Point of Post: This post is mainly concerned with laying out the definitions of the various types of morphisms in a given category.
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. , 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.
Point of Post: In this post we prove the following two common, and useful isomorphisms: and .
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.
Point of Post: This post is a continuation of this one.
Point of Post: In this post we discuss the notion of ring homomorphisms, kernels, images, etc.
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 and . 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.
Point of Post: In this post we prove that every finite dimensional inner product space is isomorphic to its dual space.
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).
Point of post: In this post we discuss the Third Isomorphism Theorem.
The third isomorphism theorem has to deal with the situation when with . It asks us if the simple arithmetic fact somehow applies to groups. In essence, it asks if ?
Point of post: In this post we prove the second isomorphism theorem.
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 of a group with normal the resulting set will be, in fact, a subgroup. Moreover, we find that . In particular we get the interesting result that if is trivial then , so you can “just cancel them!”
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 is a homomorphism then
Having already discussed the idea of quotient structures it is natural to ask “How does relate to other objects?” The first such question might be: we know that if then . Thus, the quotient group is well-defined. How exactly does this relate back to ? There must be some connection between this quotient group and the original , and indeed there is. One can think of 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 we have ‘identified’ all the problem spots with each other in the sense that in the resulting quotient group if and are such that then . 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’.