# 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

- Mathematics
- 2020

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

- Mathematics
- 2014

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

- Mathematics
- 2009

— 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

- Mathematics, Computer ScienceAppl. Categorical Struct.
- 2009

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

- Mathematics
- 2006

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

- Mathematics
- 2014

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

- Mathematics
- 1970

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

- Mathematics
- 2003

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

- Mathematics
- 2002

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

- MathematicsAlgebraic & Geometric Topology
- 2018

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…