episciences.org_7314_1674947665
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episciences.org
Logical Methods in Computer Science
18605974
07
28
2022
Volume 18, Issue 3
Uniform Envelopes
Eike
Neumann
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 socalled universal envelope which, in a welldefined 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.
07
28
2022
7314
https://creativecommons.org/licenses/by/4.0
arXiv:2103.16156
10.48550/arXiv.2103.16156
https://arxiv.org/abs/2103.16156v3
https://arxiv.org/abs/2103.16156v2
https://arxiv.org/abs/2103.16156v1
10.46298/lmcs18(3:8)2022
https://lmcs.episciences.org/7314

https://lmcs.episciences.org/9869/pdf

https://lmcs.episciences.org/9869/pdf