From lax monad extensions to Topological theories

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 quantale V = (V,⊗, k) that are fully determined by maps ξ : TV → V . We pay special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category SetT with its cartesian structure. Any such map ξ 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 presentation 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 T̂ of sets and V-valued relations (or “matrices”) which, when applied to maps or their opposites, will generally increase the value in the pointwise order of their hom-sets in comparison to an application by T ; furthermore, if T carries a monad structure, T̂ is said to laxly extend the monad if the unit and the multiplication of the monad become oplax transformations when T is replaced by T̂ . In Hofmann’s This work was partially supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT Fundação para a Ciência 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. 1 2 MARIA MANUEL CLEMENTINO AND WALTER THOLEN setting, such lax extensions get constructed from a mere function ξ : TV → V satisfying certain compatibility conditions with the monad and the quantale V (see also [8]). One of their special properties not available in the general Seal environment is that these lax extensions are dualizing, i.e., they are invariant under the involution given by inverting V-relations. But being dualizing does not characterize Hofmann’s lax monad extensions among all. In this paper we identify a stronger property, called algebraicity here, as the key ingredient to a property-based characterization of the Hofmann extensions amongst all lax extensions of a Set-monad satisfying the so called Beck-Chevalley property (BC). Algebraicity ensures that, given a V-relation r : X−→7 Y , which may be equivalently described as a V-relation r̃ : X×Y−→7 1, the value of T̂ r may actually be recovered from T̂ r̃. Such lax extensions are necessarily induced by a single map ξ : TV → V the additional properties of which are shown to correspond to known properties of the induced lax extension. Our central result establishes a 1-1 correspondence between algebraic lax extensions of a given Set-functor T satisfying BC and monotone maps ξ that are laxly compatible with the monoid structure of V (Theorem 1.4.4). Hofmann’s theories, whether in their lax or strict forms, are shown to grow naturally out of this basic correspondence (Theorems 1.6.2 and 2.2.1). While many of the key ideas and techniques of the proofs presented here are already present in [6], our presentation within the Seal context is new. We have also tried to minimize the use of “elementwise calculations”; in fact, most of our extensive calculations use exclusively the compositional structure of the monoidal-closed category V-Rel of sets and V-relations with its order enrichment. 1. Algebraic lax extensions 1.1 The symmetric monoidal-closed quantaloid V-Rel. Let V = (V,⊗, k) be a quantale which, for simplicity, is always assumed to be commutative. One associates with V the quantaloid V-Rel of sets and V-relations; here a 1A quantale is a complete lattice with a monoid structure such that the binary operation ⊗ distributes over arbitrary suprema in each variable. The neutral element k may be smaller than the top element > but it is always assumed to be larger than the bottom element ⊥. 2A quantaloid is a Sup-enriched category, where Sup is the monoidal-closed category of complete lattices and sup-preserving maps. 3A V-relation r : X−→7 Y may have different set-theoretic representations. For example, for V = 2 the two-element chain, r is usually represented by a subset of X × Y . The representation that matters in this paper is that of a function X × Y → V for which we introduce the notation −→r in (1.1.i) below whenever we want to emphasize its role as an arrow in Set in this specific form, rather than as a morphism in V-Rel. FROM LAX MONAD EXTENSIONS TO TOPOLOGICAL THEORIES 3 V-relation from a set X to a set Y is given by a function X × Y → V, written as r : X−→7 Y , and composition with s : Y−→7 Z is defined by s · r(x, z) = ∨ y∈Y r(x, y)⊗ s(y, z). When considering V as a one-object quantaloid, we obtain a homomorphism V → V-Rel, v 7→ (v : 1−→7 1), which embeds V fully into V-Rel. There is an obvious isomorphism V-Rel → V-Rel, (r : X → Y ) 7→ (r◦ : Y−→7 X) with r◦(y, x) = r(x, y), of quantaloids, which makes V-Rel self-dual. There is also a faithful functor (−)◦ : Set→ V-Rel, (f : X → Y ) 7→ (f = f◦ : X−→7 Y ) with f◦(x, y) = { k if f(x) = y, ⊥ else } , whose opposite we may compose with the above functor (r 7→ r◦) to obtain (−)◦ : Set → V-Rel, (f : X → Y ) 7→ (f◦ : Y−→7 X). Note that in V-Rel, considered as a 2-category, one has f◦ a f◦ for every map f in Set. The functor (−)◦ has a right adjoint V-Rel→ Set, (r : X−→7 Y ) 7→ (r : V X → V Y ), with r(φ)(y) = ∨ x∈X φ(x) ⊗ r(x, y) for all φ ∈ V X , y ∈ Y . This functor is represented by 1 and induces the V-powerset monad on Set (see 1.5.2 below). Here we are only interested in the counits of this adjunction, ιX : V −→7 X, ιX(φ, x) = φ(x), in particular in ι : V ∼= V 1 ι1 −→7 1, ι(v) = v. When we write a V-relation r : X−→7 Y equivalently as r̃ : X × Y−→7 1, the couniversal property of ι makes r̃ factor uniquely through the map − →r : X × Y → V :