Infinity category theory from scratch
Emily Riehl, Dominic Verity

TL;DR
This paper develops a foundational framework for $ abla$-category theory using the concept of an $ abla$-cosmos, providing a unified, 2-categorical approach to various models of $ abla$-categories and their fundamental concepts.
Contribution
It introduces the notion of an $ abla$-cosmos and develops the basic theory of $ abla$-categories and $ abla$-functors within this framework, unifying multiple models and providing new categorical insights.
Findings
Recovers Joyal and Lurie's theory of quasi-categories within the $ abla$-cosmos framework.
Establishes a 2-categorical foundation for $ abla$-category theory.
Proves model independence of key concepts like limits, adjunctions, and the Yoneda lemma.
Abstract
We use the terms "-categories" and "-functors" to mean the objects and morphisms in an "-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, -spaces, and fibered versions of each of these are all -categories in this sense. We show that the basic category theory of -categories and -functors can be developed from the axioms of an -cosmos; indeed, most of the work is internal to a strict 2-category of -categories, -functors, and natural transformations. In the -cosmos of quasi-categories, we recapture precisely the same theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts. In the first…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology
