  In this paper we study the problem of determining the homology groups of a
quotient of a topological space by an action of a group. The method is to
represent the original topological space as a homotopy limit of a diagram, and
then act with the group on that diagram. Once it is possible to understand what
the action of the group on every space in the diagram is, and what it does to
the morphisms, we can compute the homology groups of the homotopy limit of this
quotient diagram.
  Our motivating example is the symmetric deleted join of a simplicial complex.
It can be represented as a diagram of symmetric deleted products. In the case
where the simplicial complex in question is a simplex, we perform the complete
computation of the homology groups with $\mathbb Z_p$ coefficients. For the
infinite simplex the spaces in the quotient diagram are classifying spaces of
various direct products of symmetric groups and diagram morphisms are induced
by group homomorphisms. Combining Nakaoka's description of the $\mathbb
Z_p$-homology of the symmetric group with a spectral sequence, we reduce the
computation to an essentially combinatorial problem, which we then solve using
the braid stratification of a sphere. Finally, we give another description of
the problem in terms of posets and complete the computation for the case of a
finite simplex.
