Machine Space I: Weak exponentials and quantification over compact spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911592332918784 |
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| author | Faul, Peter F. Manuell, Graham |
| author_facet | Faul, Peter F. Manuell, Graham |
| contents | Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification procedure. The former are simply opens, while we call the latter \emph{machines}. Given a frame presentation $\mathcal{O} X = \langle G \mid R\rangle$ we construct a space of machines $Σ^{Σ^G}$ whose points are given by formal combinations of basic machines corresponding to generators in $G$. This comes equipped with an `evaluation' map making it a weak exponential with base $Σ$ and exponent $X$. When it exists, the true exponential $Σ^X$ occurs as a retract of machine space. We argue this helps explain why some spaces are exponentiable and others not. We then use machine space to study compactness by giving a purely topological version of Escardó's algorithm for universal quantification over compact spaces in finite time. Finally, we relate our study of machine space to domain theory and domain embeddings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_11339 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Machine Space I: Weak exponentials and quantification over compact spaces Faul, Peter F. Manuell, Graham General Topology Logic in Computer Science Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification procedure. The former are simply opens, while we call the latter \emph{machines}. Given a frame presentation $\mathcal{O} X = \langle G \mid R\rangle$ we construct a space of machines $Σ^{Σ^G}$ whose points are given by formal combinations of basic machines corresponding to generators in $G$. This comes equipped with an `evaluation' map making it a weak exponential with base $Σ$ and exponent $X$. When it exists, the true exponential $Σ^X$ occurs as a retract of machine space. We argue this helps explain why some spaces are exponentiable and others not. We then use machine space to study compactness by giving a purely topological version of Escardó's algorithm for universal quantification over compact spaces in finite time. Finally, we relate our study of machine space to domain theory and domain embeddings. |
| title | Machine Space I: Weak exponentials and quantification over compact spaces |
| topic | General Topology Logic in Computer Science |
| url | https://arxiv.org/abs/2209.11339 |