Page 109 - Textos de Matemática Vol. 46

FROM LAX MONAD EXTENSIONS TO TOPOLOGICAL THEORIES
MARIA MANUEL CLEMENTINO AND WALTER THOLEN
To Manuela, teacher and friend
Abstract. We investigate those lax extensions of a Set-monad T = (T,m,e) to the category V-Rel of sets and V-valued relations for a quan- tale V = (V,⌦,k) that are fully determined by maps ⇠ : TV ! V . We pay special attention to those maps ⇠ that make V a T-algebra and, in fact, (V, ⌦, k) a monoid in the category SetT with its cartesian structure. Any such map ⇠ forms the main ingredient to Hofmann’s notion of topological theory.
Introduction
The lax-algebraic setting, originally considered in [5] and [4] as a common syntax for the categories of lax algebras discussed in [2], was generalized by Seal in [9] and in this form adopted in [7] and studied by various authors. A very powerful specialization of the lax-algebraic setting was introduced by Hofmann [6] in the form of his topological theories, which in particular cover Barr’s pre- sentation of topological spaces [1] and the Clementino-Hofmann presentation of approach spaces (see [2, 7]). This paper carefully studies how the Hofmann notion may be characterized within the Seal setting.
Recall that, for an endofunctor T of sets and maps and a (commutative and unital) quantale V = (V, ⌦, k), Seal considers lax functors Tˆ of sets and
Received: 30 September 2014 / Accepted: 27 December 2014.
2010 Mathematics Subject Classification. 18C20, 18C15, 18D20, 18B10.
Key words and phrases. Quantale, monad, lax extension, algebraic lax extension, structure
map, topological theory.
This work was partially supported by the Centro de Matema´tica da Universidade de
Coimbra (CMUC), funded by the European Regional Development Fund through the pro- gram COMPETE and by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia under the project PEst-C/MAT/UI0324/2013. The second author is supported by the National Sciences and Engineering Council of Canada.
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