A projective representation of a group is a representation up to a central term: a group homomorphism , where is a -vector space. Via the projection , every linear representation of induces a projective representation.
By the fibration sequence
the obstruction to lift a projective representation of to a linear representation is represented by an element in .
A refinement of this idea consists in looking at the 2-groupoid . Then a functor consists of two maps and such that , and is a 2-cocycle on with values in representing the cohomology class .