Thomas Weighill (MSc Mathematics).

Linking Database Interoperability to Categorica Algebra through Grothendieck Fibrations

I am a Masters student in Mathematics at the MIH Media Lab. I am looking at a way to link the problem of database interoperability to the field of categorical algebra using a concept called a Grothendieck fibration.

The problem of whether and how databases can interoperate is an important one in the context of the large amount of heterogenous data available in both the internet and in large corporate information systems. Here, “heterogenous” means based on different ontologies or organisational templates.

This problem is very closely related to the classical problem in database theory of view updatability. Recently, view updatability has been formally defined through the use of category theory, a field of mathematics which (broadly speaking) studies structures through relationships between them. This definition relies on a fundamental concept in category theory called a Grothendieck fibration.

Grothendieck fibrations were introduced to develop descent theory in algebraic geometry, but they also have other applications in mathematics. One relatively unexplored area where they may be used is in the study of the category of groups (a group is an important algebraic structure in mathematics).

In order to understand and develop the use of Grothendieck fibrations in databases, it is necessary to understand and develop them in mathematics. In fact, it turns out that the type of Grothendieck fibrations that illuminate the category of groups is very similar to the type of fibrations found in the new approach to view updatability.

My goal is to examine these two applications of fibrations: to databases and to algebra. While these two applications seem very far apart in terms of their respective fields, they share many characteristics when viewed from the perspective of Grothendieck fibrations.

I hope to exploit this connection to develop useful theory in both areas. In databases, this will go a long way towards developing a sound mathematical framework for linking database systems.

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