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.