  We introduce the o-minimal LS-category of definable sets in o-minimal
expansions of ordered fields and we establish a relation with the semialgebraic
and the classical one. We also study the o-minimal LS-category of definable
groups. Along the way, we show that two definably connected definably compact
definable groups G and H are definable homotopy equivalent if and only if L(G)
and L(H) are homotopy equivalent, where L is the functor which associates to
each definable group its corresponding Lie group via Pillay's conjecture.
