Quantalic topological theories
From the introduction: The paper proposes a further extension of the setting that should facilitate the investigation of categories of interest in topology. For that the introduced notion of topological platform involves two monads \(\mathbb{T}\) and \(\mathbb{P}\) on a category \({\mathcal K}\) tied by a distributive law. ``Topologicity'' enters through the requirement that the Kleisli category of \(\mathbb{P}\) be a quantaloid, since the category of sets with \(\mathsf V\)-valued relations (\(\mathsf V\) denotes a quantal) is the Kleisli category of the role model \(\mathbb{P}_{\mathbf{\bigvee}}\). By contrast, \(\mathbb{T}\) should be considered the ``algebraic'' part of the notion. After a brief consideration of strict \((\mathbb{T},\mathbb{P})\)-algebras, including categories of partial \(\mathbb{T}\)-algebras, the paper establishes the fundamental fact that, for every topological platform, the forgetful functor \(\mathsf{Alg}(\mathbb{T}, \mathbb{P}) \to {\mathcal K}\) is topological (\(\mathsf{Alg}(\mathbb{T}, \mathbb{P})\) denotes the category of lax \((\mathbb{T},\mathbb{P})\)-algebras). This fact leaves open questions which are mentioned at the end of the paper. Following, it is showed how the \((\mathbb{T},\mathbb{P})\)-theory particularizes to previous work in monoidal topology. Moreover it is pointed out that the \((\mathbb{T},\mathsf{V})\)-setting should be seen as representing the monad \(\mathbb{P}\) by a quantale \(\mathsf V\) or more generally by a quantaloid Q. Finally a quantalic topological theory generalizing the notion introduced in [\textit{W. Tholen}, ``Lax distributive laws. I'', Preprint, \url{arXiv:1603.0625}] is proposed and the implication of previous Hofmann's topological theory [\textit{D. Hofmann}, Adv. Math. 215, No. 2, 789--824 (2007; Zbl 1127.18001)] is showed.
scientific article; zbMATH DE number 6825310
scientific article