Relative injectivity as cocompleteness for a class of distributors

arXiv:0807.4123v1 [math.CT] 25 Jul 2008 RELATIVE INJECTIVITY AS COCOMPLETENESS FOR A CLASS OF DISTRIBUTORS MARIA MANUEL CLEMENTINO AND DIRK HOFMANN Dedicated to Walter Tholen on the occasion of his sixtieth birthday Abstract. Notions and techniques of enriched category theory can be used to study topological struc- tures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over Set. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escard´o and B. Flagg, Semantic domains, injec- tive spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the V-enriched setting. Introduction This work continues the research line of previous papers, aiming to use categorical tools in the study of topological structures. Indeed, the perspective proposed in [2, 6] of looking at topological structures as (Eilenberg-Moore) lax algebras and, simultaneously, as a monad enrichment of V-enriched categories, has shown to be very effective in the study of special morphisms – like effective descent and exponentiable ones – at a first step [3, 4], and recently in the study of (Lawvere/Cauchy-)completeness and injectivity [5, 11, 10]. The results we present here complement this study of injectivity. More precisely, in the spirit of Kelly-Schmitt [12] we generalise the results of [10], showing that injectivity and cocompleteness – when considered relative to a class of distributors – still coincide. Suitable choices of this class of distributors allow us to recover, in the V-enriched setting, results on injectivity of Escard´o-Flagg [7]. The starting point of our study of injectivity is the notion of distributor (or bimodule, or profunctor), which allowed the study of weighted colimit, presheaf category, and the Yoneda embedding. It was then a natural step to ‘relativize’ these ingredients and to consider cocompleteness with respect to a class of distributors Φ. Namely, we introduce the notion of Φ-cocomplete category, we construct the Φ-presheaf category, and we prove that Φ-cocompleteness is equivalent to the existence of a left adjoint of the Yoneda embedding into the Φ-presheaf category. Furthermore, the class Φ determines a class of embeddings so that the injective T-categories with respect to this class are precisely the Φ-cocomplete categories. This result links our work with [7], where the authors study systematically semantic domains and injectivity characterisations with the help of Kock-Z¨oberlein monads. 2000 Mathematics Subject Classification. 18A05, 18D15, 18D20, 18B35, 18C15, 54B30, 54A20. Key words and phrases. Quantale, V-category, monad, topological theory, distributor, Yoneda lemma, weighted colimit. The authors acknowledge partial financial assistance by Centro de Matem´atica da Universidade de Coimbra/FCT and Unidade de Investiga¸c˜ao e Desenvolvimento Matem´atica e Aplica¸c˜oes da Universidade de Aveiro/FCT. 1 2 MARIA MANUEL CLEMENTINO AND DIRK HOFMANN 1. The Setting Throughout this paper we consider a (strict) topological theory as introduced in [9]. Such a theory T = (T, V, ξ) consists of: (1) a commutative quantale V = (V, ⊗, k), (2) a Set-monad T = (T, e, m), where T and m satisfy (BC); that is, T sends pullbacks to weak pullbacks and each naturality square of m is a weak pullback, and (3) a T-algebra structure ξ : T V −→V on V such that: (a) ⊗: V × V −→V and k : 1 −→V, ∗7−→k, are T-algebra homomorphisms making (V, ξ) a monoid in Set T; that is, the following diagrams T 1 !  T k / T V ξ  1 k / V T (V × V) T (⊗) / ⟨ξ·T π1,ξ·T π2⟩  T V ξ  V × V ⊗ / V are commutative; (b) For each set X, ξX : VX −→VT X, (X ϕ −→V) 7−→(T X T ϕ −→T V ξ −→V), defines a natural transformation (ξX)X : P −→PT : Set −→Ord. Here P : Set −→Ord is the V-powerset functor defined as follows. We put PX = VX with the pointwise order. Each map f : X −→Y defines a monotone map Vf : VY −→VX, ϕ 7−→ϕ · f. Since Vf preserves all infima and all suprema, it has a left adjoint Pf. Explicitly, for ϕ ∈VX we have Pf(ϕ)(y) = W{ϕ(x) | x ∈X, f(x) = y}. Examples. Throughout this paper we will keep in mind the following topological theories: (1) The identity theory I = (1, V, 1V), for each quantale V, where 1 = (Id, 1, 1) denotes the identity monad. (2) U2 = (U, 2, ξ2), where U = (U, e, m) denotes the ultrafilter monad and ξ2 is essentially the identity map. (3) UP+ = (U, P+, ξP+) where P+ = ([0, ∞]op, +, 0) and ξP+ : UP+ −→P+, x 7−→inf{v ∈P+ | [0, v] ∈x}. (4) The word theory (L, V, ξ⊗), for each quantale V, where L = (L, e, m) is the word monad and ξ⊗: LV −→V. (v1, . . .

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