Limits in functor categories
Today I want to give a great example of creation of limits that shows how useful it can be. For motivation, take a set , a monoid , and consider the set of functions from to . Then inherits a monoid structure from that on . Just define and take the function sending every element to the identity of as the identity of . We’re going to do the exact same thing in categories, but with having limits instead of a monoid structure.
As a preliminary result we need to note that if we have a set of categories for each of which has -limits, then the product category has -limits. Indeed, a functor from to the product consists of a list of functors from to each category , and each of these has a limiting cone. These clearly assemble into a limiting cone for the overall functor.
The special case we’re interested here is when all are the same category. Then the product category is equivalent to the functor category , where we consider as a discrete category. If has -limits, then so does for any set .
Now, any small category has a discrete subcategory : its set of objects. There is an inclusion functor . This gives rise to a functor . A functor gets sent to the functor . I claim that creates all limits.
Before I prove this, let’s expand a bit to understand what it means. Given a functor and an object we can get a functor that takes an object and evaluates at . This is an -indexed family of functors to , which is a functor to . A limit of this functor consists of a limit for each of the family of functors. The assertion is that if we have such a limit — a -limit in for each object of — then these limits over each object assemble into a functor in , which is the limit of our original .
We have a limiting cone for each object . What we need is an arrow for each arrow in and a natural transformation for each . Here’s the diagram we need:
We consider an arrow in . The outer triangle is the limiting cone for the object , and the inner triangle is the limiting cone for the object . The bottom square commutes because is functorial in and separately. The two diagonal arrows towards the bottom are the functors and applied to the arrow . Now for each we get a composite arrow , which is a cone on . Since is a limiting cone on this functor we get a unique arrow .
We now know how must act on arrows of , but we need to know that it’s a functor — that it preserves compositions. To do this, try to see the diagram above as a triangular prism viewed down the end. We get one such prism for each arrow , and for composable arrows we can stack the prisms end-to-end to get a prism for the composite. The uniqueness from the universal property now tells us that such a prism is unique, so the composition must be preserved.
Finally, for the natural transformations required to make this a cone, notice that the sides of the prism are exactly the naturality squares for a transformation from to and , so the arrows in the cones give us the components of the natural transformations we need. The proof that this is a limiting cone is straightforward, and a good exercise.
The upshot of all this is that if has -limits, then so does . Furthermore, we can evaluate such limits “pointwise”: .
As another exercise, see what needs to be dualized in the above argument (particularly in the diagram) to replace “limits” with “colimits”.