Benjamin Andersson: Towards Higher Algebra: A Journey from Homological Algebra to the Derived ∞-Category of R-Modules

Abstract: During recent times, (higher) category theory seem to have become increasingly relevant to know for anyone interested in more modern foundations of algebraic geometry, homotopy theory, and maybe other areas. One framework of interest might go under the name “Higher Algebra” or “Homotopical Algebra”. Motivated by our interest in this framework, we try to cover most of the homological algebra necessary for engagement with Lurie’s seminal work “Higher Algebra” ([Lur17]), by following part I of Aaron Mazel-Gee’s lecture notes “Higher Algebra: Chapter 0”. We start by the impetus for the move towards derived algebraic geometry. We then introduce important constructions such as chain complexes, chain-homotopies, quasi-isomorphisms and other concepts. We focus on how one may use homotopy co/kernels to glean information about a chain map f:M → N, and how one may use the de/suspension operator Σ together with homotopy co/kernels to create exact sequences, for an arbitrary chain map f. In the later part of the thesis, we say more about the dg-category of complexes of R-modules and how it can be viewed as enriched in a certain symmetric monoidal category Chk of complexes of k-modules with the tensor product of complexes. The last two chapters are devoted to resolutions and a brief introduction to k-linear ∞-categories. We end by saying something about more general ∞-categories, using the framework of quasicategories, and defining the derived ∞-category of R-modules as a certain ∞-categorical localization of the category ChR of chain complexes of R-modules.