====== BigObject QuotientSpace ======
//from application [[..:polytope|polytope]]//\\
\\
 A topological quotient space obtained from a ''[[..:polytope:Polytope |Polytope]]'' by identifying faces. This object will sit inside the polytope.

===== Properties =====

==== Basic properties ====
 Properties defining a quotient space.
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{{anchor:identification_action:}}
  ?  **''IDENTIFICATION_ACTION''**
  :: The group encoding the quotient space. The faces of the space are the orbits of the faces of the polytope under the group.
    ? Type:
    :''[[..:group:PermutationAction |PermutationAction]]<[[..:common#Int |Int]],[[..:common#Rational |Rational]]>''


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==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
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{{anchor:cocircuit_equations:}}
  ?  **''COCIRCUIT_EQUATIONS''**
  :: a SparseMatrix whose rows are the sum of all cocircuit equations corresponding to a fixed symmetry class of interior ridge
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''


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{{anchor:dim:}}
  ?  **''DIM''**
  :: The dimension of the quotient space, defined to be the dimension of the polytope.
    ? Type:
    :''[[..:common#Int |Int]]''


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{{anchor:faces:}}
  ?  **''FACES''**
  :: The faces of the quotient space, ordered by dimension. One representative of each orbit class is kept.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%>''


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{{anchor:face_classes:}}
  ?  **''FACE_CLASSES''**
  :: Some listing of equivalence classes of faces of the quotient space, ordered by dimension. Analogous to FACE_ORBITS, but not necessarily coming from a group
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%%%>>%%''


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{{anchor:face_orbits:}}
  ?  **''FACE_ORBITS''**
  :: The orbits of faces of the quotient space, ordered by dimension.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%%%>>%%''


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{{anchor:f_vector:}}
  ?  **''F_VECTOR''**
  :: An array that tells how many faces of each dimension there are
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


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{{anchor:n_simplices:}}
  ?  **''N_SIMPLICES''**
  :: The simplices made from points of the quotient space (also internal simplices, not just faces)
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


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{{anchor:representative_interior_ridge_simplices:}}
  ?  **''REPRESENTATIVE_INTERIOR_RIDGE_SIMPLICES''**
  :: The (//d//-1)-dimensional simplices in the interior.    
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''


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{{anchor:representative_max_boundary_simplices:}}
  ?  **''REPRESENTATIVE_MAX_BOUNDARY_SIMPLICES''**
  :: The boundary (//d//-1)-dimensional simplices of a cone of combinatorial dimension //d//
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''


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{{anchor:representative_max_interior_simplices:}}
  ?  **''REPRESENTATIVE_MAX_INTERIOR_SIMPLICES''**
  :: The interior //d//-dimensional simplices of a cone of combinatorial dimension //d//
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Bitset |Bitset]]<[[..:common#Int |Int]]%%>>%%''


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{{anchor:simplexity_lower_bound:}}
  ?  **''SIMPLEXITY_LOWER_BOUND''**
  :: A lower bound for the number of simplices needed to triangulate the quotient space
    ? Type:
    :''[[..:common#Int |Int]]''


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{{anchor:simplices:}}
  ?  **''SIMPLICES''**
  :: All simplices in the quotient space
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%>''


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{{anchor:simplicial_complex:}}
  ?  **''SIMPLICIAL_COMPLEX''**
  :: A simplicial complex obtained by two stellar subdivisions of the defining polytope.
    ? Type:
    :''[[..:topaz:SimplicialComplex |SimplicialComplex]]''


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{{anchor:symmetry_group:}}
  ?  **''SYMMETRY_GROUP''**
  :: The symmetry group induced by the symmetry group of the polytope on the ''[[..:polytope:QuotientSpace#FACES |FACES]]'' of the quotient space
    ? Type:
    :''[[..:group:Group |Group]]''


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