Thin elements and commutative shells in cubical \(\omega\)-categories. (English) Zbl 1064.18005
A cubical \(\omega\)-category is a cubical set together with some extra structure consisting of compositions and connections. A cubical \(\omega\)-category \({\mathcal X}\) is the nerve of a strict \(\omega\)-category \({\mathcal C}\) [F. A. Al-Agl, R. Brown and R. Steiner, Adv. Math. 170, 71–118 (2002; Zbl 1013.18003)]; in particular an \(n\)-dimensional element \(x\) in \({\mathcal C}\) produces an \(n\)-morphism \(x'\) in \({\mathcal X}\), and an \(n\)-shell \(y\) in \({\mathcal C}\) produces a pair of \((n-1)\)-morphisms \((y',y'')\) in \({\mathcal X}\). An \(n\)-dimensional element \(x\) is thin if \(x'\) is an identity; an \(n\)-shell \(y\) is commutative if \(y'=y''\).
The present paper describes thinness and commutativity in terms of the cubical operations and gives direct proofs of various results about them: a commutative shell has a unique thin filler; a composite of commutative shells is commutative; an element is thin if and only if it is a composite of degeneracies and connections. Given a cubical \(n\)-category \({\mathcal C}\) whose \((n-1)\)-skeleton has connections, it is shown that connection operations on \({\mathcal C}\) are equivalent to families of fillers for commutative \(n\)-shells; these families of fillers are called thin structures.
The present paper describes thinness and commutativity in terms of the cubical operations and gives direct proofs of various results about them: a commutative shell has a unique thin filler; a composite of commutative shells is commutative; an element is thin if and only if it is a composite of degeneracies and connections. Given a cubical \(n\)-category \({\mathcal C}\) whose \((n-1)\)-skeleton has connections, it is shown that connection operations on \({\mathcal C}\) are equivalent to families of fillers for commutative \(n\)-shells; these families of fillers are called thin structures.
Reviewer: Richard John Steiner (Glasgow)
MSC:
18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |