nLab join of simplicial sets
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Contents Idea The join of a pair of simplicial set (Ehlers 1993) is a simplicial set that may geometrically be thought of as a cone over with tip of shape . Topologically, it can also be thought of as the union of line segments connecting to if both are placed in general position, cf. join of topological spaces. The join operation on simplicial sets generalizes the historically earlier notion of join of simplicial complexes, cf. Spanier 1966. It is naturally discussed in view of augmented simplicial sets [Ehlers & Porter 2000] If the simplicial sets in question are quasi-categories, then their join produces the corresponding join of quasi-categories that underlies many constructions in higher category theory, such as a definition of limits in quasi-categories. Motivating examples When is the point, then the join is a genuine cone over . Or if is the discrete two-point space, the join is the suspension of . For example, consider the two cones over , the standard 2-simplex. The first picture represents , while the second represents . If you take two non-coplanar line segments in (such as and in the picture below), then join every point in one to every point in the other, you get a 3-simplex (the tetrahedron in the picture). You can think of this as being the union of all the cones on the first segment with cone points on the second one. We have that the join is .