Simplicial Sets - Sage Code

Sage is mathematical software, it implements simplicial sets as follows:

Sage Code:


AbstractSimplex represents one simplex where the vertices are numbered sequentially from 1 unless there are degeneracies specified.


sage: from sage.topology.simplicial_set import AbstractSimplex, SimplicialSet
            sage: v = AbstractSimplex(0)
            sage: w = AbstractSimplex(0)
            sage: e = AbstractSimplex(1)
            sage: X = SimplicialSet({e: (v, v)})
            sage: Y = SimplicialSet({e: (w, w)})
This is a hierarchy:

SimplicialSet_finite inherits from SimplicialSet_arbitrary and GenericCellComplex

SimplicialSet_arbitrary inherits from Parent which makes them containers.


One way to do this is find all combinations and then filter out those that are not correctly ordered.

product(A, B) returns the same as ((x,y) for x in A for y in B).

see Python code here.

We can extend this from a product to a pullback with further filtering.

Other Sage Code


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see also:

See Sage:

Correspondence about this page

Book Shop - Further reading.

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flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.


  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.


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