Rigidification of quasi-categories
type: Item
instance of: scholarly article
title: Rigidification of quasi-categories
zbMATH Open document ID: 1213.55015
author: Daniel Dugger
author: David I. Spivak
published in: Algebraic \& Geometric Topology
publication date: 2011-01-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0910.0814
review text: Quasi-categories are simplicial sets where all inner horns admit a filler (whereas Kan complexes admit fillers for all horns). They provide a model for \((\infty, 1)\)-categories where one thinks of the vertices as objects and 1-simplices as morphisms. The inner horn fillers allows one to compose morphisms, but this is defined only up to homotopy. Lurie defines in [\textit{J. Lurie}, Higher topos theory. Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press. (2009; Zbl 1175.18001)] a rigidification functor taking a quasi-category \(K\) to \(\mathfrak C(K)\), a category enriched over simplicial sets (a ``simplicial category''). The morphisms between each pair of objects form a simplicial set and composition is well-defined and associative. This functor is in fact left adjoint to the coherent nerve and part of a Quillen equivalence between the category of simplicial sets equipped with the Joyal model structure, where fibrant objects are quasi-categories, and the category of simplicial categories equipped with Bergner's model structure. The simplicial category \(\mathfrak C(\Delta^n)\) can be described or defined by nerves of posets and, since \(\mathfrak C\) is a left adjoint, \(\mathfrak C(S)\) is the colimit, taken over all simplices of the simplicial set \(S\), of simplicial categories \(\mathfrak C(\Delta^n)\). This formula does not give much concrete information because the colimit takes place in \(s\mathcal C at\)\dots The aim of the article is to provide such information by finding new models for the mapping spaces. The first observation is that the rigidification has a nerve description for necklaces, ordered wedges of \(\Delta^{n_i}\)'s where the final vertex of \(\Delta^{n_i}\) is glued to the initial one of \(\Delta^{n_{i+1}}\). The colimit over all simplices of \(S\) can thus be replaced by a colimit taken over certain morphisms from necklaces. A careful analysis of this colimit allows the authors to give a very explicit description …
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 55U40
Mathematics Subject Classification ID: 18B99
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 5837752
zbMATH Keywords: quasi-category
zbMATH Keywords: infinity category
zbMATH Keywords: rigidification
zbMATH Keywords: mapping space
zbMATH Keywords: simplicial category
zbMATH Keywords: necklace
MaRDI profile type: Publication
OpenAlex ID: W2085018767
arXiv ID: 0910.0814
cites work: Etale homotopy
cites work: Geometry of loop spaces and the cobar construction
cites work: A model category structure on the category of simplicial categories
cites work: Homotopy limits, completions and localizations
cites work: Q3956094
cites work: Vogt's theorem on categories of homotopy coherent diagrams
cites work: Function complexes in homotopical algebra
cites work: Q4790109
cites work: Quasi-categories and Kan complexes
cites work: Higher Topos Theory (AM-170)
cites work: Q4215784
cites work: Higher algebraic K-theory: I
DOI: 10.2140/AGT.2011.11.225
Recommended article: Rigidification of higher categorical structures
Recommended article: Q5662071
Recommended article: Degeneracies in quasi-categories
Recommended article: Rigid subcategories and related subcategories in extriangulated categories
Recommended article: On quasivarieties and varieties as categories
Recommended article: Categorical quasivarieties revisited
Recommended article: Quasi-uniform and syntopogenous structures on categories
Recommended article: Pre-rigid monoidal categories
Recommended article: A structural aspect of the category of quandles
Recommended article: Functors on the category of quasi-fibrations
description: scientific article