Higher category theory

higher category theory

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Higher morphisms (kk-morphisms)


In an nn-category, or most generally an \infty-category, there are many levels of morphism, parametrised by natural numbers. Those at level kk are called kk-morphisms or kk-cells.


All notions of higher category have kk-morphisms, but the shapes may depend on the model (or theory) employed.

For a simplicially based geometric model of higher categories, i.e., simplicial sets subject to some filler conditions, the kk-morphisms are literally kk-cells in the sense of a simplicial set. This applies for example to quasi-categories, weak nn-categories in the sense of Street, and the weak complicial sets of Verity. In other geometric models, based not on simplices but on other shapes such as opetopes (Baez-Dolan), multitopes (Hermida-Makkai-Power), or nn-disks (Joyal), a higher category is a presheaf

X:Shapes opSetX: Shapes^{op} \to Set

again subject to some filler conditions, and in each case kk-morphisms are elements of X(σ)X(\sigma) where σ\sigma is a shape of dimension kk. Still other shapes (e.g., cubes) are possible (see also n-fold category).

Many notions of algebraic higher category, such as those due to Batanin, Leinster, Penon, and Trimble, are algebras over certain monads acting on globular sets (such as those induced by globular operads), so that each higher category XX has an underlying globular set U(X)U(X). In that case, the kk-morphisms are the kk-cells of U(X)U(X). In such globularly based definitions, every kk-morphism ff has a (k1)(k-1)-morphism σf\sigma f as its source and a (k1)(k-1)-morphism τf\tau f as its target, and the source (k2)(k-2)-morphisms σσf\sigma \sigma f and στf\sigma \tau f must be the same, as must the target (k2)(k-2)-morphisms τσf\tau \sigma f and ττf\tau \tau f.

A 11-morphism may simply be called a morphism; a 00-morphism is an object.

For the purposes of negative thinking, it may be useful to recognise that every \infty-category has a (1)(-1)-morphism, which is the source and target of every object. (In the geometric picture, this comes as the (1)(-1)-simplex of an augmented simplicial set.)

every (non-empty? -David R) \infty-category

I think every. Up to equivalence, a kk-morphism in CC is given by a functor from the oriented kk-simplex to CC. As the (1)(-1)-simplex is empty, there is a unique such functor for every CC; thus every CC has a unique (1)(-1)-morphism.

Also note that every kk-morphism has k+1k + 1 identity (k+1)(k+1)-morphisms, which just happen to all be the same (which can be made a result of the Eckmann–Hilton argument). Thus, the (1)(-1)-morphism has 00 identity 00-morphisms, so we don't need any object. (This confused me once.)

Toby Bartels

Revised on October 17, 2011 21:16:13 by Urs Schreiber (