The long exact sequence of homotopy n-groups

Abstract

AbstractWorking in homotopy type theory, we introduce the notion of n-exactness for a short sequence $F\to E\to B$ of pointed types and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequencethat is n-exact at $\| E\|_{n-1}$ . We explain how the indexing makes sense when interpreted in terms of n-groups, and we compare our definition to the existing definitions of an exact sequence of n-groups for $n=1,2$ . As the main application, we obtain the long n-exact sequence of homotopy n-groups of a fiber sequence.