The fundamental progroupoid of a general topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004.

💡 Research Summary

The paper revisits the classical correspondence between locally constant objects (often called covering projections) in a locally connected topos 𝔈 and the classifying topos of its fundamental pro‑groupoid. In that well‑studied setting the fundamental pro‑groupoid can be taken to be a strict pro‑groupoid, equivalently a localic pro‑discrete groupoid, and its classifying topos Bπ₁(𝔈) is a Galois topos: every object of 𝔈 is a covering projection, and the pro‑groupoid represents first‑degree cohomology H¹(𝔈, G) for any localic group G.

The novelty of the present work is to drop the hypothesis of local connectedness and to develop an analogous theory for an arbitrary Grothendieck topos. The author observes that without local connectedness the usual notion of covering projection becomes too restrictive: many locally constant objects fail to be effective descent morphisms, and the classifying topos of the strict pro‑groupoid no longer captures all locally constant objects. Consequently, the fundamental object must be a genuine localic pro‑groupoid Π₁(𝔈) rather than a pro‑discrete one, and BΠ₁(𝔈) is not a Galois topos in general.

To overcome these obstacles the paper introduces a new definition of “covering projection” for an arbitrary topos. An object X ∈ 𝔈 is declared a covering projection if (i) the canonical morphism X → 1 is an effective descent morphism (so X can be reconstructed from its pull‑backs along any cover), and (ii) X is locally constant, i.e. there exists a covering family {U_i → 1} such that each pull‑back X × U_i is isomorphic to a constant object. This definition reduces to the classical one when 𝔈 is locally connected, and it is stable under pull‑back, composition, and internal equivalence.

Using this refined notion, the author constructs for each object a “localic cover” – a family of localic surjections that exhibit X as a locally constant sheaf over a localic space. By taking the inverse system of all such covers (indexed by a cofiltered diagram of localic surjections) one obtains a pro‑object in the 2‑category of localic groupoids. The resulting pro‑groupoid Π₁(𝔈) encodes simultaneously the discrete symmetry data (the groups acting on the fibers) and the topological data (the locales over which the fibers vary). Importantly, Π₁(𝔈) cannot be replaced by a single localic groupoid: the pro‑structure is essential to capture the varying degrees of local constancy present in a general topos.

The classifying topos BΠ₁(𝔈) is then defined as the topos of continuous actions of Π₁(𝔈). The paper proves that the full subcategory of covering projections in 𝔈 is equivalent to the category of Π₁(𝔈)‑actions, establishing BΠ₁(𝔈) as the correct classifying object for these generalized coverings. Because BΠ₁(𝔈) is not Galois, not every object of 𝔈 is a covering projection; the inclusion is proper, and the paper gives explicit examples (e.g., sheaf topoi over non‑connected sites) where locally constant sheaves fail to be effective descent morphisms.

A central cohomological result is that Π₁(𝔈) represents first‑degree cohomology: for any localic group G, there is a natural bijection

  Hom_{Topos}(BΠ₁(𝔈), BG) ≅ H¹(𝔈, G).

The proof proceeds by constructing, for each G‑torsor in 𝔈, a Π₁(𝔈)‑action on its total space, and conversely by showing that any Π₁(𝔈)‑action yields a G‑torsor after passing to the associated sheaf of sections. The argument relies on several technical lemmas concerning the preservation of effective descent under pull‑back, the internal characterization of localic groupoids, and the compatibility of pro‑limits with classifying toposes.

Finally, the paper discusses several applications and directions for future research. The new framework accommodates non‑locally connected situations such as the étale topos of a non‑connected scheme, the classifying topos of a non‑discrete topological group, and various “higher‑Galois” contexts where one wishes to classify torsors for higher groupoids. The author suggests that the pro‑groupoid Π₁(𝔈) could serve as a stepping stone toward a full higher‑dimensional fundamental pro‑∞‑groupoid of a topos, potentially linking to recent work on shape theory and homotopy‑type theory.

In summary, the paper successfully generalizes the classical theory of the fundamental pro‑groupoid from locally connected topoi to arbitrary topoi by (1) redefining covering projections in a way that respects effective descent, (2) constructing a genuine localic pro‑groupoid Π₁(𝔈) that classifies these objects, and (3) proving that Π₁(𝔈) continues to represent H¹. This contribution deepens the bridge between topos theory, non‑abelian cohomology, and the theory of localic groupoids, and opens new avenues for exploring Galois‑type correspondences beyond the locally connected realm.