Basic Category Theory

Basic Category TheoryJaap van OostenJaap van OostenDepartment of MathematicsUtrecht UniversityThe NetherlandsRevised, July 20021 Categories and Functors1.1 De nitions and examplesA categoryC is given by a collectionC of objects and a collectionC of arrows0 1which have the following structure. Each arrow has a domain and a codomain which are objects; one writesff : X ! Y or X ! Y if X is the domain of the arrow f, and Y itscodomain. One also writes X = dom(f) and Y = cod(f); Given two arrows f and g such that cod(f) = dom(g), the compositionof f and g, written gf, is de ned and has domain dom(f) and codomaincod(g):f g gf(X!Y !Z) 7! (X!Z) Composition is associative, that is: given f : X ! Y , g : Y ! Z andh :Z!W, h(gf) = (hg)f; For every object X there is an identity arrow id : X! X, satisfyingXid g =g for every g :Y !X and fid =f for every f :X!Y .X XExercise 1 Show that id is the unique arrow with domain X and codomainXX with this property.Instead of \arrow" we also use the terms \morphism" or \map".Examplesa) 1 is the category with one object and one arrow, id ;b) 0 is the empty category. It has no objects and no arrows.c) A preorder is a setX together with a binary relation which is re exiv e(i.e. xx for allx2X) and transitive (i.e.xy andyz imply xzfor all x;y;z2X). This can be viewed as a category, with set of objectsX and for every pair of objects (x;y) such that xy, exactly one arrow:x!y.Exercise 2 Prove this. Prove ...