Playlist - Higher Algebra

In this video, we introduce ∞-categories. This is the first of a series of videos towards a reasonably non-technical overview over stable ∞-categories and Higher Algebra.

In this video, we discuss limits in ∞-categories.

In this video, we discuss colimits and decomposition of those in ∞-categories.

In this video, we construct the ∞-categorical refinement of the derived category of an abelian category.

In this video, we provide further properties of the derived category of an abelian category. Along the way we discuss slice categories and filtered colimits.

In this video, we define and discuss derived functors between derived categories of abelian categories. Additionally we discuss the notion of adjoint functors and Kan extensions.

In this video, we discuss the notion of non-abelian derived functors and Animation. Along the way, we discuss the Yoneda lemma.

In this video, we introduce and discuss spectra (in the sense of homotopy theory). We explain how they generalise abelian groups.

In this video, we introduce the notion of a symmetric monoidal ∞-categories and give some examples.

In this video we introduce E_n-algebras in arbitrary symmetric monoidal ∞-categories. These interpolate between associated algebras (=E₁) and commutative algebras (= E_∞). We also establish some categorical properties and investigate the case of the symmetric monoidal ∞-category of spaces.

In this video we introduce the notion of p-adic completion and p-adic equivalence of spectra. We characterize those notions in concrete terms and give examples. Finally we cover the Hasse-square, which can be used to recover X from it completions and its rationalization.

In this video we introduce the Tate construction and especially Tate spectra. This is defined as the cofibre of a certain norm map, which we introduced for completely general group objects and stable ∞-categories. We then also explain what it has to do with Poncaré duality and that these constructions are lax symmetric monoidal.

In this video we discuss the Tate diagonal, which is a surprising feature of the world of spectra.