We make use of the notation introduced in category of cubes and cubical set.
Let be an integer. We denote by the full sub-category of whose objects are , , , .
We refer to as the -truncated category of cubes.
Let be an integer. The inclusion functor canonically determines a functor . We shall denote this functor by .
We refer to as the -truncation functor.
Let be an integer. The category of -truncated cubical sets is the free co-completion of .
The free co-completion of a small category can be constructed as the category of presheaves presheaves of sets on this category. Thus we can also think of the category of -truncated cubical sets as the category of presheaves of sets on .
We denote the category of -truncated cubical sets by .
An -truncated cubical set is an object of .
When we think of the category of -truncated cubical sets as the category of presheaves of sets on , we consequently think of an -truncated cubical set as a presheaf of sets on .
A morphism of -truncated cubical sets is an arrow of .
Let be an -truncated cubical set. Let be an integer. By an -cube of , we shall mean an -cube of , where is -skeleton functor defined in Notation .
Let be an integer. By left Kan extension, the functor admits a left adjoint . We shall denote this functor by .
We refer to as the -skeleton functor.
Let be an integer. By right Kan extension, the functor admits a right adjoint . We shall denote this functor by .
We refer to as the -coskeleton functor.
Last revised on July 8, 2024 at 10:04:50. See the history of this page for a list of all contributions to it.