I’m not going to reiterate the definitions of limits and co-limits from the last post, so just look them up if you’re new here. They’re not too hard.
This post is mainly just about some random cool things I’ve noticed/ “remembered” / come across whilst playing with the notions of limit and co-limit in various categories.
Say we’re working in R-Mod for some ring , and let be -modules. Taking the limit of the “discrete” diagram gives us the “product diagram: ” . We can then take the co-limit of this diagram, which is the quotient , where is the module generated by elements of the form for . It then follows that the pushout is trivial. Strange.
EDIT: for some clarification, I want to show that for any diagram of the form , then the colimit is the object , where is the submodule generated by the elements . Clearly, it does fit into the appropriate diagram. Now let be any other module with maps , such that . Then, via the universal property of the direct sum, there is a unique map such that and (where and are the canonical inclusions). Then we have that , so for any . Hence , i.e. . Via the universal property of the quotient, there is a unique homomorphism that makes the whole diagram commute.
In the case where and and , , so that the quotient .
The dual construction is (obviously) similar, where we first take the co-limit of the discrete diagram, then the limit of the “co-product diagram.” It is also the zero module.
Adjunction spaces in Top, the category of topological spaces. Let be topological spaces, (represented as a monic ) be a subspace. Let be a continuous function. Then the adjunction space obtained by gluing to along is just the co-limit of the diagram .
Let be a topological space, which has a natural poset structure on its collection of open sets. Formally, we turn into a category X whose objects are the open sets of and the morphisms are determined via iff . Let be elements of X such that and . Then the limit of the diagram is just the intersection .
Limits as functors. Turns out you can replace the notion of a “diagram in C” (where C is the category we’re looking at) with a functor , where I is a small category. Think about it! The limit of such a diagram is denoted by .
Quick note: Adjoint Pairs of Functors
Say we have two functors and . We say form an adjoint pair if, for all , , we have a bijection that is natural in and . Furthermore, we say is left adjoint to , and similarly is right adjoint to . Also, is a right adjoint functor if it has a left adjoint, and likewise for left adjoint functors.
Right adjoint functors commute with Limits. Let and be an adjoint pair of functors, and let be a diagram. The statement is then that
Awesome. The proof is actually pretty straightforward abstract nonsense, just take the definition of as a limiting cone, apply , get a map . Then, use adjunction to get a map for all objects in I. The universal property of gives us a map by applying adjunction again. These maps are quickly seen to be inverses of each other (keep looking through universal properties and such). A good outline of this proof can be found in Paolo Aluffi’s “Algebra: Chapter 0”
Similarly, we have that Left adjoints commute with co-limits.
Math is awesome.