  We describe a fully faithful embedding of the category of (reflexive)
globular sets into the category of counital cosymmetric $R$-coalgebras when $R$
is an integral domain. This embedding is a lift of the usual functor of
$R$-chains and the extra structure consists of a derived form of cup coproduct.
Additionally, we construct a functor from group-like counital cosymmetric
\mbox{$R$-coalgebras} to $\omega$-categories and use it to connect two
fundamental constructions associated to oriented simplices: Steenrod's cup-$i$
coproducts and Street's orientals. The first defines the square operations in
the cohomology of spaces, the second, the nerve of higher-dimensional
categories.
