  We define a comonad cohomology of track categories and we show it is linked
by a long exact sequence to its Dwyer-Kan-Smith cohomology . Under mild
hypothesis on the track category, we show that its comonad cohomology
coincides, up to dimension shift, with its Dwyer-Kan-Smith cohomology,
therefore obtaining an algebraic formulation of the latter. We also specialize
our results to the case where the track category is a $2$-groupoid.
