higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A projective space is a quotient
by the action of a multiplicative group object (for a given line object).
If instead of forming the quotient one forms the weak quotient/action groupoid, one speaks of the projective stack
The definition appears for instance as def 2.1.1 in