Gavin J. Seal

Monoidal Topology: A Categorical Approach to Order, Metric, and Topology

Preface 1. Introduction Robert Lowen and Walter Tholen 2. Monoidal structures Gavin J. Seal and Walter Tholen 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen Bibliography Tables Index.

A Kleisli-based Approach to Lax Algebras

By exploiting the description of topological spaces by either neighborhood systems or filter convergence, we obtain a neighborhood-like presentation of categories of lax algebras. The simplicity of this presentation pinpoints the importance of the Kleisli extension, which is introduced as a particular lax extension of the associated monad functor.

Cartesian Closed Topological Categories and Tensor Products

Abstract The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X.

Exponential Kleisli Monoids as Eilenberg-Moore Algebras

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Free Modules over Cartesian Closed Topological Categories

Abstract The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category.

Embeddings of polar spaces

In this article, two different notions of embeddings of polar spaces are compared. By using existing results in the field, a statement for a Fundamental Theorem of Polar Geometry is then obtained.