Related papers: $\mathcal{S}$-categories, $\mathcal{S}$-groupoids,…
Notes from lectures given at the Autumn School on Algebraic and Arithmetic Geometry at the Johannes Gutenberg-Universit\"at Mainz in October 2017.
These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand…
We survey some aspects of the theory of Hopf-Galois objects that may studied advantageously by using the language of cogroupoids. These are the notes for a series of lectures given at Cordoba University, may 2010. The lectures are part of…
Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the…
This Note describe my own recollection of the first 30 years of Category Theory, it is not the result of any historical investigation. The choice of concepts and its evaluation is my own, necessarily subjective. It follows a chronological…
These are the notes from lectures I gave at the Oberwolfach Seminar "Tensor Triangular Geometry and Interactions" which was held in October 2025. The aim of these notes is twofold: We develop notions of support for triangulated categories,…
These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different…
These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of…
These are lecture notes for a short course about spectral sequences that was held at M\'alaga, October 18--20 (2016), during the "Fifth Young Spanish Topologists Meeting". The approach was to illustrate the basic notions via fully computed…
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of…
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central…
These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the…
We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal…
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…
In this paper we study categorical properties of the category of abelian hypergroups that leads to the notion of hyper (almost) preadditive and hyper (almost) abelian categories. Our goal is to create a path towards a general theory of…
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…
This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…
We review recent developments in the theory of compact hyper-K\"ahler varieties, from the viewpoint of Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories. These notes originated from the lecture by the second…
We give an elementary introduction to the theory of triangulated categories covering their axioms, homological algebra in triangulated categories, triangulated subcategories, and Verdier localization. We try to use a minimal set of axioms…
The purpose of these notes is to give a categorical presentation/analysis of homotopy type theory. The notes are incomplete as they stand (October 2017). The chapter on univalent tribes is missing. The references are not always connected to…