Locally anisotropic toposes (Q684088)

An object \(X\) of a topos \({\mathcal E}\) is said to be \textit{anisotropic} if the slice topos \({\mathcal E}/X\) is anisotropic: \(\forall z\in Z, x\in X (xz=x \Rightarrow z=u)\), where \(Z\) is the isotropy group of \({\mathcal E}\) and \(u\) the unit of \(Z\). The authors call a topos \({\mathcal E}\) \textit{locally anisotropic} if it has a globally supported anisotropic object (an object \(X\) is \textit{globally supported} when the map \(X \to 1\) is an epimorphism). An object \(X\) of \({\mathcal E}\) is called an \textit{isotropy torsor} when \(X\) is globally supported and the universal action \(\theta_X: X\times Z\to X\) is free and transitive. An object \(X\) is \textit{isotropically trivial} if \(\theta_X\) is trivial. Some equivalent conditions for a topos to be anisotropic or locally anisotropic are found. The structure theorem of the latter states that \({\mathcal E}\) is locally anisotropic iff \({\mathcal E}\) has a globally supported isotropically trivial object \(O\) such that \({\mathcal E}/O\simeq {\mathcal B}({\mathcal F}; G)\), where \(G\) is a group internal to an anisotropic topos \({\mathcal F}\) and \({\mathcal B}({\mathcal F}; G)\) is the topos of right \(G\)-objects in \(\mathcal F\). It is also proved that a topos \({\mathcal E}\) has an isotropy torsor if and only if \({\mathcal E}\) is equivalent to \({\mathcal B}\)(\({\mathcal F}; G)\). The isotropy group \(Z_G\) of \({\mathcal B}\)(\({\mathcal E}; G)\) and its universal action are explicitly described. Relations between the structure theorem and free isotropy algebras are shown: an object \(X\) of a topos is anisotropic iff \(\widehat{X}\) (the free isotropy algebra on \(X\)) is isotropically trivial in the sense that its inverse image functor preserves the isotropy group. A detailed explanation is provided about how the developed theory is interpreted for inverse semigroups. It turnes out, for example, that an inverse semigroup \(S\) has an isotropy torsor iff \(S\cong T\ltimes G\), where \(G\) is an étale \(T\)-group for a fundamental inverse semigroup \(T\).