• Corpus ID: 119320216

Composition + Homotopy = Cubes

@article{Grignou2018CompositionH,
  title={Composition + Homotopy = Cubes},
  author={Brice Le Grignou},
  journal={arXiv: Category Theory},
  year={2018}
}
The goal of this article is to emphasize the role of cubical sets in enriched categories theory and infinity-categories theory. We show in particular that categories enriched in cubical sets provide a convenient way to describe many infinity-categories appearing in the context of homological algebra. 

References

SHOWING 1-10 OF 19 REFERENCES
A cubical approach to straightening
For a suitable choice of the cube category, we construct a topology on it such that sheaves with respect to this topology are exactly simplicial sets (thus establishing simplicial sets as a
Univalent universes for elegant models of homotopy types
We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for
La catégorie cubique avec connexions est une catégorie test stricte
— The aim of this paper is to prove that the category of cubes with connections, introduced by R. Brown and Ph. J. Higgins, is a strict test category in Grothendieck’s sense. In particular this
On Combinatorial Model Categories
  • J. Rosický
  • Mathematics, Computer Science
    Appl. Categorical Struct.
  • 2009
TLDR
Some new results about homotopy equivalences, weak equivalences and cofibrations in combinatorial model categories are contributing to this endeavour by some new resultsabout homotope equivalences.
CATEGORICAL HOMOTOPY THEORY
This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The
A Model Structure for Enriched Coloured Operads
We prove that, under certain conditions, the model structure on a monoidal model category $\mathcal{V}$ can be transferred to a model structure on the category of $\mathcal{V}$-enriched coloured
On closed categories of functors
Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, concerning the relationship of Kan functor extensions to closed structures on functor
CUBICAL SETS AND THEIR SITE
Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite
Les Pr'efaisceaux comme mod`eles des types d''homotopie
Grothendieck introduced in Pursuing Stacks the notion of test category . These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes. A well
Cubical rigidification, the cobar construction and the based loop space
We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the
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