of objects and morphisms erminology and fine points If f: A to B in a category, A is called the domain or source of f, and B is called the codomain or target of f. text{Hom}(A,B) is called a hom class or a hom set if it is indeed a set. In general a hom set may be empty, but for any object A, text{Hom}(A,A) is not empty because it contains the identity morphism. The hom class text{Hom}(A,B) may be denoted by text{Hom}_mathcal{C}(A,B) or mathcal{C}(A,B) if it is necessary to specify which category is referred to. Morphisms may also be called maps. This does not mean that every morphism in any category is a set function (see #Baby examples and #Preorders). Arrow is a less misleading name. The composite gf may be written gcirc f. It might be more natural to write the composite of f:Ato B and g:Bto C as f g instead of g f but the usage given here is by far the most common. This stems from the fact that if the arrows are set functions and xin A, then (gf)(x)=g(f(x)). Thus g f is best read as "do g after f".$