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Categorical product of simplicial complexes

asked 2014-07-02 17:25:08 -0500

Does Sage have some function that takes the categorical product of two finite simplicial complexes? I see that the SimplicialComplex library has a product() function that appears to take the topologicalproduct of two complexes (details here: http://www.sagemath.org/doc/reference...), but this isn't what I'm looking for.

The example given on the linked webpage is Simplex(1).product(Simplex(1)), which returns [('L0R0', 'L0R1', 'L1R1'), ('L0R0', 'L1R0', 'L1R1')], or a square with a diagonal through it. This is what I would expect from a topological product, since the product of two lines is a square. However, the categorical product of two complexes is different and is in general not homeomorphic to their topological product. The categorical product of two edges (1-simplexes) should be a tetrahedron and not a square.

Is there a Sage function that will do this for me? I'm not familiar with the markdown syntax on this forum so sorry about the poor formatting.

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The product of two 1-simplices isn't a 2-simplex (triangle)? Sorry if this is naive.

kcrisman gravatar imagekcrisman ( 2014-07-03 12:00:51 -0500 )edit

No, see my comment on John Palmieri's answer. The (categorical) product of two 1-simplexes is the standard simplex on (1+1)*(1+1) = 4 vertices, or the standard 3-simplex, or a tetrahedron.

Vikram Saraph gravatar imageVikram Saraph ( 2015-09-19 17:38:13 -0500 )edit

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answered 2014-07-03 11:13:24 -0500

There is no Sage function to do this. It shouldn't be hard to write, though. Can you provide a description of the categorical product, in terms of the (maximal) simplices of the two factors?

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I completely forgot I posted this question! Anyways, the categorical product of two complexes A and B is a complex with vertex set V(A) \times V(B), and a set of vertices S in V(A) \times V(B) is a simplex if pi_1(S) is a simplex in A and pi_2(S) is a simplex in B, where pi_i are projections onto each coordinate.

If A is the standard n simplex, and B is the standard m simplex, then A \times B is the standard (n+1)(m+1) - 1 simplex.

This definition can be found in Dmitry Kozlov's Combinatorial Algebraic Topology.

Vikram Saraph gravatar imageVikram Saraph ( 2015-09-19 17:36:28 -0500 )edit

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Asked: 2014-07-02 17:25:08 -0500

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Last updated: Jul 03 '14