New York Journal of Mathematics
Volume 25 (2019), 627-641


Krzysztof Kapulkin, Zachery Lindsey, and Liang Ze Wong

A co-reflection of cubical sets into simplicial sets with applications to model structures

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Published: August 9, 2019.
Keywords: cubical sets, simplicial sets, model categories, (∞,1)-categories.
Subject: Primary: 55U35, Secondary: 18G55, 55U40.

We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can transfer any cofibrantly generated model structure in which cofibrations are monomorphisms to cubical sets, thus obtaining cubical analogues of the Quillen and Joyal model structures.


We wish to thank Christian Sattler and the anonymous referee for helpful comments.

Author information

Krzysztof Kapulkin:
Middlesex College 116
London, Ontario N6A 5B7, Canada


Zachery Lindsey:
Middlesex College 133
London, Ontario N6A 57B, Canada


Liang Ze Wong:
Department of Mathematics
University of Washington
C-34 Padelford Hall, Seattle, WA 98195-4350, USA