Characterizations of modalities and lex modalities

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:226

Article

Christensen, J. Daniel¹  Rijke, Egbert² 

[1] Univ Western Ontario, Dept Math, London, ON, Canada

[2] Univ Ljubljana, Dept Math, Jadranska Ul 21, Ljubljana 1000, Slovenia

关键词: Reflective subuniverse;    Localization;    Modality;    Lex modality;    Factorization system;    Homotopy type theory;   

DOI  :  10.1016/j.jpaa.2021.106848

来源: Elsevier

【 摘 要 】

A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of infinity-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse L: (1) the associated subuniverse L' of L-separated types is a modality; (2) L is a modality; (3) L is a lex modality; and (4) L is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to L, such as the L-etale maps, the L equivalences, the L-local maps, the L-connected maps, the unit maps eta(X), and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free