Newly discovered Potentials for Topos Theory from Grothendieck's Handwritten Notes: Functorial Correspondences and Topos Duality: A Reconstruction from Pages (Cote 115)

 This abstract delivers a remark related to Topos Duality. We base ourselves on Alexander
 Grothendieck’s handwritten manuscript [from 1982] 103 Functorial "correspondences". Duality
 of topos: handwritten notes (nd). Rating No. 115 (14 p.), preserved at the Université de Mont
pellier archives (see https://grothendieck.umontpellier.fr/archives-grothendieck/#).
 We create our case directly from these 14 page scans, included in this abstract; we will use
 arrows, 2-cells and triangles that are present in the handwriting and we will transcribe them
 into bicategorical diagrams and formalize them in the ∞-categorical language accepted today.
 Bridging logics and geometry we believe our abstract can help advance Topos theory, through
 a deeper understanding of modern categorical logic, we regard a topos as the semantics of the
 theory. Duality then trades geometric morphisms for theory morphisms. In practice this informs
 howweinternalize our construct (e.g. synthetic algebraic geometry, cohesive/synthetic homotopy
 theory). Please note, Grothendieck through his notes, seems to sketch the clear functoriality
 required for such a bridge.
 Topos-theoretic Galois theory. Dualities between atomic/Boolean topoi and profinite group
 actions inspire contemporary refinements: étale homotopy types, profinite or condensed avatars,
 and generalized Galois categories. The handwritten notes do indicate when a topos is governed
 by a “Galois object,” which we reinterpret to study fundamental groupoids in ∞-topoi and
 stratified settings.
 His notes also appear to cover Morita equivalence for theories.( Two sites can present the
 same topos; two theories can be Morita-equivalent.) Grothendieck his notes emphasize the
 principle that equivalence of topoi, not presentation, is the relevant invariant for structural
 claims. We build further upon said angle, and provide new structural claims.