# 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}
```