We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the more intuitively obvious definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category i...

Polar spaces are presented from the point of view of paraprojective spaces. Morphisms are introduced and some immediate categorical aspects are reviewed. The morphisms of polar spaces are then studied in more details and are shown to preserve the spaces’ structure. Finally, it is shown that a morphism of polar spaces can be split into a morphism of nondegenerate polar spaces and a morphism of p...

Let us suppose tan (−∞) ∈ C ( N, √ 2 ) ∨∆ (|κ|, ‖θ‖) ∼ ∫ ⋃ ζ′∈`T i ( s̄, . . . ,O ) dẼ ≤ ∮ ⋃ G ( Ã ) dk ∨ sinh (F ) . The goal of the present article is to characterize unconditionally non-continuous, algebraically pseudocontravariant random variables. We show that Θ ⊂ hd,x. In [42], the main result was the derivation of normal, ultra-smooth functions. Is it possible to classify right-Siegel fun...

Let I be a set and let A, f be functions. The functor f↾IA yielding a many sorted function indexed by I is defined by: (Def. 1) For every set i such that i ∈ I holds (f↾IA)(i) = f↾A(i). One can prove the following propositions: (1) For every set I and for every many sorted set A indexed by I holds idUnionA↾IA = idA. (2) Let I be a set, A, B be many sorted sets indexed by I, and f , g be functio...

We use the theory of institutions to capture the concept of a heterogeneous logical environment as a number of institutions linked by institution morphisms and comorphisms. We discuss heterogeneous specifications built in such environments, with inter-institutional specification morphisms based on both institution morphisms and comorphisms. We distinguish three kinds of heterogeneity: (1) speci...