Abstract Nonsense

Functors (Pt. II)

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

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December 27, 2011

Types of Morphisms

Point of Post: This post is mainly concerned with laying out the definitions of the various types of morphisms in a given category.

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Motivation

As of now, with just the bare definition of categories, we have little to work with. From our current standpoint, all morphisms are created equal. But, keeping in mind that categories are modeled off the common categories we usually work with (e.g. $\mathbf{Top,Grp,Ring,Set}$, etc.) we know that this is certainly not true. Namely, the ideas of group epimorphisms, topological embeddings, and the like quickly show us that morphisms come in many different varieties. This post shall lay out the definitions to make these differences clear.

December 27, 2011

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

Product of Rings (Pt. II)

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

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July 11, 2011

Ring Homomorphisms (Pt. II)

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

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June 18, 2011

Ring Homomorphisms (Pt. I)

Point of Post: In this post we discuss the notion of ring  homomorphisms, kernels, images, etc.

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Motivation As is standard in math, especially in algebra after defining a structure and its subobjects we define the morphisms between the two objects. In particular, we would now like to define the morphisms between two rings $R$ and $R'$. What do we want these morphisms to do? Well, we clearly want the morphisms to preserve the ring structure. In other words, we’d like it to be additive (be a group homomorphism for the group structure) and multiplicative (be a semigroup homomorphism for the semigroup structure). After this we define the obvious notions of kernel and image and show that they are, in fact, subrings ,etc.

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June 18, 2011

Canonical Isomorphism Between a Finite Dimensional Inner Product Space and its Dual

Point of Post: In this post we prove that every finite dimensional inner product space is isomorphic to its dual space.

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Motivation

We have seen in the past the proof that every finite dimensional vector space is isomorphic to its double dual. We know of course since dimension is preserved under taking duals for finite dimensional vector spaces (this is, in fact, a characterization of finite dimensionality) but there was no canonical (free of basis choice) way of defining the mapping. In this post we prove the scene is different if the vector space is supplied with an inner product (or more generally a non-degenerate bilinear form).

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June 4, 2011

Review of Group Theory: The Third Isomorphism Theorem

Point of post: In this post we discuss the Third Isomorphism Theorem.

Motivation

The third isomorphism theorem has to deal with the situation when $K, H\trianglelefteq G$  with $K\subseteq H$. It asks us if the simple arithmetic fact $\displaystyle \frac{\frac{a}{b}}{\frac{c}{b}}=\frac{a}{c}$ somehow applies to groups. In essence, it asks if $\left(G/K\right)/\left(H/ K\right)\cong G/H$?

January 2, 2011

Review of Group Theory: The Second Isomorphism Theorem

Point of post: In this post we prove the second isomorphism theorem.

Motivation

As was stated in the post on the First Isomorphism Theorem there other isomorphism theorems which, using the first one, aren’t too hard to prove. The second one in essence says that it says that if you multiply two subgroups $A,B$  of a group $G$ with $B$ normal the resulting set  $AB$ will be, in fact, a subgroup. Moreover, we find that $AB/B\cong A/(A\cap B)$. In particular we get the interesting result that if $A\cap B$ is trivial then $AB/B\cong A$, so you can “just cancel them!”

January 2, 2011

Review of Group Theory: The First Isomorphism Theorem

Point of post: In this post we prove the first isomorphism theorem which, in essence says that for any homomorphism the image is isomorphic to the domain with a small perturbation. Precisely what this perturbation turns out to be is the kernel of the homomorphism. Explicitly we prove that if $\phi:G\to G'$ is a homomorphism then

$G/\ker\phi\cong\phi\left(G\right)$

Motivation

Having already discussed the idea of quotient structures it is natural to ask “How does $G/N$ relate to other objects?” The first such question might be: we know that if $\phi\in\text{Hom}\left(G,G'\right)$ then $\ker\phi\trianglelefteq G$. Thus, the quotient group $G/\ker\phi$ is well-defined. How exactly does this relate back to $\phi$? There must be some connection between this quotient group and the original $\phi$, and indeed there is. One can think of $\phi$ as an “almost isomorphism” in the sense that it’s an isomorphism satisfying  the temporary issue of non-injectivity. Consequently, one may ask “is there a way to ‘throw out’ the problem elements”? This is a familiar ideology to those working in analysis where one ‘mods out’ by violators of the positive semi-definitness of a metric or norm. It turns out that, for all intents and purposes, the answer is yes. Moreover, the way one does it is kind of what ‘seems natural’, especially if one is familiar with general topology or the above ideas of analysis (or more generally the set-theortic notion of kernel). In essence, we’ll see that by considering $G/\ker\phi$ we have ‘identified’ all the problem spots with each other in the sense that in the resulting quotient group if $g$ and $g'$ are such that $\phi(g)=\phi(g')$ then $g\ker\phi=g'\ker phi$. Thus, the resulting space will be one for which there is a canonical ‘reduction’ of the original surjective homomorphism but one for which the ‘disease’ of injectivity is ‘cured’.

January 2, 2011