[Paper Review] $\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories

Timothy Porter|ArXiv.org|Jan 21, 2004

Homotopy and Cohomology in Algebraic Topology30 references3 citations

TL;DR

This paper provides a comprehensive overview of $γ$-categories, $γ$-groupoids, Segal categories, and quasicategories, unifying classical abstract homotopy theory with modern simplicial methods. It establishes connections between Dwyer-Kan localizations, homotopy coherent nerves, and Tamsamani's weak $n$-categories, offering a framework for understanding higher categorical structures in homotopy theory through simplicial enrichment and coherence conditions.

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 Laguña, 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.

Motivation & Objective

• To reframe classical abstract homotopy theory using modern simplicial and enriched category tools, particularly $γ$-categories and quasicategories.
• To unify and clarify the relationships between key homotopy-theoretic structures: Dwyer-Kan localizations, Segal categories, and Joyal's quasicategories.
• To investigate how homotopy coherent nerves and higher coherence data (e.g., from simplicial groups) model higher-dimensional algebraic structures.
• To explore the potential of Tamsamani's weak $n$-categories as a foundation for higher stacks and homotopy types, especially in the context of the Grothendieck program.
• To identify open problems and conceptual gaps in the coherence and comparison of higher categorical models, especially in relation to model categories and geometric intuition.

Proposed method

• Uses simplicially enriched categories ($γ$-categories) as the foundational framework, where hom-sets are simplicial sets satisfying coherence axioms.
• Applies the Dwyer-Kan hammock localization to construct $γ$-groupoids from categories with weak equivalences, capturing homotopy types.
• Introduces the homotopy coherent nerve construction to relate $γ$-categories to quasicategories, generalizing the classical nerve functor.
• Employs Segal categories as models for $(∞,1)$-categories via the Segal condition on multi-simplicial sets.
• Utilizes Tamsamani's construction of weak $n$-categories via multi-simplicial sets and their Poincaré groupoid analogs.
• Relies on the Dold-Kan correspondence and simplicial resolutions to model chain complexes and crossed complexes as $γ$-categories.

Experimental results

Research questions

• RQ1 What is the precise relationship between the Dwyer-Kan $γ$-groupoid and homotopy coherent simplices, and how do they encode higher interchange laws?
• RQ2 Can the Tamsamani construction be generalized to non-space inputs (e.g., simplicial groups or categories), and what do the resulting Poincaré weak $n$-groupoids reveal?
• RQ3 How closely related is the Dwyer-Kan hammock localization of $β^{ℕ}$ to the mapping space $\mathcal{S}(S(ℕ), Ner_{h.c.}(\mathcal{B}))$ when $\mathcal{B}$ is locally Kan?
• RQ4 Can a homotopy coherent nerve be defined for general Segal categories, and what is the precise relationship between Segal categories and quasicategories?
• RQ5 Is it possible to construct a geometrically intuitive, non-technical framework for 2-stacks using Segal category machinery, avoiding excessive model category or $∞$-category machinery?

Key findings

• The homotopy coherent nerve of a $γ$-category yields a quasicategory, providing a bridge between simplicially enriched categories and Joyal’s $(∞,1)$-categories.
• The Dwyer-Kan hammock localization of a category with weak equivalences produces a $γ$-groupoid that models its homotopy category with coherent higher morphisms.
• Segal categories are equivalent to quasicategories in the homotopy category of $(∞,1)$-categories, establishing a key comparison result.
• Tamsamani’s weak $n$-categories generalize the notion of a bigroupoid and can be seen as a multi-simplicial model for higher groupoids with coherent composition.
• The diagonal and Artin-Mazur codiagonal of a Tamsamani weak $n$-category may relate to hypercrossed complexes, suggesting a link to non-abelian homotopy theory.
• The paper identifies a conceptual gap between technical machinery (e.g., model categories) and geometric intuition in higher stack theory, advocating for simpler, more intuitive frameworks.