Uniform Envelopes

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