The mathematical basis of maps allow for a number of optimizations[[optimization]]s. If one has <code>map f . map g</code> ('.' is function application; <code>map f (map g xs)</code> is equivalent) then it is the same as the simpler <code>map (f . g)</code>; this particular simplification and optimization is a "map fusion". MapsMap functions can beand often are defined in terms of a [[Fold (higher-order function)|fold]] such as <code>foldr</code>, which means one can do a "map-fold fusion": <code>f z . map g</code> is equivalent to <code>foldr (f . g) z</code>.
