Uniform Envelopes

Abstract: In the author's PhD thesis (2019) universal envelopes were introduced as a tool for studying the continuously obtainable information on discontinuous functions. To any function $f \colon X \to Y$ between $\operatorname{qcb}_0$-spaces one can assign a so-called universal envelope which, in a well-defined sense, encodes all continuously obtainable information on the function. A universal envelope consists of two continuous functions $F \colon X \to L$ and $\xi_L \colon Y \to L$ with values in a $\Sigma$-split injective space $L$. Any continuous function with values in an injective space whose composition with the original function is again continuous factors through the universal envelope. However, it is not possible in general to uniformly compute this factorisation. In this paper we propose the notion of uniform envelopes. A uniform envelope is additionally endowed with a map $u_L \colon L \to \mathcal{O}^2(Y)$ that is compatible with the multiplication of the double powerspace monad $\mathcal{O}^2$ in a certain sense. This yields for every continuous map with values in an injective space a choice of uniformly computable extension. Under a suitable condition which we call uniform universality, this extension yields a uniformly computable solution for the above factorisation problem. Uniform envelopes can be endowed with a composition operation. We establish criteria that ensure that the composition of two uniformly universal envelopes is again uniformly universal. These criteria admit a partial converse and we provide evidence that they cannot be easily improved in general. Not every function admits a uniformly universal uniform envelope. We can however assign to every function a canonical envelope that is in some sense as close as possible to a uniform envelope. We obtain a composition theorem similar to the uniform case.