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Register a new userExtending binary operations to funtor-spaces
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Given a continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subsemigroup of all elements of TX that have finite support.
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For a commutative quantale $mathcal{V}$, the category $mathcal{V}-cat$ can be perceived as a category of generalised metric spaces and non-expanding 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 $T_{mathcal{V}}$ on $mathcal{V}-cat$. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation $Xcong TX$ 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 $mathcal{V}$, the `discrete functor $D:mathsf{Set}to mathcal{V}-cat$ from sets to categories enriched over $mathcal{V}$ is $mathcal{V}-cat$-dense and has a density presentation that allows us to compute left-Kan extensions along $D$.
Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence ${bf LKHaus}simeq{bf CLCA}^{rm op}$ that was obtained by the first author more than a decade ago. Unlike the morphisms of $bf CLCA$, the morphisms of the new category and their composition law are very natural and easy to handle.
Let $mathcal{C}$ be a finitely bicomplete category and $mathcal{W}$ a subcategory. We prove that the existence of a model structure on $mathcal{C}$ with $mathcal{W}$ as subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where $mathcal{C}$ is a partial order and show that these are determined by the homotopy categories.
Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of simplicial metric thickenings whose objects have a natural realization as metric spaces. The properties of this category allow us to prove that, for a large class of thickenings including Vietoris-Rips and Cech thickenings, the product of metric thickenings is homotopy equivalent to the metric thickenings of product spaces, and similarly for wedge sums.
In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
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