In the section “References – Internal to more general categories” (here) I have expanded out the references on internalization in finitely complete categories:

Anders Kock,

*Fibre bundles in general categories*, Journal of Pure and Applied Algebra**56**3 (1989) 233-245 [doi:10.1016/0022-4049(89)90059-5]Anders Kock,

*Generalized fibre bundles*, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics**1348**(2006) 194-207 [doi:10.1007/BFb0081359]

and I have taken the liberty of adding:

]]>added pointer to:

- Loring Tu, Parts I-II in:
*Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies**204**, AMS 2020 (ISBN:9780691191744)

added pointer to:

- Gerd Rudolph, Matthias Schmidt, Section 1.1 of:
*Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 (doi:10.1007/978-94-024-0959-8)

(this maths-phys text trumps every pure math textbook account that I have seen regarding exposition of the theory, both in coherent conceptual breadth and systematic account of the details)

]]>hyperlinking the *shear map*

I fixed the link to

*Séminaire Henri Cartan*1949-1950 (numdam:SHC_1948-1949__1)

but I notice that this is still not a very useful pointer, since there are 11 *exposés* behind this link, none of which has a title that would suggest it introduces principal bundles.

I made the remark on “Cartan principal bundles” (Palais’ terminology for the notion without the local triviality condition) a numbered Remark environment (here) and added more precise pointer to where in Palais 61 it says so (namely Def. 1.1.2)

]]>Questions related to the existence slices of G-spaces, of sections of $G$-bundles and conditions for properness of some related maps are treated in

- Richard S. Palais,
*On the existence of slices of actions of non-compact Lie groups*, Ann. Math. 73:2 (1961) pdf

added pointer to

- Ralph Cohen,
*The Topology of Fiber Bundles*, Standord University (2017) (pdf, OMN:201707.110706)

added more publication data to some of the references

]]>seems we want right G-spaces…

]]>True, that was an old remnant. Good that you spotted it, I didn’t read the Idea-section anymore!

I have now replaced it with a different discussion. I hope to find time now to bring the entire entry a bit more up to speed.

]]>I agree with Zoran.

]]>I find a bit strange the instistence on the distinction between torsors and principal bundles put in this article (in sentence 1 in the idea, as if it were central, while it is just a matter of local culture). To me, torsor and principal bundle is the same thing, except that torsor is usually used in algebraic context, with respect to Grothendieck topologies like flat or etale. My principal references are Husemoller, Eells, Postnikov etc. The equivalence of the two words is also accepted for the noncommutative generalization by most practitioners in noncommutative geometry. The article says that principal bundle includes local triviality while torsor does not. In my experience, both words can be used in locally trivial and not locally trivial case. By convention, one may drop the locally trivial prefix, if it is assumed throughout a text and mentioned at the beginning of the article.

]]>I have added to *principal bundle*

a remark on their

*definition*As quotients;statements about (classes of) (counter-)examples of quotients

Thanks for pointers to the literature from this MO thread!

]]>