2022/11/09 by Jacopo Garofali, Garofali, Jacopo
Computer Science · Mathematics · #Advanced Topology and Set Theory #Category Theory (math.CT) #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2211.05260
openalex publication_date 2022/11/09 · openalex created_date 2022/11/16 · openalex updated_date 2026/07/28
In the present work we define and study the classifying (or "quotient") site [X/Σ] for any small site X with (countable) coproducts endowed with an action of a (countable) semigroup Σ. A simple case (the most relevant to our applications) is the case Σ=ℕ, on which, therefore we concentrate. Our main result consists in establishing an equivalence of the corresponding Tòpos with the category of sheaves on X with ``Σ-action''. We prove also that there is a spectral sequence computing sheaf cohomology in [X/ℕ] and we deduce some topological properties of this site, such as its fundamental group. We finally apply the above formalism in Holomorphic Dynamics, giving a Tòpos-theoretic interpretation of Epstein's work on the Fatou-Shishikura Inequality and Infinitesimal Thurston's Rigidity.