James reduced product
= James reduced product
{c}
{title2=$JX=\coprod_{k\geq0}X^k/\text{basepoint deletion}$}
Finite words in a based space, with occurrences of its basepoint deleted. Word concatenation defines a multiplication. For connected based CW spaces, the natural map to $\Omega\Sigma X$ is a <weak homotopy equivalence>. With free integral <homology>, the <Bott–Samelson theorem> identifies its <homology> algebra with the <tensor algebra> on reduced <homology>.