# 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

#### 12 Citations

t-structures on stable infinity-categories

- Mathematics
- 2020

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 reflective… Expand

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 results… Expand

Equivariant Gorenstein Duality

- Mathematics
- 2018

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 framework… Expand

On the embeddings of quasi-categories into prederivators

- Computer Science
- 2019

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

- 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]),
On… Expand

Coalgebras in the Dwyer-Kan localization of a model category

- Mathematics
- 2020

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

- Mathematics
- 2020

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 in… Expand

Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $\operatorname{S}$-constructions

- Mathematics
- 2019

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 higher… Expand

DERIVATORS, POINTED DERIVATORS, AND STABLE DERIVATORS

- 2012

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, the… Expand

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