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Find a way to view a product as the limit of a diagram. 9. (a) Construct an example of a category and a diagram which has no limit. (b) A discrete category is one in which the only morphisms are identities. What is the limit of a diagram whose shape is discrete? 2. Limits and Colimits 34 Colimits. The discussion of limits, of course, dualizes. A colimit of a diagram F : I -+ C is an object P E C and a morphism of diagrams F -+ Ap that is initial among all morphisms F -+ AX for X E C. That is, given solid arrow diagram morphisms AP'**""** ..................

19. Let L : C -4 D and R : D -+ C be an adjoint pair of functors. (a) Suppose that every diagram in C has a colimit. Then for every F E C', L(colim F) is a colimit for L o F E Dz. (b) Suppose that every diagram in DT has a limit. Then for every F E Dz, R(lim F) is a limit for R o F E C'. 20. 19. 4. Special Kinds of Limits and Colimits Some diagram shapes are particularly useful, and their limits and colimits have special names. 2We consider the object colim F as coming with structure maps to the diagram F.

Show that there are natural maps X -+ RLX and LRX -+ X and that RX is naturally a retract of RLRX. Chapter 2 Limits and Colimits There are two important ways to define new objects using ones you already have. The first of these is called taking the limit of a diagram-a limit is defined by its properties as the target of morphisms. The dual is the colimit, which is defined in terms of its properties as the domain of morphisms. 1. Diagrams and Their Shapes We begin our discussion by revisiting diagrams from our more sophisticated category-theoretical point of view.