4 1. Categories and Functors Each arrow has a domain and a target—thus X f Y is a simple diagram with a single arrow f whose domain is X and whose target is Y . If g is another arrow with domain Y , then we can form the ‘composite arrow’ g f with domain X and target Y . The triangle X f h ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ Y g Z is commutative if h = g f. If the diagram above is commutative, then we say that h factors through f and through g we also say that h factors through Y . If X and Y are objects in a diagram, we may be able to use the various arrows and their composites to obtain many potentially distinct arrows from X to Y for example, in the cube diagram, there are precisely 6 different composites from X1 to Y4. Each of these paths represents an arrow X1 Y4, but it can happen that different paths become, on composition, the same arrow. If it turns out that, for each pair X, Y of objects in the diagram, all of the possible composite paths from X to Y are ultimately the same arrow, then we say that the diagram is commutative. We can expand a given (not necessarily commutative) diagram D by drawing as arrows all of the composites of the given arrows, as well as ‘iden- tity arrows’ from each ‘vertex object’ to itself, which compose like identity maps. We’ll refer to the expanded diagram as D. Exercise 1.1. Show that D is commutative if and only if D is commutative. It is frequently helpful to express complicated definitions and properties in terms of diagrams. Here’s an example. Let F be a field, and let F E be a field extension. Then there is an inclusion map f : F E, which just carries an element α F to the same element, but thought of as being an element of E. Now an algebraic closure for F is an algebraic field extension a : F A such that for any other algebraic extension f, there is a unique map ¯ making the diagram F a f A E ∃! ¯ commutative. Here you should observe that we use dotted arrows to denote arrows that we do not know exist. Also, this definition gives the algebraic closure as a solution to a ‘universal problem’.
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