Algebras

Introduction In the last blog post we introduced and studied adapted homology theories. Given a stable $\infty$-category $\mathcal{C}$ and $\mathcal{A}$ an abelian category with enough injectives together with a local grading $[1]\colon \mathcal{A}\longrightarrow \mathcal{A}$, an adapted homology theory is a functor $$H\colon \mathcal{C}\longrightarrow \mathcal{A}$$such that $H$ is additive, sends fiber sequences to long exact sequences, sends the suspension $\Sigma$ of $\mathcal{C}$ to the shift $[1]$ of $\mathcal{A}$ and such that we can lift injectives in $\mathcal{A}$ into $\mathcal{C}$ through $H$. We explored which categories $\mathcal{A}$ can actually exist in this setting, and saw that they had to be closely related to the Freyd envelope $A(\mathcal{C})$ of $\mathcal{C}$. More specifically, $\mathcal{A}$ had to be the sheafification of $A(\mathcal{C})$ with respect to a topology on $\mathcal{C}$ determined by $H$, which we called the $H$-epimorphism topology. ...

I have recently handed in and defended my master thesis in mathematics, so I though I would go through its abstract and try to explain what it’s all about. We look at formality of DG-algebras, Massey products, A_infinity-algebras and how we can use these to some interesting results.

The first post on this blog is titled “geometric intuition”, and discusses the geometry behind Noether’s normalization lemma. When I wrote it I didn’t yet understand all the pieces, as I was not very comfortable working with algebraic geometry. One year later, I’m still not comfortable, but a bit more than last year. So, I thought I would update last years post with my new knowledge, as well as generalize the intuition to schemes - which we introduced in the last post. ...

The first two posts ([1],[2]) I ever did on this blog - now over a year ago - were posts about algebraic geometry. In particular we explored the geometric implications of some of the algebraic results I was learning in my commutative algebra class. Last summer I also wrote a post about sheaves, and left it off by claiming to soon write about schemes. If you scroll through the blog we have covered a bunch of different topics, but the blog post on schemes, seems to have fallen through the cracks. Today we will rectify this situation. I have my algebraic geometry exam this week, so this is both an explainer-post, and a “making sure I understand the course material”-post. These types of posts have in fact become common on this blog, but hopefully that is ok. ...

In the two last posts we have been discussing operations that are associative up to homotopy, and where such operations might arise naturally in topology. One claim I made, which I later realized was maybe a bit unmotivated and in need of some clarification was how some higher arity maps actually defined (or were defined by) homotopies between combinations of the lower arity maps. We also purely looked at this in a topological setting, but in algebraic topology we often translate to algebraic structures, so I also wanted to see clearly that the same constructions hold in that setting. To be more precise I am talking about the claim that a map we denoted by $m_3$ was a homotopy between $m_2(id\otimes m_2)$ and $m_2(m_2 \otimes id)$, where $m_2$ was a product induced through a homotopy equivalence. Don’t worry if you don’t recall the definitions and this problem, we will go through it again shortly. Today we in fact upgrade this earlier homotopy equivalence slightly such as to have a bit more to work with. As said we also take a turn away from standard topology and make our choice of “space” for this post to be chain complexes of vector spaces. I will not cover in detail why this is a reasonable thing to do but I will mention that the de Rham complex of a manifold, the rational singular cochains on a topological space and the rational cohomology of a topological space are all such structures. So, if we believe that algebraic topology is a nice way to study spaces, then studying these should be highly relevant. ...

For those that don’t know I am a fifth year mathematics student at NTNU, meaning I am finishing my masters degree after this semester. During my time at NTNU I have had some wonderful classes, and some wonderful teachers. Since most I post about on this blog is related to topology, it is very safe to assume that some of my most memorable courses are exactly the topology courses. I very recently looked at my notes from my first topology course, or rather one of the two first, as I took two in parallel during my fourth semester. The course was focused on differential topology and the study of smooth manifolds. It was taught by my now supervisor, but on a couple of the last lectures we had some guest appearances from the other topology professors at NTNU. One of these guest lectures is the focus of todays blog post. ...

Since one of my main mathematical interests is homotopy theory, im bound to often bump into things that require the use of base-points. This has long been the classical way to study spaces, especially in terms of homotopy groups. When I was introduced to these so-called pointed spaces, I couldn’t help but feel that these we less natural, or more ad hoc, than regular spaces. I didn’t know much about categories then, but have since learned it is usually in this context that some form of naturality occur. It turns out that pointed spaces actually come from a very nice natural categorical construction, which of course is the focus of this post. I will assume introductory knowledge of categories, and I will try to keep this short for once. ...