Triangulated Categories

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. But, for the remainder of my PhD I will solely focus on research and outreach, hence I will hopefully have some more time to write and think. This post has been a long time coming and features the precise area of mathematics where I do most of my research, namely exotic algebraic models. I will throughout this post, and its sequels, explain what these are and connect it to almost all the previous blog posts I have made for the last couple of years. This will also set up some of the needed background for presenting my own research, which I will do once I am done writing the paper presenting it. ...

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. ...

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. ...

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. ...

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. But the question is, what is it the homotopy category of? As we remarked in the post on tensor triangulated categories, it is the homotopy category of the category of spectra, and it is here that the different approaches lie. What exactly is the category of spectra, and which spectra are we even talking about? Is it the sequential spectra? or maybe the orthogonal spectra? or perhaps the symmetric ones? maybe $S$-modules or excisive functors? All these names of course means nothing to us yet, as we haven’t properly looked at any of them. We did however meet the $\Omega$-spectrum in an earlier post, but which of the above types does it belong to? ...

For the last five years mathematics has been my passion, as well as my main focus in life. This passion for mathematics will hopefully not diminish, as I am now heading into four more years of studies and research through a PhD in mathematics at NTNU. I am joining a project, called Tensor triangulated geometry in Trondheim, so today I thought I would explore the definition of one of the main players in this theory, namely tensor triangulated categories. We have already met half of this structure during our several encounters of monoidal categories, but the other half remains, as well as how to glue them together into a cohesive joint structure. ...