The homotopical structures we are studying are ‘categorically algebraic’, in the sense that they are based on endofunctors and ‘operations’ on them (natural transformations between their powers), much in the same way as in the theory of monads.

This is why such a structure can generally be lifted from a ground category A to a categorical construction on the latter, yielding a second category E equipped with a forgetful functor U : E → A, or with a family of functors Ui : E → A.

We treat thus: categories of diagrams and sheaves (Section 5.1), slice categories (Section 5.2), categories of algebras for a monad (Sections 5.3 and 5.4) and categories of differential graded algebras (Sections 5.5–5.7).

Applying – for instance – these results to dTop, the symmetric dIP4-homotopical category of d-spaces, we obtain that any category of diagrams dTopS and any slice category dTop\A or dTop/B is a symmetric dIP4-homotopical category (Sections 5.1.5 and 5.2.6). Furthermore, any category of sheaves Shv(S, dTop), over any site S, is a symmetric dP4-homotopical category (Section 5.1.5).

The same is also true of any category of algebras dTopT, for every monad T on dTop which is made consistent with the path functor, in a natural sense. This yields the homotopical structure of d-topological semigroups, or d-topological groups, or d-spaces equipped with an action of a fixed d-topological group, etc. (Section 5.4).

Similarly, the homotopy structure of Cat can be lifted to the category of strict monoidal categories (Section 5.4.6).