In the fall of 2024, I gave a local mini-course on synthetic spectra. It consisted of six two-hour lectures; more details below.

I am currently in the process of writing an expository paper on synthetic spectra based on these lectures. In the meantime, I have put preliminary versions of the lecture notes below. A big omission in the current notes is the lack of references to the literature and the historical context.

I hope even these temporary notes are useful, but use them at your own risk: there may be some inaccuracies or wrong statements in these very early versions. (If you’d like to use results in them more seriously that cannot be found elsewhere, please contact me.) If you spot mistakes or typo’s, or have other forms of feedback, then please let me know!

I roughly covered the following topics:

• Spectral sequences from the point of view of filtered spectra
• Bockstein spectral sequences
• “What’s up with $\tau$?”
• Synthetic techniques for computing differentials.
• Goerss-Hopkins and Toda obstruction theories.
• Construction and variants of synthetic spectra.

Lecture 1: Spectral sequences | Notes v0.1
22 October

I will explain the overall goals of the mini-course. The rest of the lecture will be an introduction to spectral sequences, from the perspective of filtered spectra. (One can replace ‘spectrum’ by ‘chain complex’ in this talk, if that is more familiar.) The perspective will be slightly nonstandard, and in particular, will focus on the role of the magical element tau.

v0.1: The current lecture notes do not yet contain the comparison and overview of the literature I want to give, so referenes are severly lacking. Also, the notes only contain (most of) the informal discussion I gave of spectral sequences. A section making everything formal is in the works, but not ready to be shared publicly yet.

Lecture 2: Tau in filtered spectra | Notes v0.2
29 October

After discussing some examples of spectral sequences (the Atiyah-Hirzebruch and Bockstein spectral sequences), We will lift the map tau from the previous lecture to the level of filtered spectra. This leads to the ‘synthetic perspective’ on spectral sequences, reinterpreting a spectral sequence in terms of the cofibre of tau. Finally, we will discuss the universal property of filtered spectra, roughly describing it as the universal category on the element tau.

v0.1: The lecture notes contain additional material on the truncated Bockstein spectral sequence, which I did not cover in the lecture. I also included a short discussion to make the Whitehead filtration functorial.

v0.2: Corrected parts of the Omnibus Theorem (but the statement is not yet entirely correct!), and fixed a number of minor mistakes.

Lecture 3: Basics of synthetic spectra | Notes v0.1
5 November
Change of time: 10:00-12:00 (still in HFG 7.07B)

This lecture can be thought of as an introduction to the Adams spectral sequence from the modern point of view. We will discuss the basic categorical properties of synthetic spectra (but for the moment without discussing the explicit construction). We will import the variable $\tau$ from filtered spectra, and see how this leads to a description of synthetic spectra as encoding a particular kind of spectral sequence. We then compute that, for synthetic analogues, this recovers the Adams spectral sequence in the usual sense.

v0.1: Conforms to what I covered in the lecture, with a bunch of added remarks and warnings. Detailed comparison and references to the literature, and citations of proofs, are missing. Currently using the nonstandard symbol よ to denote the ‘spectral Yoneda embedding’, which is denoted by $Y$ in most sources. (Please let me know if you have suggestions for another symbol for this functor!)

No lecture on 12 November.

Lecture 4: Computations with synthetic spectra | Notes v0.0
19 November

This lecture demonstrates the benefits of using synthetic spectra to compute Adams spectral sequences. As a first application, we compute the first 14 stems of the synthetic sphere spectrum. This computes the first 14 stems of the homotopy groups of the sphere spectrum, but also keeps track of the filtration and the differentials that arise. Secondly, we briefly discuss the benefit of computing Toda brackets synthetically rather than for spectra. Lastly, we give an overview of the approach used by Isaksen-Wang-Xu to use MU-synthetic spectra to compute the Adams spectral sequence through to the 90-stem.

v0.0: I did not yet have the time to type this lecture, so I attached my handwritten notes for now. An Adams chart is provided on the last page of the document.

Lecture 5: Obstruction theories | Notes v0.0
26 November

We set up an obstruction theory to answer the following type of question: if $M$ is an $E_*E$-comodule, when does there exist a spectrum $X$ with $E_*(X)\cong M$? This lecture is based on the papers by Pstrągowski-VanKoughnett and Barkan.

v0.0: I did not yet have the time to type this lecture, so I attached my handwritten notes for now.

Lecture 6: Construction and variants of synthetic spectra | Notes v0.0
3 December

The goal of this final lecture is to describe the construction of synthetic spectra, and to explore generalisations and variants of it. We begin by discussing a general version of Adams spectral sequences, based on adapted homology theories. This serves as an overall organising principle for later constructions: roughly, a synthetic category should categorify the Adams spectral sequence arising from an adapted homology theory. The most general construction of this sort we discuss is the derived $\infty$-category of a stable $\infty$-category with respect to an adapted homology theory. Most of this lecture is based on the paper by Patchkoria-Pstrągowski, while other parts use the original synthetic paper by Pstrągowski.

v0.0: I did not yet have the time to type this lecture, so I attached my handwritten notes for now.