
Codensity Lifting of Monads and its Dual
We introduce a method to lift monads on the base category of a fibration...
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Monads and Quantitative Equational Theories for Nondeterminism and Probability
The monad of convex sets of probability distributions is a wellknown to...
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Directed Homotopy in NonPositively Curved Spaces
A semantics of concurrent programs can be given using precubical sets, i...
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A Probability Monad as the Colimit of Finite Powers
We define a monad on the category of complete metric spaces with short m...
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Alternation diameter of a product object
We prove that every permutation of a Cartesian product of two finite set...
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A metric for sets of trajectories that is practical and mathematically consistent
Metrics on the space of sets of trajectories are important for scientist...
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Effective local compactness and the hyperspace of located sets
We revisit the definition of effective local compactness, and propose an...
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Extending set functors to generalised metric spaces
For a commutative quantale V, the category Vcat can be perceived as a category of generalised metric spaces and nonexpanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor T_V on Vcat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover wellknown notions such as the PompeiuHausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation X TX in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base. Mathematically, the heart of the matter is to show that, for any commutative quantale V, the "discrete" functor Set →Vcat from sets to categories enriched over V is Vcatdense and has a density presentation that allows us to compute leftKan extensions along D.
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