Towards a General Description of Physical Invariance in Category Theory

Invariance is one of the most important notions in applications of mathematics. It is one of the key concepts in modern physics, is a computational tool that helps in solving complex equations, etc. In view of its importance, it is desirable to come up with a definition of invariance which is as general as possible. In this paper, we describe how to formulate a general notion of invariance in categorial terms. Invariance is important. Invariance is one of the most important concepts in applications of mathematics. In addition to its role as a computational tool in the solution of complex equations (see, e.g., [3, 5]), invariance (symmetry) is perhaps the most important notion in the conceptual foundations modern physics; see, e.g., [4, 11]. Invariance has a central role in contemporary metaphysics insofar as it relates to the problem of individuation. For example, Robert Nozick describes objectivity in terms of invariance under transformation and describes necessary truths as those which are invariant in all possible worlds [8]. While this paper treats invariance only in physical contexts, our analysis is conducted with an eye to basic metaphysical questions of the type that Nozick informally addressed. It is important to provide a general definition of invariance. Since the notion of invariance plays such a central role in foundational research, it is desirable to provide a formal definition of invariance which is as general as possible. In mathematics, such general definitions are usually provided by category theory; see, e.g., [1, 7]. In this paper, we will therefore attempt to describe how to formulate a general notion of invariance in categorial terms.