  Symmetric function theory is a key ingredient in the Schubert calculus of
Grassmannians. Quasisymmetric functions are analogues that are similarly
central to algebraic combinatorics, but for which the associated geometry is
poorly developed. Baker and Richter (2008) showed that $\textrm{QSym}$
manifests topologically as the cohomology ring of the loop suspension of
infinite projective space or equivalently of its combinatorial homotopy model,
the James reduced product $J\mathbb{C}\mathbb{P}^\infty$. In recent work, we
used this viewpoint to develop topologically-motivated bases of $\textrm{QSym}$
and initiate a Schubert calculus for $J\mathbb{C}\mathbb{P}^\infty$ in both
cohomology and $K$-theory.
  Here, we study the torus-equivariant cohomology of
$J\mathbb{C}\mathbb{P}^\infty$. We identify a cellular basis and introduce
double monomial quasisymmetric functions as combinatorial representatives,
analogous to the factorial Schur functions and double Schubert polynomials of
classical Schubert calculus. We also provide a combinatorial
Littlewood--Richardson rule for the structure coefficients of this basis.
  Furthermore, we introduce an algebro-geometric analogue of the James reduced
product construction. In particular, we prove that the James reduced product of
a complex projective variety also carries the structure of a projective
variety.
