Abstract Nonsense

Crushing one theorem at a time

Limits, Colimits, and Representable Functors (Pt. I)


Point of Post: This post is mostly focused on motivating and defining limits and colimits, and defining representable functors only to make this process easier.

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Motivation

We have, up until this point, discussed various kinds of “constructions”, in particular we have discussed products and equalizers and their appropriate dual notion. Despite their apparent differences, all of these constructions were made via a very similar procedure. Namely, we had some “diagram” in mind, and we were looking for the object and arrows to (or from) that object to make the diagram a reality. For example, the product of X_1 and X_2  could be phrased as trying to find an object X_1\times X_2 and maps X_1\overset{\pi_1}{\leftarrow}X_1\times X_2\overset{\pi_2}{\rightarrow}X_2 such that given an object Y and maps X_1\overset{f_1}{\leftarrow}Y\overset{f_2}{\rightarrow}X_2 we have (uniquely) the following commuting diagram

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Similarly, we could write down a certain diagram for equalizers involving known objects/arrow, desired objects/arrows, and variable objects/arrows which tells us (unsurprisingly) what we have, what we want, and what our wanted things should do. This is the general idea of a limit, or perhaps better, the “limit of a diagram”. Namely, we have some set of objects and maps between these objects and some diagram (such as the one above) in mind for which there is some imaginary (desired) object and a variable object that tells us what this imaginary object does (in relation to the known objects, and maps from the known objects to the variable objects) [was this the same sentence as two sentences ago?]. Of course, this is all intuition and we need a firm, rigorous grounding to set this heavy intuition upon. So, how do we do this? The key was the observation we made during our discussion of equalizers, that an equalizer could very well be thought of as a “representing element” that takes a \mathbf{Set}-valued functor and turns it into a covariant (or contravariant) Hom functor. Roughly the idea behind this, is that we can state in \mathbf{Set} what diagram we want with very little problem (because \mathbf{Set} is such a manageable category) and finding a Hom functor that “represents” this diagram (which, secretly, is itself a functor) is equivalent to finding an object that makes everything work out the way we want it to.

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The above was very wishy-washy, and perhaps (to some–or most) not at all helpful, but it’s what makes sense in my head. If the above doesn’t suffice as motivation perhaps some we’ve-secretly-already-done-this magic will help. Indeed, the limits and colimits we are about to consider can be thought of as generalizations to general categories (and more general diagrams) of the notion of direct and inverse limits of modules or rings.

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April 13, 2012 Posted by | Algebra, Category Theory | , , , , , , | 3 Comments

Tensor Products Naturally Commute with Direct Limits


Point of Post: In this post we give a proof that, roughly, \varinjlim (M_\alpha\otimes_R N)\cong (\varinjlim M_\alpha)\otimes_R N, and moreover we show that this isomorphism is natural.

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Motivation

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In this post we begin the long succession of instances where the adjointness of Hom and tensor in conjunction with Yoneda’s lemma will be useful. In particular, we will show that tensor and direct limits naturally commute, a fact that shall be supremely useful in calculations involving direct limits (most prominently, coproducts).

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January 19, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , | 5 Comments

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. IV)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , | 1 Comment

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. III)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , | 2 Comments

Category of Directed/Inverse Systems and the Direct/Inverse Limit Functor (Pt. II)


Point of Post: This is a continuation of this post.

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December 28, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , , | 3 Comments

Inverse Limits of Modules (Pt. II)


Point of Post: This is a continuation of this post.

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December 9, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , | 4 Comments

A Different Way of Looking at Direct Limits


Point of Post: In this post we discuss a perhaps more intuitive way to think about direct limits.

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Motivation

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So, we have already discussed what it means to have a “directed system” of modules and what a “direct limit” of this directed system may look like. To recant,  we can think of a directed system of modules as being a “lattice” (I mean this only in the visualization sense) diagram of modules with the property that whenever one module M_\alpha “sits below” another module M_\beta there is a map f_{\alpha,\beta}:M_\alpha\to M_\beta. Intuitively, we thought about these maps as being embeddings, so what we could view M_\alpha as sitting inside M_\beta (while this is true most of the time, we see that there is no actual need to require these maps be embeddings). What we were then curious about (especially in the case that our indexing set was directed) was what the “limiting behavior” of this diagram looked like. Namely, while the diagram may not have a “highest member” (i.e. while our indexing set might not have a largest member, so that there is a module sitting at the “top” of the diagram) we see that, at least intuitively, going up the diagram gives us modules that are getting closer together. Indeed, assume for a second that our indexing set is directed. Then, at least intuitively going high enough allows one to put any two modules “close together”, since if we are dealing with M_\alpha,M_\beta by assumption that our preordered set is directed we can find some \gamma\geqslant\alpha,\beta and so M_\alpha\to M_\gamma and M_\beta\to M_\gamma putting them “close together”. Thus, we see that going up the lattice diagram gives us modules that seem to be “approximating” some module, but most often this module isn’t part of the diagram–the direct limit is this module, the ideal “top module” which ‘should’ be there. One can completely analogize this situation to completing a metric space so that all Cauchy sequences converge. To make the translation more apparent, we add in the direct limit \varinjlim M_\alpha to our lattice diagram to “complete it”, so that that our “Cauchy sequence” of modules actually has a limit–the direct limit.

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That said, even with all of this intuition about what the direct limit “does” it’s hard to get a firm grasp on what it should “look like”. That is the goal of this post, to describe direct limits over directed systems of modules (over directed sets, with injective put-in maps) as “directed unions”. What are directed unions? And, why do they help us understand direct limits? Well, directed unions are just, in essence, unions of submodules, but the collection of submodules (over which we are unioning) is such that we actually get a submodule. I think this will be more clear if one recalls that for two submodules L,N\leqslant M one has that L\cup N is a submodule of M if and only if, without loss of generality, L\subseteq N. Thus, we shall see that having a set \{N_\alpha\} of submodules of some fixed module M is exactly the condition one needs so that \displaystyle \bigcup_{\alpha\in\mathcal{A}}N_\alpha is actually a submodule of N–moreover (and we have actually already proven this) the union is precisely the direct limit of the directed system of submodules! Now, this construction is easy to picture, it’s just a union. The point of this post shall be to show that (for sufficiently nice examples) all direct limits are of this form! Roughly, once we put our directed system of modules “inside the same module” we will see that the direct limit of this system really is just the directed union of these submodules.

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December 9, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , | 2 Comments

Direct Limit of Rings (Pt. III)


Point of Post: This is a continuation of this post.

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December 4, 2011 Posted by | Algebra, Ring Theory | , , , , , , | Leave a comment

Direct Limit of Rings (Pt. II)


Point of Post: This is a continuation of this post.

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December 4, 2011 Posted by | Algebra, Ring Theory | , , , , , , , , , | Leave a comment

Direct Limit of Rings (Pt. I)


Point of Post: In this post we discuss the direct limit of rings, prove their universal characterization, and give some examples.

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Motivation

Unsurprisingly, we can take the direct limit of rings in much the same way we can take the direct limit of modules. This reflects that the notion of a direct limit can truly be defined in any category, though we shall not take that view (yet). All the intuition is going to be the same for that of modules, and so as to not just repeat it verbatim, we omit it in the below material. We shall see that many, many interesting constructions can be described as certain direct limits.

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December 4, 2011 Posted by | Algebra, Ring Theory | , , , , , , | 3 Comments