Catalog of simplicial sets#

This provides pre-built simplicial sets:

  • the \(n\)-sphere and \(n\)-dimensional real projective space, both (in theory) for any positive integer \(n\). In practice, as \(n\) increases, it takes longer to construct these simplicial sets.

  • the \(n\)-simplex and the horns obtained from it. As \(n\) increases, it takes much longer to construct these simplicial sets, because the number of nondegenerate simplices increases exponentially in \(n\). For example, it is feasible to do simplicial_sets.RealProjectiveSpace(100) since it only has 101 nondegenerate simplices, but simplicial_sets.Simplex(20) is probably a bad idea.

  • \(n\)-dimensional complex projective space for \(n \leq 4\)

  • the classifying space of a finite multiplicative group or monoid

  • the torus and the Klein bottle

  • the point

  • the Hopf map: this is a pre-built morphism, from which one can extract its domain, codomain, mapping cone, etc.

  • the complex of a group presentation.

All of these examples are accessible by typing simplicial_sets.NAME, where NAME is the name of the example. Type simplicial_sets.[TAB] for a complete list.

EXAMPLES:

sage: RP10 = simplicial_sets.RealProjectiveSpace(8)                                 # needs sage.groups
sage: RP10.homology()                                                               # needs sage.groups sage.modules
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: C2, 6: 0, 7: C2, 8: 0}

sage: eta = simplicial_sets.HopfMap()
sage: S3 = eta.domain()
sage: S2 = eta.codomain()
sage: S3.wedge(S2).homology()                                                       # needs sage.modules
{0: 0, 1: 0, 2: Z, 3: Z}

>>> from sage.all import *
>>> RP10 = simplicial_sets.RealProjectiveSpace(Integer(8))                                 # needs sage.groups
>>> RP10.homology()                                                               # needs sage.groups sage.modules
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: C2, 6: 0, 7: C2, 8: 0}

>>> eta = simplicial_sets.HopfMap()
>>> S3 = eta.domain()
>>> S2 = eta.codomain()
>>> S3.wedge(S2).homology()                                                       # needs sage.modules
{0: 0, 1: 0, 2: Z, 3: Z}