Involutive weak globular higher categories: Leinster's approach

This dissertation explores an involutive version of the weak globular higher category concept, employing the framework introduced by Leinster T. The primary objective is to establish the cartesian nature of the free strict involutive globular higher category monad on the category of ω-globular sets. A detailed construction of an involutive globular operad, in relation to this monad, is explicitly presented. The concept of a Leinster’s contraction is then enhanced with the introduction of the involutive globular operad. Demonstrating the categorical properties, weestablish that the category of involutive globular operads-with-contraction admits an initial object. This initial object serves as the foundation for defining an involutive weak globular higher category, which is characterized as an algebra for the aforementioned initial involutive operad-with-contraction. The dissertation concludes with the provision of significant examples showcasing the diversity and relevance of involutive weak globular higher categories in variousmathematical contexts.