hombicategory

hombicategory is a term that can refer to a few related concepts within mathematics, primarily in the fields of category theory and algebraic topology. In its most general sense, it describes a category whose objects are related to higher dimensional structures, often involving simplices. A common interpretation is a category where the objects are simplices and the morphisms are face and degeneracy maps. This structure allows for the study of topological spaces through their homological properties.

More specifically, the term can appear in the context of simplicial sets. A simplicial set is a presheaf of sets on the simplex category, meaning it assigns a set to each simplex and functions to the structure-preserving maps between simplices. The simplex category itself is a key component in the construction of simplicial sets, and a "hombicategory" might be used to refer to this foundational category or related structures.

Another related idea is found in higher category theory, where "hombicategory" might allude to categories with a more complex internal structure, possibly involving objects that are themselves categories or higher dimensional structures. The focus is often on capturing aspects of dimension and structure beyond that of a standard category. The precise definition can vary depending on the specific research context, but the underlying theme involves categories that encode dimensional information, often through a relationship with simplices.