A Skeletal Model for $$2\hbox {d}$$ Conformal AQFTs

A simple model for the localization of category $\mathbf{CLoc}_2$ oriented and time-oriented globally hyperbolic conformal Lorentzian $2$-manifolds at all Cauchy morphisms is constructed. This provides an equivalent description $2$-dimensional algebraic quantum field theories (AQFTs) satisfying time-slice axiom in terms only two algebras, one Minkowski spacetime flat cylinder, together with a suitable action copies orientation preserving embeddings $1$-manifolds. The latter result used to construct adjunctions between categories chiral AQFTs whose right adjoints formalize generalize Rehren's observables.