Tell me about transformation in a category theory and representation theory sense In both [[Category theory|category theory]] and [[Representation theory|representation theory]], [[Transformation|transformation]] refers to the process of relating different mathematical structures or objects. In category theory, a transformation is commonly referred to as a morphism or an arrow between two objects in a category. It provides a way to connect and compare different objects within the same category. For example, in the category of sets, a transformation can be seen as a function mapping elements from one set to another. In this context, transformations are used to study relationships and properties between objects in various mathematical disciplines. Representation theory, on the other hand, focuses on studying how abstract algebraic structures can be represented as linear transformations on vector spaces. In this context, transformations refer to linear maps that preserve certain algebraic properties of these structures. For example, in the representation theory of groups, transformations are used to study how group elements act on vector spaces through linear maps that preserve the group structure. Transformations also play a crucial role in connecting different areas of mathematics. For instance, in algebraic geometry, transformations between categories known as functors are used to relate geometric objects with algebraic structures. These functors provide a way to translate problems and results from one domain to another and establish connections between seemingly unrelated mathematical concepts. ## What is natural transformation? In category theory, a natural transformation is a fundamental concept that describes the relationship between two functors. ![[natural transformation]] # Conclusion Overall, transformation in both category theory and representation theory serves as a fundamental tool for understanding relationships between mathematical objects and structures across various mathematical disciplines. # References ```dataview Table title as Title, authors as Authors where contains(subject, "transformation") or contains(title, "transformation") or contains(subject, "representable") or contains(title, "representable") or contains(subject, "Representable") or contains(title, "Representable") ```