A promoter increases a catalyst's activity without itself being catalytically active; $Al_2O_3$ (with added $K_2O$ in practice) is the classic promoter for the iron catalyst used in Haber's ammonia synthesis, while $As_2O_3$ and $H_2S$ are poisons, not promoters.

In the industrial Haber process, $N_2+3H_2 \rightleftharpoons 2NH_3$, finely divided iron acts as the heterogeneous catalyst. Small amounts of certain oxides, called promoters (here mainly $Al_2O_3$, alongside a little $K_2O$), are mixed with the iron; these substances have no catalytic activity of their own but improve the iron catalyst's activity and selectivity — believed to work by increasing the surface area/porosity and preventing sintering (crystal growth) of the active iron particles at reaction temperature. By contrast, $As_2O_3$ and $H_2S$ a …

Acid-catalysed hydrolysis of sucrose is homogeneous because the $H^+$ catalyst and the dissolved sucrose are both in the same (aqueous) phase; the other three processes listed all use a solid catalyst acting on gaseous reactants, which is heterogeneous catalysis.

Hydrogenation of oil uses a solid Ni catalyst with liquid oil and gaseous $H_2$ — different phases, heterogeneous. Haber's process for ammonia uses solid finely-divided Fe catalyst with gaseous $N_2$/$H_2$ — heterogeneous. The Contact process for sulphuric acid uses solid $V_2O_5$ catalyst with gaseous $SO_2$/$O_2$ — heterogeneous. In contrast, inversion (hydrolysis) of sucrose, $C_{12}H_{22}O_{11} + H_2O \xrightarrow{H^+} C_6H_{12}O_6 + C_6H_{12}O_6$, …