Universality of Coproducts in Categories of Lax Algebras

Abstract

Categories of lax \((T,V\,)\)-algebras are shown to have pullback-stable 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.