A short course on $\infty$-categories
Moritz Groth

TL;DR
This paper provides a non-technical overview of $$-categories, introducing foundational concepts and key structures like monoidal, stable, and algebraic $$-categories, to facilitate understanding of advanced topics in higher category theory.
Contribution
It offers an accessible survey of $$-categories, connecting foundational ideas with applications in spectra and derived algebraic geometry, based on the work of Joyal and Lurie.
Findings
Introduces basic $$-categorical notions
Discusses model structures for $(,1)$-categories
Explores applications in spectra and derived geometry
Abstract
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 -categories. We also discuss monoidal -categories and algebra objects, as well as stable -categories. These notions come together in Lurie's treatment of the smash product on spectra, yielding a convenient framework for the study of -ring spectra, -ring spectra, and Derived Algebraic Geometry.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Intracranial Aneurysms: Treatment and Complications · Algebraic structures and combinatorial models
