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When building a cold frame, do you leave the ground inside clear or do you cover it with a mat something ?

Depends on what you want to use it for. If you are using it just to start off/protect seedlings until its time to plant out then I would cover the floor to stop slugs getting to the plants (unless its on a hard surface already).
Some people use cold frames for growing tender crops or very early/late crops - in that case you would be wanting to grow things in the earth...

Lining the inside with a membrane covered with pea-shingle is recommended by many.

That's what we'll be doing in a week or so's time when we get round to chippery-pokery  :)