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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness, with pointer to * [[Alexander Grothendieck]] et al., 16.5.15 in: *&#201;l&#233;ments de g&#233;om&#233;trie alg&#233;brique* IV_4. &#201;tude locale des sch&#233;mas et des morphismes de sch&#233;mas (Quatri&#232;me partie) Inst. Hautes &#201;tudes Sci. Publ. Math. 32 (1967), 5&#8211;361. Ch.IV.&#167;16&#8211;21 ([numdam:PMIHES_1967__32__5_0](http://www.numdam.org/item/PMIHES_1967__32__5_0)) <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/1">v1</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   for completeness, with pointer to

   • Alexander Grothendieck et al., 16.5.15 in: Éléments de géométrie algébrique IV_4. Étude locale des schémas et des morphismes de schémas (Quatrième partie) Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5–361. Ch.IV.§16–21 (numdam:PMIHES_1967__32__5_0)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Laurent Moret-Bailly]], slide 5 of: *Principal Bundles over Valued Fields*, Oberwolfach 2013 ([[MoretBaillyPrincipalBundlesValuedFields.pdf:file]]) <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/2">v2</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   added pointer to:

   • Laurent Moret-Bailly, slide 5 of: Principal Bundles over Valued Fields, Oberwolfach 2013 (MoretBaillyPrincipalBundlesValuedFields.pdf:file)

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded Grothendieck's alternative "formally principal homogeneous space" Also added pointer to pages 73-74 in * Alessandra Bertapelle, Cristian D. Gonzalez-Aviles, *The Greenberg functor revisited*, European Journal of Mathematics volume 4, pages 1340–1389 (2018) ([arXiv:1311.0051](https://arxiv.org/abs/1311.0051), [doi:10.1007/s40879-017-0210-0](https://doi.org/10.1007/s40879-017-0210-0)) <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/3">v3</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   added Grothendieck’s alternative “formally principal homogeneous space”

   Also added pointer to pages 73-74 in

   • Alessandra Bertapelle, Cristian D. Gonzalez-Aviles, The Greenberg functor revisited, European Journal of Mathematics volume 4, pages 1340–1389 (2018) (arXiv:1311.0051, doi:10.1007/s40879-017-0210-0)

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Alexander Grothendieck]], *Technique de descente et théorèmes d’existence engéométrie algébrique. I. Généralités. Descente parmorphismes fidèlement plats*, Séminaire N. Bourbaki, 1960, exp. no190, p. 299-327 ([numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)) which already has the term "formally principal" on p. 15 (without the "homogeneous") <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/3">v3</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   added pointer to

   • Alexander Grothendieck, Technique de descente et théorèmes d’existence engéométrie algébrique. I. Généralités. Descente parmorphismes fidèlement plats, Séminaire N. Bourbaki, 1960, exp. no190, p. 299-327 (numdam:SB_1958-1960__5__299_0)

   which already has the term “formally principal” on p. 15 (without the “homogeneous”)

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexSo I guess the category of pseudotorsors for $G$ is (equivalent to) the result of adding a formal initial object to the groupoid $\mathbf{B}G$?

   So I guess the category of pseudotorsors for $G$ is (equivalent to) the result of adding a formal initial object to the groupoid $\mathbf{B}G$?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt's that trivial only internal to Sets, but the point of the exercise is that it's much more interesting when internalized in richer ambient categories. Such as schemes. Or G-spaces. Or G-schemes....

   It’s that trivial only internal to Sets, but the point of the exercise is that it’s much more interesting when internalized in richer ambient categories. Such as schemes. Or G-spaces. Or G-schemes….

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded one more source for the terminology "formally principal homogeneous": * [[Alexander Grothendieck]], p. 9 of _Exemples et Complements_ ([doi:10.1007/BFb0058666](https://link.springer.com/chapter/10.1007/BFb0058666), [pdf](https://link.springer.com/content/pdf/10.1007%2FBFb0058666.pdf)), which is p. 293 in: [[Alexander Grothendieck]], [[Michèle Raynaud]], *Revêtements étales et groupe fondamental* ([[SGA]] 1), Lecture Notes in Mathematics **224**, Springer 1971 ([arXiv:math/0206203](https://arxiv.org/abs/math/0206203), [doi:10.1007/BFb0058656](https://link.springer.com/book/10.1007/BFb0058656)) <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/3">v3</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   added one more source for the terminology “formally principal homogeneous”:

   • Alexander Grothendieck, p. 9 of Exemples et Complements (doi:10.1007/BFb0058666, pdf), which is p. 293 in:

     Alexander Grothendieck, Michèle Raynaud, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer 1971 (arXiv:math/0206203, doi:10.1007/BFb0058656)

   diff, v3, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 23rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexcross-linked with *[empty heap](https://ncatlab.org/nlab/show/heap#empty)* <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/4">v4</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   cross-linked with empty heap

   diff, v4, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexMade explicit ([here](https://ncatlab.org/nlab/show/pseudo-torsor#PseudoPrincipalProjectionIsEffectiveIfItIsQuotient)) the observation that a pseudo-principal bundle $P \to X$ is an effective epi iff it is the $G$-quotient projection. <a href="https://ncatlab.org/nlab/revision/diff/pseudo-torsor/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/pseudo-torsor/7">v7</a>, <a href="https://ncatlab.org/nlab/show/pseudo-torsor">current</a>

   Made explicit (here) the observation that a pseudo-principal bundle $P \to X$ is an effective epi iff it is the $G$-quotient projection.

   diff, v7, current