The goal of these notes is to provide motivation for the construction of synthetic spectra. They were originally constructed in , but the story presented here is largely based on . This is aimed towards readers who are familiar with the formal properties of synthetic spectra and want to understand the construction of the category.
The main idea is that the category of synthetic spectra is the derived category of spectra with respect to -homology. To understand this perspective, we begin with the classical notion of the derived category of an abelian category.
Let be an abelian category. One motivation for the derived category is that quotienting often loses information, and we want to somehow remember that information. For example, the cokernel of the map is just itself, and the source has been completely forgotten. In the derived category, we want to “remember” this, and the cofiber of this map is the chain complex
We can think of this as freely adding cokernels, so that in the derived category, the cokernel does not lose information.
However, freely adding all cokernels is not quite what we want. For example, the cokernel of should still be , because this quotient does not lose any information. That is, we want short exact sequences to remain exact (fiber sequences) in the derived category. 1
Thus, the construction of the derived category breaks into two steps — freely add some colimits, and then force certain sequences to remain exact. The construction of synthetic spectra will follow similar footsteps, but with various modifications so that the resulting category has good categorical properties.