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Generalised Atiyah’s theory of principal connections. (English) Zbl 07655746

Summary: This is a condensed report from the ongoing project aimed on higher principal connections and their relation with higher differential cohomology theories and generalised short exact sequences of \(L_\infty\) algebroids. A historical stem for our project is a paper from sir M. F. Atiyah [Trans. Am. Math. Soc. 85, 181–207 (1957; Zbl 0078.16002)] who observed a bijective correspondence between data for a horizontal distribution on a fibre bundle and a set of sections for a certain splitting short exact sequence of Lie algebroids, nowadays called the Atiyah sequence. In a meantime there was developed quite firm understanding of the category theory and in the last two decades also the higher category/topos theory. This conceptual framework allows us to examine principal connections and higher principal connections in a prism of differential cohomology theories. In this text we cover mostly the motivational part of the project which resides in searching for a common language of these two successful approaches to connections. From the reasons of conciseness and compactness we have not included computations and several lengthy proofs.

MSC:

18N60 \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories

Citations:

Zbl 0078.16002
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References:

[1] Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), no. 1, 181-207 · Zbl 0078.16002 · doi:10.1090/S0002-9947-1957-0086359-5
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