Sheaves on the category of periodic observation (Q1592619)

A category \(\mathbb{N} ^{\circlearrowleft}\) based on the natural numbers \(\mathbb{N} \) and division in \(\mathbb{N} \) is defined and called the category of ``observers with different time units''. A presheaf on \(\mathbb{N} ^{\circlearrowleft}\) with values in discrete dynamical systems is defined. The notion \(J(n)\) of sieve on \(n\) in \(\mathbb{N} \) is introduced and \(J\) is shown to define a Grothendieck topology on \(\mathbb{N} ^{\circlearrowleft}\). For a discrete dynamical system \(D = (X, J)\), there is an induced presheaf \(P_{D}\) on \(\mathbb{N} ^{\circlearrowleft}\). The authors show that, to \(D\), one can associate a discrete dynamical system \(\hat{D}= (\hat{X},\hat{J})\) so that the sheafification \(P^{+}_{D}\) of \(P_{D}\) satisfies \(P^{+}_{D}(n) = \widehat{(X, J^{n})}\).