$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories

TL;DR

This paper explores advanced concepts in homotopy theory, including Segal categories, quasicategories, and their connections to classical ideas, aiming to clarify and unify various approaches in the field.

Contribution

It introduces Segal categories and Joyal's quasicategories, linking them with classical homotopy theories and providing new perspectives on their relationships.

Findings

Revisits classical homotopy and enriched categories.

Links Segal categories and quasicategories with older theories.

Raises open questions about the connections among these concepts.

Abstract

The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Lagu\~{n}a, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicial ly enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; (ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas; (iii) to introduce Joyal's quasicategories, and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and Porter; and finally to ask lots of questions of myself and of the reader.