Algebraic 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...

Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation f...

Recursion-Closed Algebraic Theories

A class of algebraic theories called “recursion-closed,” which generalize the rational theories studied by J. A. Goguen, J. W. Thatcher, E.G. Wagner and J. B. Wright in [in “Proceedings, 17th IEEE Symposium on Foundations of Computer Science, Houston, Texas, October 1976,” pp. 147-158; in “Mathematical Foundations of Computer Science, 1978,” Lecture Notes in Computer Science, Vol. 64, Springer-...

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

Algebraic Presentations of Dependent Type Theories

In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and different properties of the resulting theory may be deduced from properties of the basic ones. We define a category of algebraic dependent type theories which...

From torsion theories to closure operators and factorization systems

Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].