# A short course on ∞-categories

@inproceedings{Groth2020ASC,
title={A short course on ∞-categories},
author={Moritz Groth},
year={2020}
}
In this short survey we give a non-technical introduction to some main ideas of the theory of ∞-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides the basic ∞-categorical notions leading to presentable ∞-categories, we mention the Joyal and Bergner model structures organizing two approaches to a theory of (∞, 1)-categories. We also discuss monoidal ∞-categories and algebra objects, as well as stable ∞-categories. These notions come together in… Expand
t-structures on stable infinity-categories
The present work re-enacts the classical theory of t-structures reducing the classical definition given in *Faisceaux Pervers* to a rather primitive categorical gadget: suitable reflectiveExpand
Tilting theory via stable homotopy theory
• Mathematics
• 2018
We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these resultsExpand
Equivariant Gorenstein Duality
This thesis concerns the study of two flavours of duality that appear in stable homotopy theory and their equivariant reformulations. Concretely, we look at the Gorenstein duality frameworkExpand
On the embeddings of quasi-categories into prederivators
This thesis describes how to construct two kinds of embeddings of the former into the latter and sees that there exists a model structure on prederivators which is equivalent to the one presented in the first chapter. Expand
Higher symmetries in abstract stable homotopy theories
• Mathematics
• 2019
This survey offers an overview of an on-going project on uniform symmetries in abstract stable homotopy theories. This project has calculational, foundational, and representation-theoretic aspects,Expand
Quasi-categories vs. Segal spaces: Cartesian edition
• Nima Rasekh
• Mathematics
• Journal of Homotopy and Related Structures
• 2021
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), OnExpand
Coalgebras in the Dwyer-Kan localization of a model category
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories.Expand
The coalgebraic enrichment of algebras in higher categories
We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras inExpand
Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $\operatorname{S}$-constructions
These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higherExpand
DERIVATORS, POINTED DERIVATORS, AND STABLE DERIVATORS
We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. Stable derivators are shown to canonically take values in triangulated categories. Similarly, theExpand

#### References

SHOWING 1-10 OF 275 REFERENCES
A Survey of (∞, 1)-Categories
In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are allExpand
• Mathematics
• 2015
Abstract We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, ifExpand
∞-Categories for the Working Mathematician
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,Expand
ON AUTOEQUIVALENCES OF THE (∞, 1)-CATEGORY OF ∞-OPERADS
We study the (∞, 1)-category of autoequivalences of ∞-operads. Using techniques introduced by Toën, Lurie, and Barwick and Schommer-Pries, we prove that this (∞, 1)-category is a contractibleExpand
Whirlwind Tour of the World of ( ∞ , 1 )-categories
This introduction to higher category theory is intended to a give the reader an intuition for what (∞, 1)-categories are, when they are an appropriate tool, how they fit into the landscape of higherExpand
• Mathematics
• 2013
We introduce the dendroidal analogues of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to theExpand
Abelian Categories
Abelian categories are the most general category in which one can develop homological algebra. The idea and the name “abelian category” were first introduced by MacLane [Mac50], but the modernExpand
Homotopical algebraic geometry. I. Topos theory.
• Mathematics
• 2002
This is the rst of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this rst part we investigate a notion of higher topos.Expand
Spectra and symmetric spectra in general model categories
Abstract We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A 1Expand
Algebras and Modules in Monoidal Model Categories
• Mathematics
• 1998
In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictlyExpand