Extension (simplicial set)

In higher category theory in mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty and strong applications in homotopical algebra. Among the most well-known is its application in the construction of Kan complexes from arbitrary simplicial sets, which often enables without loss of generality to take the former for proofs about the latter. It is furthermore very well compatible with the Kan–Quillen model structure and can for example be used to explicitly state its factorizations or to search for weak homotopy equivalences.