# 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}