A Universe of Sorts

We need an ordered simplex, so there is a total ordering on the vertices. This is to split a chain apart at number $k$ .
Takes $i$ cocahins and $k$ chains to spit out a $k - i$ chain given by $\xi \frown \gamma \equiv \sum_a \gamma_a \xi (a_{\leq i}) a_{\geq i}$ .
The action of the boundary on a cap product will be $\partial (\xi \frown \gamma) \equiv (-1)^i [(\xi \frown \partial \gamma) - (\partial \gamma \frown \gamma)]$ .
Consequence: cocycle cap cycle is cycle.
coboundary cap cycle is boundary.
cocyle cap boundary is boundary.
Cap product will be zero if the chain misses the cochain.
Cap product will be nonzero if the chain must always intersect the cochain.
This is why it's also called as the intersection product, since it somehow counts intersections.

§ Cap Product [TODO ]