Universality of Coproducts in Categories of Lax Algebras

Abstract. 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 pres...

• Type: Journal
• Year: 2006
• Where: ACS
• Authors: Mojgan Mahmoudi, Christoph Schubert, Walter Tholen
• Keywords: ACS 2006, ACS 2007, Coproducts, Inverse Images, Topological Categories