Contents
Definition
Definition
The category of cubical sets is the free co-completion of , the category of cubes.
Notation
We denote the category of cubical sets by .
Definition
A cubical set is an object of .
Definition
A morphism of cubical sets is an arrow of .
Monoidal structure
The strict monoidal structure of gives rise to a (non-strict) monoidal structure on , by Day convolution. The unit of the monoidal structure is , in the notation of Notation . Whenever we use the symbol when working with cubical sets or morphisms of cubical sets, we shall always refer to the functor of this monoidal structure.
Notation
Free standing -cube, and an -cube of a cubical set
Notation
Let denote the Yoneda embedding functor. Let be an integer. We denote the cubical set by .
Terminology
We refer to as the free-standing -cube.
Terminology
Let be a cubical set. Let be an integer. By an -cube of , we shall mean a morphism of cubical sets .
Notation
Let be a 1-cube of . We shall often depict as or as follows.
In this case, is to be understood to be the -cube of , and is to be understood to be the -cube of .
Notation
Let be a 2-cube of . We shall often depict as follows.
In this case, is to be understood to be the -cube of , is to be understood to be the -cube of , is to be understood to be the -cube of , and is to be understood to be the 1-cube of .
It can be checked that this notation is consistent with Notation .
Boundary of the free standing -cube, and of an -cube of a cubical set
Notation
Let be an integer. We denote by the functor given by defined by , where is the - truncation functor for cubical sets, and is the - skeleton functor for cubical sets.
Terminology
Let be an integer. We refer to as the boundary of .
Notation
We also denote by (recalling that is, by construction, co-complete) the initial object of .
Notation
Let be an integer. We denote by the morphism of cubical sets corresponding, under the adjunction between and described at cubical truncation, skeleton, and co-skeleton, to the identity arrow in .
Terminology
Let be an integer, and let be an -cube of a cubical set . We refer to the morphism of cubical sets
as the boundary of .
Notation
Let be a 2-cube of as follows.
We shall often depict the boundary of as follows.
Horns of the free-standing -cube
Notation
We define inductively, for any integer , any integer , and any integer , a cubical set and a morphism of cubical sets .
When , we define both and to be . We define to be , and define to be .
Suppose that, for some integer , we have defined and a morphism of cubical sets for all integers , and all integers . For , we define (recalling that is co-complete by construction) to be a cubical set fitting into a co-cartesian square in as follows.
We denote by the canonical arrow determined, by means of the universal property of , by the following commutative square in .
We define to be a cubical set fitting into a co-cartesian square in as follows.
We denote by the canonical arrow determined, by means of the universal property of , by the following commutative square in .
Terminology
We refer to together with the morphism as a horn of .
Morphism from to
Notation
We denote by the arrow of , making use of the fact that is , since is the unit of the monoidal structure of .
Model structure
The category of cubical sets admits a Cisinski model structure, which admits a Quillen equivalence to the Kan–Quillen model structure on simplicial sets. See the article model structure on cubical sets for more information.
The category of cubical sets also admits a Joyal-type model structure, which admits a Quillen equivalence to the Joyal model structure on simplicial sets. See the article model structure for cubical quasicategories for more information.
Expository material
For Expository and other material, see cubical set - exposition.
General
Applications of cubical sets
In higher category theory
Theory of cubical sets
References
The original reference for cubical sets (based on the 1950 paper by Samuel Eilenberg and J. A. Zilber on simplicial sets) is
- Daniel M. Kan, Abstract homotopy. I, Proceedings of the National Academy of Sciences 41:12 (1955), 1092–1096. doi.
Kan switched to simplicial sets in Part III of the series.