Abstract: A dynamical system is a map of spaces $X \times S \to X$, and a map of dynamical systems $X \to Y$ over $S$ is an $S$-equivariant map. There is both an injective and a projective model structure for this category.

These model structures are special cases of injective and projective model structures for space-valued diagrams $X$ defined on a fixed category $A$ enriched in simplicial sets. Simultaneously varying the parameter category $A$ (or parameter space $S$) along with the diagrams $X$ up to weak equivalence is more interesting, and requires new model structures for $A$-diagrams having weak equivalences defined by homotopy colimits, as well as a generalization of Thomason's model structure for small categories to a model structure for simplicial categories.