Categorical Homotopy Theory

Categorical Homotopy Theory studies the relationships between shapes, spaces, and transformations using the language of categories—abstract structures that capture relationships and mappings. It extends classical ideas of deformation (homotopy) to a broad, algebraic framework, allowing mathematicians to analyze complex topological features in a highly organized way. By focusing on how objects can be continuously transformed into one another within a category, this theory provides powerful tools to understand shape equivalences, invariants, and structural properties across diverse mathematical contexts.