  For any collection of spaces A, we investigate two non-negative integer
homotopy invariants of maps: l_A(f), the A-cone length of f, and L_A(f), the
A-category of f. When A is the collection of all spaces, these are the cone
length and category of f, respectively, both of which have been studied
previously. The following results have been obtained: (1) For a map of one
homotopy pushout diagram into another, we derive an upper bound for I_A and L_A
of the induced map of homotopy pushouts in terms of I_A and L_A of the other
maps. This has many applications including an inequality for I_A and L_A of the
maps in a mapping of one mapping cone sequence into another. (2) We establish
an upper bound for I_A and L_A of the product of two maps in terms of I_A and
L_A of the given maps and the A-cone length of their domains. (3) We study our
invariants in a pullback square and obtain as a consequence an upper bound for
the A-cone length and A-category of the total space of a fibration in terms of
the A-cone length and A-category of the base and fiber. We conclude with
several remarks, examples and open questions.
