Related papers: A categorical approach to the maximum theorem

This paper investigates continuity properties of value functions and solutions for parametric optimization problems. These problems are important in operations research, control, and economics because optimality equations are their…

We extend Berge's Maximum Theorem to allow for incomplete preferences. We first provide a simple version of the Maximum Theorem for convex feasible sets and a fixed preference. Then, we show that if, in addition to the traditional…

We use double categories to obtain a single theorem characterizing certain exponentiable morphisms of small categories, topological spaces, locales, and posets.

This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the results from E.A. Feinberg, P.O. Kasyanov, N.V. Zadoianchuk, "Berge's theorem for noncompact image sets," J. Math. Anal. Appl. 397(1)(2013),…

We discuss generalised duality theory for monoidal categories and its applications to the categories of exact endofunctors, graded vector spaces, and topological vector spaces.

We prove that the Baire Category Theorem is equivalent to the following: Let G be a topological groupoid such that the unit space is a complete metric space, and there is a countable cover of G by neighbourhood bisections. If G is…

Dependence on the parameter is continuous when perturbations of the parameter preserves strict preference for one alternative over another. We characterise this property via a utility function over alternatives that depends continuously on…

For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this…

We prove a model theoretic Baire category theorem for $\tilde\tau_{low}^f$-sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in…

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…

Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces).…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

We have established various criteria for the topological transitivity of families of continuous (holomorphic) functions. Furthermore, by leveraging the properties of expanding families of meromorphic functions, we offer an alternative proof…

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and…

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…

We prove a topological completeness theorem for the modal logic GLP containing operators $\langle\lambda\rangle$ for $\lambda \in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that,…

We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be…

We use the recently introduced \'etale open topology to prove several facts about large fields. We show that these facts lift to a very general topological setting.

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in…

A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of…