Let $R$ be a ring and $f:B\to C$ be a morphism of $R$-modules. The image of $f$ is of course
$${\rm im}(f) = \{ f(x) : x\in B \}.$$The image of $f$ is a submodule of $C$. It is pretty much self-evident that $f$ factors as
$$B\xrightarrow{e} {\rm im}(f)\xrightarrow{m} C$$where $e$ is a surjective homomorphism and $m$ is an injective homomorphism. In fact, there is nothing special about working in the category of $R$-modules at all. The same thing holds in the category of sets and a proof for the category of sets works perfectly well for the category of $R$-modules. This set-theoretic reasoning is very natural.

However, we can't always work with categories whose objects are sets with additional structure and whose morphisms are set functions that respect the additional structure (concrete categories). Sometimes we have to work with abelian categories. What's an abelian category? Briefly, it is a category $\Acl$ such that:
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Let $ \mathcal{A}$ be a small category and $ \mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $ \mathbf{B}\mathcal{A}$ and $ \mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen's paper Higher Algebraic K-Theory: I", so grab the notes or read the summary below:

Given any functor $ f:\mathcal{A}\to\mathcal{B}$, and an object $ B\in\mathcal{B}$, we can consider the category $ f^{-1}(B)$ consisting of objects $ A\in \mathcal{A}$ such that $ f(A) = B$. The morphisms of $ f^{-1}(B)$ are defined to be all the morphisms that map to $ 1_B$ under $ f$. Let us apply this to the following situation:

Given any small (or skeletally small) category $ \mathcal{A}$, we can construct another category $ S(\mathcal{A})$ as follows: the objects of $ S(\mathcal{A})$ are the arrows $ X\to Y$ of $ \mathcal{A}$, and a morphism $ (X\to Y)\to (X'\to Y')$ is a pair of morphisms $ X'\to X$ and $ Y\to Y'$ in $ \mathcal{A}$ making the obvious square commute. Now, we can consider the functor $ s:S(\mathcal{A})\to \mathcal{A}$ sending the object $ X\to Y\in S(\mathcal{A})$ to the object $ X\in\mathcal{A}$.
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Mostly to take a break from marking exams, I thought I'd start a new recurring series here about mathematics papers and books that I find, both new and old. The "new" will consist mainly of preprints that look interesting (to encourage me to browse the arXiv) and the "old" will consists of papers I will likely read (to encourage me to read, or at least skim, more papers).

New Preprints

  1. Clark Barwick, "On the algebraic K-theory of higher categories": For a while I've wanted to learn a bit about higher category theory, but I've not yet found any application or motivation for it in something I'm already really interested in (including the stuff here). Perhaps this paper by Barwick will change my mind: algebraic K-theory may be best viewed as a functor with a universal property that generalises the well-known universal properties known for the classical K groups.
  2. Booker, Hiary, and Keating, Detecting squarefree numbers": Under the Generalised Riemann Hypothesis the authors propose an algorithm to test whether an integer is squarefree, without needing the number's factorisation.
  3. Shalit, "A sneaky proof of the maximum modulus principle": This is a proof of the maximum modulus principle in complex analysis, now with even less complex analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as well.

View the whole post to reveal the hidden classic:
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In this post we shall see a natural example that should give some motivation for derived functors of functors on the homotopy category of cochain complexes in an abelian category $ \mathcal{A}$. At first glance, the formalism of derived functors in this setting may seem less intuitive than the formalism of classical derived functors, partially because of the need to consider hypercohomology (cohomology of complexes), but hopefully in this and in future posts we will see some examples that show the elegance of the theory.
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Posted by Jason Polak on 26. June 2011 · Write a comment · Categories: category-theory

Category theory is tremendously useful now, so it's amusing to read the following paragraph from the introduction to Mitchell's "The Theory of Categories":

A number of sophisticated people tend to disparage category theory as consistent as others disparage certain kinds of classical music. When obliged to speak of a category they do so in an apologetic tone, similar to the way some say, "It was a gift–I've never even played it" when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisite that the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind.