Functorial ML

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...

A Functional, Functorial Approach to the Model-View-Controller Software Architecture for Command-oriented Programs in ML

We describe a functional, functorial approach to the modelview-controller software architecture for command-oriented programs in ML. In our approach, a controller’s algorithm is apparent; it doesn’t have to be mentally pieced together from a set of event handlers. Furthermore, what would be the state of a model (domain-specific part of the program) in an object-oriented setting becomes argument...

A Functorial Approach to the Infinitesimal Theory of Groupoids

Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere’s functorial notion of algebraic theory and Ehresmann’s functorial notion of theory called sketc...

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

Functorial Quasi-Uniformities on Frames

In this paper we present a unified study of functorial frame quasiuniformities by means of Weil entourages and frame congruences. In particular, we use the pointfree version of the Fletcher construction, introduced by the authors in a previous paper, to describe all functorial transitive quasi-uniformities.

Suspension Splittings and Hopf Invariants for Retracts of the Loops on Co-h-spaces

James constructed a functorial homotopy decomposition ΣΩΣX ' W∞ n=1 ΣX (n) for path-connected, pointed CW -complexes X. We generalize this to a functorial decomposition of ΣA where A is any functorial retract of a looped co-H space. This is used to construct Hopf invariants in a more general context. As well, when A = ΩY is the loops on a co-H space, we show that the wedge summands of ΣΩY furth...