1.2. Categories 5 Exercise 1.2. (a) Take some time to convince yourself that the given definition of algebraic closure actually does define what you think of as algebraic closure. (b) The isomorphism theorems of elementary group theory can be written down in terms of diagrams. Do it! (c) Rewrite the statement that the cube-shaped diagram above is commu- tative without using any diagrams at all.1 1.2. Categories Informally, a category is simply a ‘complete’ list of all the things you plan to study together with a complete list of all the allowable maps between those things. So an algebraist might work in the category of groups and homomorphisms, while a topologist might work in the category of topological spaces and continuous functions, and a geometer could work in the category of subsets of the plane and rigid motions. Formally, a category C consists of two things: a collection2 ob(C), called the objects of C, and, for each X, Y ob(C), a set morC(X, Y ), called the set of morphisms from X to Y . These are subject to the following conditions: (1) If X, Y, Z ob(C), f morC(X, Y ), g morC(Y, Z), then there is another morphism g f morC(X, Z) (which you should think of as the composite of f and g).3 (2) The composition operation is associative: f (g h) = (f g) h. Diagrammatically, this says that the diagram W h g◦h X g f◦g Y f Z is commutative. (3) For each X ob(C), there is a special morphism idX morC(X, X) which satisfies idX f = f for any f morC(W, X) and g idX = g 1 Thanks to Jason Trowbridge for this idea. 2 The vague word ‘collection’ is intended to gloss over some technical set-theoretical issues. The idea is that the collection of objects should be allowed to be larger than any set, so we can’t call it a set of objects. Many authors use a class of objects, which is a well-defined concept in set theory (or logic) but one of the go-to books on category theory (Mac Lane [110]) uses a set-theoretical trick to get around classes. 3 This rule makes it meaningful to ask whether a given diagram of objects and arrows in C is commutative or not.
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