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We first compute James’ sectional category (secat) of the Ganea map gk of any map ιX in terms of the sectional category of ιX: we show that secat gk is the integer part of secat ιX/(k+1). Next we compute the relative category (relcat) of gk. In order to do this, we introduce the relative category of order k (relcatk) of a map and show that relcat gk is the integer part of relcatkιX/(k+1). Then we establish some inequalities linking secat and relcat of any order: we show that secat ιX⩽relcatkιX⩽secat ιX+k+1 and relcatkιX⩽relcatk+1ιX⩽relcatkιX+1. We give examples that show that these inequalities may be strict

Sectional Category of the Ganea Fibrations and Higher Relative Category