Because we need to think about smaller animals coming in from above, ie, wild birds, squirrels, snakes (for eggs, for seeds, for water . . . whatever!) we're covering our combination coop/run top with hardware cloth. Actually, we're covering the whole run with hardware cloth, and what's not covered in wire are cedar siding walls. . .
Because we ALSO live in a climate that gets a ton of snow and rain (and other interesting weather, like tornadoes!), we are putting a true roof on over the hardware cloth using 2x8x8 boards and Sun Tuf polycarbonate sheets (the manufacturer claims they're 110% "indestructable no matter what") that will cut down on UV exposure as well as provide protection from snow, rain and hail, primarily, though of course it's another layer of predator control too.
Ventilation of the coop will be provided by the space between the hardware cloth and the Sun Tuf sheets, basically, the entire 6x3 top of the coop will have a good few inches, we hope it's enough. If not, we can always cut windows out from the cedar!
The Sun Tuf sheets are expensive ($25/sheet, including their own spacers) but worth it. In fact, they are the only thing on this coop/run that we're buying new!
Whitewater