Functorial Semantics for Relational Theories

Functorial semantics of topological theories

Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology. The cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. We introduce t...

Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories

Functorial Semantics for Higher-order Logic Dissertation Abstract

This dissertation investigates what may be termed the model theory of higher-order logic using the methods of category theory. Of course, there is no such eld of logic as \higher-order model theory," and so our rst concern in chapter I will be to specify the basic objects under investigation, viz. higher-order logical theories and their models. This is a fairly straightforward generalization|in...

Functorial Data Migration

In this paper we present a simple database definition language: that of categories and functors. A database schema is a category and a state is a set-valued functor. We show that morphisms of schemas induce three “data migration functors” that translate states from one schema to the other in canonical ways. Database states form a topos of which the classical “relational algebra” is a fragment. ...

Functorial Semantics of Second-Order Algebraic Theories

The purpose of this work is to complete the algebraic foundations of second-order languages from the viewpoint of categorical algebra as developed by Lawvere. To this end, this paper introduces the notion of second-order algebraic theory and develops its basic theory. A crucial role in the definition is played by the second-order theory of equality M, representing the most elementary operators ...