In classical algebraic topology, homotopy equivalence between ‘spaces’ gives rise to a plain equivalence of their fundamental groupoids; therefore, the categorical skeleton provides a minimal model of the latter.

But the study of homotopy invariance in directed algebraic topology is far richer and more complex. Our directed structures have a fundamental category ↑Π1 (X), and this must be studied up to appropriate notions of directed homotopy equivalence of categories, which are more general than categorical equivalence.

We shall use two (dual) directed notions, which take care, respectively, of variation ‘in the future’ or ‘from the past’: a future equivalence in Cat is a future homotopy equivalence (Section 1.3.1) satisfying two conditions of coherence; it can also be seen as a symmetric version of an adjunction, with two units. Its dual, a past equivalence, has two counits. Then we study how to combine these two notions, so as to take into account both kinds of invariance. Minimal models of a category, up to these equivalences, are then introduced to better understand the ‘shape’ and properties of the category we are analysing, as well as of the process it represents.

Within category theory, the study of future (and past) equivalences is a sort of ‘variation on adjunctions’: they compose as the latter (Section 3.3.3) and, moreover, two categories are future homotopy equivalent if and only if they can be embedded as full reflective subcategories of a common one (Theorem 3.3.5); therefore, a property is invariant for future equivalences if and only if it is preserved by full reflective embeddings and by their reflectors.