Exotic algebraic models

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....

March 14, 2024 · 20 min · Torgeir Aambø

Hopf algebroids

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 $H$....

May 9, 2023 · 29 min · Torgeir Aambø

Adapted homology

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...

March 24, 2023 · 20 min · Torgeir Aambø

Universal homology

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....

January 31, 2023 · 15 min · Torgeir Aambø

Bousfield localization

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?...

December 2, 2021 · 19 min · Torgeir Aambø