Ordered products and based transport

A chronological chart path is $f_0,f_1,\dots ,f_m$, with each overlap component specified. The last transition acts on the left:

The based-loop package supplies a common collar basepoint and affine frame, a chosen connecting path to each loop, its transport, and a marked primitive positive peripheral class. A connecting path is data, not a topologically unique path.

If $Q$ transports the common reference coordinates into the starting chart and $W_f$ is the chart-based loop word, then

Peripheral equality always means equality after transport to the same basepoint/frame.