## § Cap product [TODO ]

• 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 $\partial (\xi \frown \gamma) \equiv (-1)^i [(\xi \frown \partial \gamma) - (\partial \gamma \frown \gamma)]$.