Simplicial sets in topology, category theory, and beyond
type: Item
instance of: scholarly article
title: Simplicial sets in topology, category theory, and beyond
zbMATH Open document ID: 1506.18001
author: Julie Bergner
published in: La Matematica
publication date: 2022-12-08T00:00:00Z
review text: This is a motivated introduction to simplicial sets (ssets) with a view toward applications in higher category theory. It is suitable for students, or experts in distant branches, wishing to make a first acquaintance with the topic and to receive some pointers to current research. As such, the author begins with simplicial complexes, a more directly visual precursor to ssets, as models for topological spaces. The move to ssets is then made in two steps, by solving compatibility issues with limits and colimits, respectively. Namely, products of simplicial complexes and topological spaces are shown to be different, which motivates oriented simplicial complexes; then the latter are argued to model quotients poorly, which motivates ssets. A detailed and fully motivated exposition of ssets follows. The author then gives a quick introduction to basic category theory that reaches the definition of an sset as a presheaf on the simplex category. The singular-chains-geometric-realisation Quillen equivalence, and the fact that the singular complex of a space is a Kan complex, are discussed intuitively in low dimensions, without going into details. The paper then takes another categorical turn, embedding category theory into sset theory via the nerve construction. The definition of a quasi-category is motivated by the characterisation of nerves of categories by the version of the weak Kan property with unique lifts. The author ends by briefly mentioning some other developments that use simplicial objects. Overall, this is a worthwhile and timely addition to the literature as a well-written and undaunting source that builds up from lines and triangles to a well-motivated definition of quasi-categories, including a discussion of the topological as well as categorical aspects of the weak Kan property and its variants.
reviewed by: Ödül Tetik
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 18N50
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 7629584
zbMATH Keywords: simplicial sets
zbMATH Keywords: simplicial complexes
zbMATH Keywords: categories
zbMATH Keywords: topological spaces
MaRDI profile type: Publication
full work available at URL: s44007-022-00029-8
OpenAlex ID: W4295047917
Wikidata QID: Q114215842
cites work: A whirlwind tour of the world of (∞,1)-categories
cites work: A Survey of (∞, 1)-Categories
cites work: The Homotopy Theory of (∞,1)-Categories
cites work: Homotopy invariant algebraic structures on topological spaces
cites work: Q4896538
cites work: Homotopy theoretic methods in group cohomology
cites work: Survey article: an elementary illustrated introduction to simplicial sets
cites work: Simplicial homotopy theory
cites work: A short course on ∞-categories
cites work: Q4819371
cites work: Quasi-categories and Kan complexes
cites work: Some Recent Trends in Motivic Homotopy Theory
cites work: Higher Topos Theory (AM-170)
cites work: Q5550390
cites work: Q4485837
cites work: Category theory in context
DOI: 10.1007/S44007-022-00029-8
description: scientific article; zbMATH DE number 7629584