# Categorical product of simplicial complexes

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 *topological*product 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.

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

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.