Or: just what is a Milnor square? A Milnor square is a certain pullback in the category of associative rings. It happens when we have a ring $R$, a two-sided ideal $I\subseteq R$ and a ring homomorphism $f:R\to S$ such that the restriction $f:I\to f(I)$ is an isomorphism of rings without identity. Since $I$ is […]
I’ve been talking a little about abelian categories these days. That’s because I’ve been going over Weibel’s An Introduction to Homological Algebra. It’s a book I read before, and I still feel pretty confident about the material. This time, though, I think I’m going to explore a few different paths that I haven’t really given […]
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 […]
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 […]
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 […]
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 […]
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 […]