What is modal logic

Real and Complex Analysis

Preface 1. Introduction Robert Lowen and Walter Tholen 2. Monoidal structures Gavin J. Seal and Walter Tholen 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen Bibliography Tables Index.

Monoidal topology : a categorical approach to order, metric, and topology

It is known since 1973 that Lawvere’s notion of Cauchy-complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper, we introduce the corresponding notion of Lawvere completeness for $(\mathbb{T},\mathsf{V})$
 -categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones it means weak sobriety while for the latter it means Cauchy completeness. Further, we show that $\mathsf{V}$
 has a canonical $(\mathbb{T},\mathsf{V})$
 -category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; this structure permits us to define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$
 -categories.

http://sweet.ua.pt/dirk/artigos/2009/paper_complete_rev.pdf

Lawvere Completeness in Topology

Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

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.

Universality of Coproducts in Categories of Lax Algebras

In this survey we discuss the generalisation of stochastic Kripke models for general modal logics through predicate liftings for functors over general measurable spaces. We derive results on expressivity and show that selection arguments allow us to incorporate the discussion of bisimilarity, provided the underlying spaces are assumed to be standard Borel.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0960129510000526

Coalgebraic logic over general measurable spaces ??? a survey

We use the Kolmogorov Consistency Theorem from Measure Theory to construct terminal coalgebras for a large class of functors on the category of measurable spaces. In particular, we construct terminal stochastic relations and terminal labelled Markov processes. We use this constructions to provide extended expressivity results for canonical interpretations of modal and temporal logics in these structures.

Christoph Schubert

Papers

Universality of Coproducts in Categories of Lax Algebras

Coalgebraic logic over general measurable spaces ??? a survey

Terminal Coalgebras for Measure-Polynomial Functors

Coalgebraic logic for stochastic right coalgebras

Coalgebraic Logic over Measurable Spaces: Behavioral and Logical Equivalence