Abstract
We introduce a framework for studying homotopy theory through neutrosophic multi-valued functions. The basic idea is to combine the set-valued nature of multivalued mappings with the three-component description of truth, indeterminacy, and falsity characteristic of neutrosophic set theory. A neutrosophic multi-valued function is represented by a family of nonempty subsets together with three membership profiles describing truth, indeterminacy, and falsity. We define neutrosophic continuity, neutrosophic homotopy, neutrosophic homotopy equivalence, neutrosophic selectors, and admissible neutrosophic multi-valued functions. We establish elementary properties of neutrosophic homotopy, including reflexivity, symmetry, transitivity, compatibility with composition, and product constructions. We further introduce a Vietoris-type representation and use it to formulate induced maps on homology. The resulting framework contains ordinary homotopy theory and ordinary multivalued homotopy theory as special cases. It also provides a natural setting for describing topological phenomena in which several possible images coexist with different degrees of truth, indeterminacy, and falsity. The paper concludes with a proposed neutrosophic homotopy category and several problems concerning fundamental groups, higher homotopy groups, fixed point theory, and categorical extensions.