For any collection of spaces ${\cal A}$, we investigate
two non-negative integer homotopy invariants of maps: $L_{\cal A}(f)$,
the ${\cal A}$-cone length of $f$, and
${\cal L}_{\cal A}(f)$, the ${\cal A}$-category of $f$. When ${\cal A}$ 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 are obtained:
(1) For a map of one homotopy pushout diagram into another, we derive
an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the induced map of homotopy
pushouts in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the other maps. This has
many applications, including an inequality for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the
maps in a mapping of one mapping cone sequence into another.
(2) We establish an upper bound for $L_{\cal A}$ and ${\cal L}_{\cal A}$ of the product
of two maps in terms of $L_{\cal A}$ and ${\cal L}_{\cal A}$
of the given maps and the ${\cal A}$-cone length of their domains.
(3) We study our invariants in a pullback square and obtain as a
consequence an upper bound for the ${\cal A}$-cone length and ${\cal A}$-category
of the total space of a fibration in terms of the
${\cal A}$-cone length and ${\cal A}$-category of the base and fiber.
We conclude with several remarks, examples and open questions.