Machine Space I: Weak exponentials and quantification over compact spaces
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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 machines. Given a frame presentation 𝒪 X = ⟨ G | R⟩ 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 for Σ^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.
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