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We develop foundations for the category theory of $infty$-categories parametrized by a base $infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G$ a finite group when the base is chosen to be the orbit category of $G$. We apply this theory to show that the $G$-$infty$-category of $G$-spaces is freely generated under $G$-colimits by the contractible $G$-space, thereby affirming a conjecture of Mike Hill.
We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams.
We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a
We discuss an approach to the emph{covering} and emph{vanishing} theorems for the comparison map from bounded cohomology to singular cohomology, based on the observation that the comparison map is the coassembly map for bounded cohomology.
Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an algebraic pattern, bywhich we mean an $infty$-category equipped with a factorization system and a collection of elementary objects. Examples
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a higher homotopy structure for $X$ (in terms of chain complexes of spaces). This structure is then used to defi