We establish a large class of homotopy coherent Morita-equivalences Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical correspondence between $\infty$-categories simplicial objects and connective chain complexes. Our results generalize many known 1-categorical equivalences such as classical cor...

The aim of this paper is to construct functors associated with topological groups as well as to investigate these functors. More precisely, we prove that for a given topological groups G there always exists a contravariant functor F (G) from the homotopy category of pointed topological spaces and homotopy classes of base point preserving continuous maps to the category of groups and homomorphis...

Homotopy type theory (HoTT) is a new branch of mathematics that connects algebraic topology with logic and computer science, and which has been proposed as a new language and conceptual framework for mathematical practice. Much of the power of HoTT lies in the correspondence between the formal type theory and ideas from homotopy theory, in particular the interpretation of types, tokens, and equ...