  We present general techniques for constructing functorial factorizations
appropriate for model structures that are not known to be cofibrantly
generated. Our methods use "algebraic" characterizations of fibrations to
produce factorizations that have the desired lifting properties in a completely
categorical fashion. We illustrate these methods in the case of categories
enriched, tensored, and cotensored in spaces, proving the existence of
Hurewicz-type model structures, thereby correcting an error in earlier attempts
by others. Examples include the categories of (based) spaces, (based) G-spaces,
and diagram spectra among others.
