This blog-post is dedicated to this day, $\pi$-day (14th of march), where we celebrate $\pi_*$, the stable homotopy groups. As has been the case a couple of times already, when faced with an increased workload I tend to neglect writing on this blog. It is only natural that increased amounts of work in one section lead to a decreased amount of work in another — there is, after all, only a finite amount of time given to us....
Introduction In the last post we studied homology theories as abstract functors from stable $\infty$-categories to abelian categories. We showed that for every stable $\infty$-category $\mathcal{C}$ there is a universal homology theory which all others factor through, namely the Yoneda embedding into the Freyd envelope, $$ y\colon \mathcal{C}\longrightarrow A(\mathcal{C}). $$ The fact that this is universal means that for any homology theory $H\colon \mathcal{C}\longrightarrow \mathcal{A}$ there is an essentially unique factorization...
New year; same me; new math. During the fall I said that I wanted to post on this blog monthly, but that did not happen for some reason. I thought I’d try again this semester, but maybe I am setting this not that high bar still too high for my self. For some reason there seems to be fewer hours in the day than it used to. The cover page for the post is generated by Dalle·2 using the prompt “A man finding the universal piece, old painting” – a true work of art....
It has been some time since we studied at the correlation between formal group laws, which were certain power series that looked like Taylor expansion of multiplication on a Lie group, and complex oriented cohomology theories. In particular, we learned that these two completely separate notions had a common universal object. The universal formal group law over the Lazard ring was the same as the formal group law determined by the universal complex oriented cohomology theory — complex cobordism cohomology....
This semester I am taking part in a seminar on $\infty$-categories, administered by Rune Haugseng. So far we have covered roughly: the basic definitions, fibrations, limits, colimits, Joyal’s lifting theorem, equivalences, straightening, Yoneda lemma, adjunctions and Kan extensions. This week it is my turn to give a talk on stable $\infty$-categories, and this blog post will hopefully be some sort of lecture notes for this talk. The intersection of things in this post and the contents of the talk should at least be non-empty....
Recently my friend Elias started his own math blog adventure, and his first post gave a nice introduction to spectral sequences. Reading it I remembered that I should really understand some of the parts better myself, because a lot of the arguments one makes in chromatic homotopy theory are based on spectral sequences. There is a framework for constructing spectral sequences that are not covered in my old post on them, as well as Elias’ post, and that is creating spectral sequences from exact couples....
Over the holidays sadly Edgar H. Brown passed away. He was one of the influential men behind many of the concepts this blog has featured and will feature in the future. This post is in particular focused on one of these concepts, namely Brown-Peterson cohomology and the Brown-Peterson spectrum. In the last post we developed the category of $p$-local spectra, and in the post before that we explored complex cobordism cohomology....
Topology, particularly homotopy theory, is hard. The scenes where these kind of mathematics happen are immensely complicated; the category of topological spaces; the category of spectra. The problem is that there is simply too much information to try to capture by using simple tools that we can actually understand properly. Trying to classify topological spaces or spectra is a feat that many deem impossible, it is simply too difficult. So, how can we try to fix this?...
In the next couple years I will need to understand the ins and outs of different cohomology theories and the spectra that represents them. Some of the most important of these (for my research) can be described using $MU$ — the complex cobordism spectrum. We briefly met this spectrum — or at least its cohomology theory — when we discussed formal group laws. There we explained briefly a theorem of Quillen, stating that the universal formal group law over the Lazard ring corresponds to complex cobordism cohomology....
A little while ago we discussed the definition of a tensor triangulated category, and in that post we mentioned an example that we didn’t explicitly define, namely the stable homotopy category. The goal for todays post is to fix this. There are many ways of defining it, and some are actually better than others. As the name suggests, the stable homotopy category is a homotopy category, which we have discussed before in the fibration series....