---
author:
- Timothy Porter
title: '$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories'
---

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 Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, , or any equivalent text if one can be found!

What do the notes set out to do?

“Aims and Objectives!” or should it be “Learning Outcomes”? {#aims-and-objectives-or-should-it-be-learning-outcomes .unnumbered}
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-   To revisit some oldish material on abstract homotopy and simplicially 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;

-   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;

-   To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his 