Universality of Coproducts in Categories of Lax Algebras

Categories of lax (T, V )-algebras are shown to have pullbackstable coproducts if T preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax (T, V )-algebras, with T preserving inverse images. Moreover, we show that any such category of (T, V )-algebras has a concrete, coproduct preserving functor into the category of topological spaces. Universality of coproducts is a property that distinguishes Set-based topological categories: while in many “everyday” topological categories coproducts are stable under pullback (topological spaces, preordered sets, premetric spaces, approach spaces), some others fail to enjoy the property (uniform spaces, proximity spaces, nearness spaces, merotopic spaces, see [6]). All of the topological categories in the first group happen to be presentable as categories of lax algebras, that is: they are of the form Alg(T, V ), for a suitable extension T of a Set-monad T0 = (T0, e,m) and a complete lattice V that comes with an associative and commutative binary operation ⊗ preserving suprema in each variable, and a ⊗-neutral element k (distinct from the bottom element ⊥), [5]. In this note, we show that this observation is not coincidental, that is: coproducts in Alg(T, V ) are always stable under pullback, making Alg(T, V ) in fact an (infinitely) extensive category, provided that T0 preserves inverse images. This condition is weaker than preservation of the Beck–Chevalley Property, as used by Clementino and Hofmann (see [4]); for an example, see [7, Example 1.3(a)]. Of special importance are open morphisms of lax algebras, as defined by Möbus [9] in the context of relational algebras in a category, and by Clementino and Hofmann [4] in the context used in this paper. These generalize open morphisms in the category Top of topological space. We characterize coproducts in Alg(T, V ) by the fact that all injections of the underlying Set-coproduct are open morphisms in Alg(T, V ). Moreover, we construct a concrete functor Alg(T, V ) → Top which preserves open embeddings and, hence, coproducts. This result generalizes a construction of Manes [8]. Date: May 8, 2006. 1991 Mathematics Subject Classification. 18C99, 18A30.