Abstract Nonsense

Crushing one theorem at a time

Finite Limits and Colimits as Products/Coproducts and Equalizers/Coequalizers


Point of Post: In this post we prove the result that a category has all finite products/coproducts and equalizers/coequalizers if and only if it has all limits/colimits.

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Motivation

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In this post we prove a pretty neat fact that, at least for finite sets (we shall state a more general result, but we don’t care about it as much) limits/colimits are the same thing as equalizers/coequalizers and products/coproducts. This is often cool because it reduces having to prove some result holding for all finite limits/colimits to proving some problem holds for equalizers/coequalizers and finite products/coproducts. In particular, it will tell us that a category admits finite limits/colimits if and only if it admits finite products/coproducts.

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April 13, 2012 Posted by | Algebra, Category Theory | , , , , , , , , , , , | 1 Comment

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


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

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April 13, 2012 Posted by | Algebra, Category Theory | , , , , , , , | 1 Comment

Categorical Products


Point of Post: In this post we discuss the notion of products in the sense of category theory.

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Motivation

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We have indicated in our description of universal arrows that one of their key features was the ability to take constructions found in various parts of mathematics that, while ostensibly unrelated, are really the same thing when viewed as the solution to certain universal mapping problems. In this post we give a typifying example of this to show how the notion of “product” found throughout mathematics (e.g. module theory, ring theory, group theory, topology, etc.) is really just a guised version of general, categorical, notion of “product” which is going to be the solution to a certain universal mapping property. This should be seen as a triumph since not only does it allow us to understand why the particular definitions of “product” mentioned before make sense, but it also allows us to be able to smartly define “product” in any new category we wish to explore. So, what exactly are categorical products? Roughly the product of two objects x,y should be an object p for which the arrows z\to p  into this third object are in one-to-one correspondence with pairs of arrows z\to x,z\to y into the factor objects.

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January 24, 2012 Posted by | Algebra, Category Theory | , , , , , | 4 Comments

Homomorphism Groups of Products and Coproducts (Pt. I)


Point of Post: In this post we prove the following two common, and useful isomorphisms: \displaystyle \text{Hom}_R\left(\bigoplus_{\alpha\in\mathcal{A}}M_\alpha,N\right)\cong\prod_{\alpha\in\mathcal{A}}\text{Hom}_R(M_\alpha,N) and \displaystyle \text{Hom}_R\left(M,\prod_{\alpha\in\mathcal{A}}N_\alpha\right)\cong\prod_{\alpha\in\mathcal{A}}\text{Hom}_R(M,N_\alpha).

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Motivation

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We shall in the future have many occasions to deal with homomorphism groups (modules). In particular, we shall often deal with one of the Hom functors, and surprisingly often we shall have that in the free variable there is a product or coproduct. Consequently, it would be nice if we had some way of simplifying such Hom’s in terms of nicer groups. That is precisely the content of this post. It shall be good practice for us applying our notions of product and coproduct.

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November 12, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , | 6 Comments

Product of Modules


Point of Post: In this post we discuss the product of modules, including their characterizations via univeral mapping properties.

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Motivation

We now consider the product of modules, which as always, is just endowing the Cartesian product of a set of modules with operations that turn the resulting (set) product into a module. That said, while we have mentioned before that products of things can be characterized via certain universal mapping properties (e.g. for rings and groups) here we shall actually start with thinking of products in terms of these universal mapping properties and then define the “natural product”  only to prove existence of such modules. Why? What precisely is the point of doing? Well, we all have an intuitive idea about what products are, we have seen them as objects for much of our mathematical careers. That said, we understand them intuitively in a concrete “I can see them” sense. What we are now more currently interested is understanding them in a “how do they act” sense–what makes “products” products from the view of mappings. Well, this is precisely what the universal characterization of products tells us. Stated it says that “a product of the set \left\{M_\alpha\right\}_{\alpha\in\mathcal{A}} of left R-modules is a left R-module M together with a set of maps f_\alpha:M\to M_\alpha with the following property: given any left R-module N and maps g_\alpha:N\to M_\alpha there exists a unique map g:N\to M such that f_\alpha\circ g=g_\alpha.” Ok, so fine, but what is this big-long-horrible definition really telling us about this M? The existence of these f_\alpha‘s tell us that M is “put together” in some way via the M_\alpha‘s. The fact that any map N\to M is determined by its values on M_\alpha (translated via the “put together maps”, f_\alpha) tells us that M is put together in a fairly minimal way–i.e. this M isn’t put together in such a way that there is much “room outside the M_\alphas” for movement. That said, we see that the M_\alpha‘s are living inside of M in a fairly faithful manner, in the sense that a functions values on each of the M_\alphas are independent of one another–this tells intuitively that we didn’t squish the M_\alpha‘s, or that we didn’t underepresent any of the M_\alphas at any stage of construction. Thus, with this in mind, it’s clear why we’d consider products of modules.

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November 11, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , | 10 Comments

Halmos Sections 34 and 35:Products and Polynomials (Pt. II)


Point of post: This post is a continuation of

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November 29, 2010 Posted by | Fun Problems, Halmos, Linear Algebra, Uncategorized | , , , , , | Leave a comment