Limits, Colimits, and Representable Functors (Pt. II)
Point of Post: This is a continuation of this post.
Similarly to cones we can define a cocone of to be an object and arrows which make the resulting diagrams
I don’t think I need to tell you how to define a morphism of cocones and that we have a category . We define then that a colimit of shall be a terminal object of . It’s pretty easy to see that a cocone of is nothing more than a cone of the functor considered as the composition .
Since initial and terminal objects are unique up to isomorphism we shall often denote the objects associated to the limit and colimit (if they exist–and this isn’t a vaccuous statement) as and respectively–occasionally we may write and .
So, let’s get our hands a little dirty and take a look at some real live limits and colimits.
Ok, let be the discrete category on two objects. What then is a limit for a functor ? Well, we have two objects and and corresponding to the two objects in and no non-identity arrows so that we have the (pretty unexciting) diagram
We see then that a limit of is an object of with arrows with the property that given any pair of arrows and there exists a unique arrow such that . Thus, we see that colimit of is nothing more than a product of and . As is probably not hard to guess, a colimit of is a coproduct of and .
This simple example shows that limits/colimits need not exist in general for, as we have seen, products need not always exist in categories. This is something to keep in mind–not all diagrams have limits.
Ok, let be the category shown in . What is a limit of this diagram? Well, we see that it would need to be an object along with arrows and such that and moreover any time we have an object along with arrows and such that there exists a unique arrow such that and . Take a minute to convince one’s self that the maps and were incidental (there inclusion as information is redundant since we know what they have to be–) and that the only thing was that . From this make the leap to conclude that a limit of this diagram is nothing more than an equalizer of . Shocking as this may come to you [sorry for those with a weak heart] but a colimit of this diagram is a coequalizer of .
Ok, so what do limits and colimits have to do with making something work in and using this to define the object in ?
Well, let’s begin by showing that given an arbitrary functor that exists. Indeed, define
We claim that along with the functions given by . What we claim is that is actually a limit of .
To see that is even a cone we need to check that for all . But, this is simple. Let be arbitrary, then from where the conclusion follows. Now, we need to prove that this cone really is a universal object in . To do this, suppose that is another cone of and define by . What we now need to check is that for all , but this is trivial since . Ok, so the last thing to check is that such a mapping is unique. But, this is clear by the definition of the product of sets–if two functions agree on the projections (for that’s all these are) for all possible coordinates the functions are equal. Thus, we see that .
Ok, so now the idea is to use the fact that limits exist in to define them in arbitrary categories. The basic idea should be that we should be able to phrase the properties of a purported limit of a diagram merely by talking about sets–namely the Hom sets. How to do this?
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