In models of homotopy theory by homotopical categories it is a common concept that morphisms may be replaced by fibrant resolutions or cofibrant resolutions. These are in general far from being unique. Moreover, generic ways of constructing them, notably via a small object argument, yield resolutions which are “very large” in that they are relative cell complexes (or co-cell complexes, respectively) with large transfinite cardinality of the set of cell attachments.
In contrast, a minimal (co-)fibration is one which is “as small as possible” in some precise sense. Typically this minimality is witnessed by the property that any weak equivalence between minimal (co-)fibrations is already an isomorphism.
The discussion of the concept of minimal (co-)fibrations is often restricted to particular choices of ambient homotopical categories (see the list of Examples below), prominent examples being the minimal Kan fibrations in the classical model structure on simplicial sets and the minimal Sullivan algebras in the projective model structure on dgc-algebras. A general concept of minimal models is considered in Roig 93.
A general concept of minimal fibrations and minimal models (in general model categories) is discussed in