Relative category and monoidal topological complexity
  If a map $f$ has a homotopy retraction, then Doeraene and El Haouari conjectured that the sectional category and the relative category of $f$ are the same. In this work we discuss this conjecture for some lower bounds of these invariants. In particular, when we consider the diagonal map, we obtain results supporting Iwase-Sakai's conjecture which asserts that the topological complexity is the monoidal topological complexity. 
J.G. Carrasquel-Vera, J.M. Garc\'ia Calcines and L. Vandembroucq
math.AT