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Smooth coalgebra: testing vector analysis. (English) Zbl 1387.18007

The paper is related with the application of coalgebras in some questions of justification of differential geometry and Newtonian mechanics. Let Set be the category of sets and mappings. For an arbitrary functor \(F: \mathrm{Set} \to \mathrm{Set}\), a coalgebra over \(F\) (or \(F\)-coalgebra) is a pair \((X, \alpha)\), where \(X\) is a set, and \(\alpha: X \to FX\) is a mapping. Coalgebras are used for modeling processes with a discrete state space \(X\). In general, we can take any well category \(S\) instead of Set. In the paper presented, it is the category \(\mathrm{Man}_{\infty}\) of manifolds and smooth mappings. In this case, the state space is locally homeomorphic to a normed vector space and carry a differential structure. Dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over \(\mathrm{Man}_{\infty}\). A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high-level view of these fundamental structures, which are implemented rather intricately in their standard form. The brief coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton’s laws.
The introduction contains the history of the question. It mentions the coalgebraic theory of processes [J. J. M. M. Rutten, Theor. Comput. Sci. 249, No. 1, 3–80 (2000; Zbl 0951.68038)], semantic connections of algebras and coalgebras [D. Pavlovic et al., Lect. Notes Comput. Sci. 4019, 308–322 (2006; Zbl 1236.68065)], syntetic differential geometry [A. Kock, Synthetic differential geometry. 2nd ed. Cambridge: Cambridge University Press (2006; Zbl 1091.51002)].
Section 2 is devoted to a general overview of testing correlations. The authors use the notion of testing correlation developed in [Moore (1956)] and [Pavlovic (2006)]. Let \(\Sigma\) be a family of systems, \(\Theta\) a family of tests, and \(\Omega\) a type of observations. A testing correlation (or just testing) is a mapping \({\mathbb T}: \Sigma\times \Theta\to \Omega\). The observation \({\mathbb T}(S,t)\) is often writen by \(S\vdash t\). The set \(\Omega\) can be equal to \(\{true, false\}\), or \([0,1]\), or \(\mathbb R\). If the two systems induce the same observations for all tests, then they are called observationally indistinguishable, and we write \(S \sim R \Leftrightarrow \forall t\in \Theta. (S\vdash t)= (R \vdash t)\).
To each \(S\in \Sigma\) there corresponds a mapping \(S\vdash (-): \Theta \to \Omega\). This gives a mapping \(\vdash: \Sigma \to \Omega^{\Theta}\). We denote its image by \(L\). The main feature of this representation is that the elements of \(L\), with a suitable coalgebraic structure, can be used to build the canonical minimal representatives of the behaviours the system in \(\Sigma\), in so far as they are observable under testing by the tests from \(\Theta\).
In Section 3, the authors summarize the basic ideas about manifolds and their tangents.
In Section 4, they apply testing correlations in the context of manifolds to provide the semantic reconstructions of the tangent bundle functors and of the cotangent bundle functors. For any manifold \(M\) is introduced the presheaves \(\Sigma M, \Theta M: OM^{op}\to Set\) (Definition 4.2) and the testing correlation \(\mathbb T: \Sigma M \times \Theta M\to {\mathbb R}\) (Definition 4.7). The testing correlation leads to morphisms \(\vdash: \Sigma M \to {\mathbb R}^{\Theta M}\) and \(\dashv: \Theta M \to {\mathbb R}^{\Sigma M}\) which admit epi-mono factorizations in \(Top\): \((\vdash: \Sigma M \to {\mathbb R}^{\Theta M}) = (\Sigma M \twoheadrightarrow T_*M \rightarrowtail {\mathbb R}^{\Theta M})\) and \((\dashv:\Theta M \to {\mathbb R}^{\Sigma M})= (\Theta M \twoheadrightarrow T^*M \rightarrowtail {\mathbb R}^{\Sigma M}) \).
The main result (Theorem 4.11) asserts that these epi-mono factorizations determine the functors \(T_*, T_{\sharp}: \mathrm{Man}_{\infty}\to \mathrm{Man}_{\infty}\) and \(T^*:\mathrm{Man}^{op}_{\infty}\to \mathrm{Man}_{\infty}\) with \(T_{\sharp}M=T^*M\) on the objects, and such that \(T_*\) is a monad, \(T_{\sharp}\) is a comonad and \(T^*\) is self-adjoint.
The coalgebras for these endofunctors are the usual cross sections of the bundle projections, and thus respectively correspond to vector fields (or dynamical systems) and to differential forms. The testing correlations over manifolds thus provides a categorical view of the practice of integration of differential systems over differential forms.
Section 5 is devoted to Newton’s Second Law as an example of the coalgebraic treatment. Interestingly, the structural duality of the tangent and the cotangent bundles, displayed in the categorical treatment, immediately points beyond Newton, and into relativity theory. An overview of the standard definitions from vector analysis is provided in Appendix A.

MSC:

18B20 Categories of machines, automata
51K10 Synthetic differential geometry
58A32 Natural bundles
53Z05 Applications of differential geometry to physics
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