Last post, I talked about computing a direct limit on some cofinal subset. In this post, I want to prove what I asserted.
Note that direct limits are a special case of colimits, and what I’ll talk about can easily be generalized. In particular, colimit may also be computed on cofinal subcategories of their indexing category, and yield equal (“isomorphic up to a unique canonical isomorphism”) colimits. However, in this post I’ll stay in the particular case of direct limits, and I’ll try to define things first in terms of elements, to stay as concrete as possible.
Recall that a poset is called directed if any pair of elements in has an upper bound. For instance, the integers with the usual order are a rather trivial example of a directed set. Another example which is quite important in algebraic geometry is the set of all open neighborhoods of a given point in a topological space, ordered by reverse inclusion: a natural choice for an upper bound for two such open neighborhoods is given by their intersection.
When is a poset, any subset is also a poset in the order . Such a subset is said to be cofinal in if, for any , there exists some with . More formally: . If we dualize, we obtain the statement: . Hence the concept of being cofinal is dual to the concept of having an upper bound. That was just a quick side remark, it’s not really important as far as I know. If is a directed set and is cofinal in , then an important fact is that is also a directed set (hint: two arbitrary elements in are also in , so they have an upper bound in ; by cofinality there’s an element of that’s larger than this upper bound).
Let be a category that’s “algebraic”, i.e. objects are “sets with some optional extra structure” and arrows are “set functions which respect that extra structure”. For instance, the category itself fits this description, as does the category of groups, or the category of modules over some ring, etc. In these categories, objects have elements, so we can be very concrete in our definitions.
Let be a directed set, which we will use to index objects and arrows in as follows: we consider a collection of objects of , and a collection of arrow for all pairs with , such that:
- is the identity on for all ; and
- for all .
You may recognize that this data is exactly a (covariant) functor from the indexing set (any poset may be interpreted as a category) to , or in other terminology a diagram of shape in . In this article, however, we will not use this terminology, and simply say that the “pair” is a direct system over .
The direct limit of a direct system is defined as a set by the equation where is the equivalence relation generated by: two elements and verify if and only if there is an upper bound of and such that . In other words, two elements are declared to be equal when they are “eventually equal” at some large enough index .
Because is a directed set, the underlying set we just defined for the direct limit can always be equipped with the appropriate algebraic structure so that it is an object of :
In the category of groups, define multiplication on in the following way. Write and for two equivalence classes that are elements of , so that and . Because is a directed set, there exists some such that both and . Now define the operation by using the group operation in : This is well-defined. First, we show this definition is independant of the chosen upper bound . Let be any other upper bound for and . Then, because is a directed set, there exists some that is an upper bound of and . But now whence in . Second, we show the definition is independant of the representants of the equivalence classes. Suppose and . By definition, this means there are such that is an upper bound for and , is an upper bound for and , and we have and . Now choose any which is an upper bound for and . In particular, is an upper bound for , , and . By the first part, the product does not depend on , and neither does the product . But we have hence the product is well-defined, as claimed.
In the category of rings, addition and multiplication may be done “representative-wise”, in the same way as what we did with group, simply by sending the two representatives to a common ring using the fact that is a directed set. This gives a well-defined multiplication and addition, and they respect the axioms of a ring since they are defined in terms of elements in a ring.
Etc.
The direct limit always come with its canonical arrows, or canonical morphisms, which are the projections sending an element to its equivalence class. In fact, the algebraic operations are defined on to be the “free-est”, or less constrained, operations such that the canonical arrows are morphisms. Also, for any , we have the commutativity condition . One can show that together with its canonical morphisms is in fact the colimit in under the diagram of shape given by the data of a directed system as above. This means that for any object in , and any collection of morphisms such that when , there exists a unique morphism such that for every . Also, the fact the direct limit is a colimit shows that it is unique up to a unique canonical isomorphism.
By its construction, the direct limit is in some sense the “smallest upper bound”, the idea being that it’s the smallest object (i.e. the one having the least amount of internal constraints) which approximates from above all of the ’s. Category theory teaches us that we can learn about the internals of an object by studying its external relations.
Now we get to the main point. If is a direct system over , and if is cofinal in , then we can “restrict” the direct system over to a direct system over by considering only objects and arrows with . Now the fact we’re trying to prove can be expressed as:
Let be a direct system over , and let be cofinal in . Then where the equality symbol is to be interpreted, as usual, as “isomorphic up to a unique isomorphism which makes a certain diagram commute”, the diagram being the expected one with canonical morphisms .
Let’s prove this. We will construct a function between the two direct limits and show it is a bijection. Take any element in . Then is an element of . This is what we define to be: This is a well-defined function. Indeed, suppose . Then there exists some which is an upper bound of both and , such that . Now, the commutativity condition of the canonical morphisms tells us that so is well-defined as claimed. Moreover, if the objects are groups, rings, anything with algebraic structure, then we see that respects this structure, since it’s defined in terms of the canonical morphisms. Now suppose . Then in , so there exists some upper bound of both and such that . Because is cofinal in , we can actually suppose . Then this implies in the direct limit over , so is injective. To show it is surjective, pick any element in . Again, because is cofinal in , there exists some with . Of course, , so and is surjective.