boundary of a simplex
Paths and cylinders
In as far as the simplicial -simplex (a simplicial set) is a combinatorial model for the -ball, its boundary is a combinatorial model for the -sphere.
The boundary of the simplicial -simplex is the simplicial set generated from the simplicial set minus its unique non-degenerate cell in dimension .
This may equivalently be described to be degreewise the coequalizer
defined by the (induced coproduct maps of the) simplicial identities .
Regarding as the presheaf on the simplex category that is represented by , then this means that is the simplicial set generated from minus the identity? morphism .
There is a canonical monomorphism
the boundary inclusion .
The geometric realization of this is the inclusion of the -sphere as the boundary of the -disk.
Simplicial boundary inclusions are one part of the cofibrant generation of the classical model structure on simplicial sets.
For low the boundaries of -simplices look as follows (see also the illustrations at oriental)
Revised on March 12, 2012 21:14:27
by Urs Schreiber