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    • a bare list of references, to be !include-ed into the References of related entries

      v1, current

    • just for completeness, since elsewhere I need this as a link

      v1, current

    • added a bunch more items to “Selected writings”

      diff, v4, current

    • I factored in an observation by Daniel that a passage in the Jäsche-Logik points out that the difference between negative and infinite judgments vanishes in the context of the excluded middle.

      diff, v24, current

    • I have edited fibration

      • promted by an email question I have added more information on when the pullback of a fibration is a homotopy pullback;

      • in the discussion of “transport” in topological spaces I added a pointer to Flat ∞-parallel transport in Top which gives details;

      • I fixed a mistake where quasifibration was mentioned and pointed to, but, fibration in the Joyal model structure was meant (despite the previous warning of exactly this trap…)

      • I added a subsection “Related concepts”

      • I added loads and loads of hyperlinks to the keywords.

      FInally, I noticed that the following old discussion was sitting there, which hereby I move fromthere to here

      begin forwarded discussion

      +–{.query} Tim: I do not quite agree with ’transport’ as being the main point of fibrations. Rather ’lifting’ is the main point, in particular lifting of homotopies, at least in topological situations. For transport, one needs connections of some sort to get things working well, but in many cases there is only a very weak notion of action, so perhaps that should be derived as a property rather than taken as a ’defining property’ in some sense.

      Perhaps a reference to Stasheff and Wirth

      James Wirth & Jim Stasheff

      Homotopy Transition Cocycles

      math.AT/0609220.

      and the discussion

      http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html

      on the cafe would be a good idea to add.

      Urs: In situations where one wants to talk of transport, the fibration usually arises as the pullback of some “universal fibration”, a generalized universal bundle. For instance (split op-)fibrations of categories are precisely the pullbacks of the universal CatCat-bundle Cat *→CatCat_* \to Cat along a functor F:C→CatF : C \to Cat.

      If one looks at this kind of situation where we do have an established notion of (parallel) transport one sees:

      • it is the classifying functor F:C→CatF : C \to Cat which should be addressed as the “(parallel) transport”, while the corresponding fibration is its “action object” as in action groupoid, i.e. the thing whose objects are all possible things that the parallel transport can transport and whose morphisms take these things to the image of that transport. So it’s a subtle difference, but an important one.

      For instance, to make this more concrete, consider the category of smooth groupoids (which is a category of fibrant objects), let for any manifold XX the groupoid P 1(X)P_1(X) be the groupoid of smooth thin-homotopy classes of paths in XX, let GG be any Lie group, BG\mathbf{B} G the corresponding one-object Lie groupoid and consider the _universal fibration _ EG→BG\mathbf{E} G \to \mathbf{B}G – the groupoid incarnation of the universal GG-bundle as described at generalized universal bundle. Then

      Theorem: GG-bundles with connection on XX are equivalent to functors tra:P 1(X)^→BGtra : \widehat{P_1(X)} \to \mathbf{B}G out of acyclic fibrations P 1(X)^→P 1(X)\widehat{P_1(X)} \to P_1(X) over P 1(X)P_1(X) (i.e. smooth anafunctors P 1(X)→BGP_1(X) \to \mathbf{B}G). These functors are literally the corresponding “parallel transport”: indeed, evaluated on a path γ\gamma in XX there is locally a 1-form A∈Ω 1(X,Lie(G))A \in \Omega^1(X, Lie(G)) such that the group element tra(γ)tra(\gamma) is the traditional parallel transport of that 1-form, tra(γ)=Pexp(∫ γA)tra(\gamma) = P exp(\int_\gamma A).

      Now, we can form the fibration which is associated with this parallel transport, namely the pullback

      tra *EG → EG ↓ ↓ P 2(X)^ →tra BG ↓ P 2(X). \array{ tra^* \mathbf{E} G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \widehat{P_2(X)} &\stackrel{tra}{\to}& \mathbf{B}G \\ \downarrow \\ P_2(X) } \,.

      This fibration tra *EG→P 2(X)^tra^* \mathbf{E}G \to \widehat{P_2(X)} is what is properly speaking the action groupoid of tratra acting on the fibers of the principal GG-bundle.

      Mike: Can you clarify the distinction between “lifting” and “transport”? In what way does the lifting of a path ff starting at a point ee not transport ee along ff? Certainly in geometric situations to get a parallel notion of transport, you need a connection, but I see that as a stronger requirement.

      forwarded discussion continued in next entry

    • starting something. Not done yet, but need to save.

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • to record Bar-Natan95’s “linear duals of marked surfaces”

      v1, current

    • created an entry beable

      (Surprisingly, this keyword does not have a Wikipedia entry…)

    • starting something – not done yet, but need to save

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • collecting some references, not done yet

      v1, current

    • starting something - not done yet, but need to save

      v1, current

    • Since its creation, the entry étale map started out as wanting to be about the abstraction of the notion of local homeomorphism and étale morphism to more general contexts.

      I have added references to such axiomatizations now. But I am not sure that it is good to have “étale map” be interpreted so much differently from “étale morphism”. I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to “étale morphism of schemes”.

      Finally, I changed the wording of the Idea-section at the beginning of étale map: it used to say that an étale map is like a “local isomorphism”. As we have recently seen in discussion, this is a very misleading thing to say, since it is not like a local isomorphism but like a local homeomorphism. These two concepts, maybe unfortunately termed, are not about the same idea.

    • a little table to be !include-ed in relevant entries, for ease of cross-linking

      v1, current

    • bare reference-list to be !include-ed into relevant entries, for ease of updating

      v1, current

    • I changed the title to Montague semantics, which is the traditional terminology. Also added some references.

      diff, v2, current

    • added references

      The original article:

      Review:

      • Marc Bourdon, Mostow type rigidity theorems, Handbook of Group Actions (Vol. IV) ALM 41, Ch. 4, pp. 139–188 (pdf)

      diff, v2, current

    • Reading over section 5 of her

      I thought it worth a reference to

      I don’t think we have anything on this refinement.

      diff, v4, current

    • This is my new website; the originally entered one no longer exists. Tristan Hübsch

      Tristan Hübsch

      diff, v2, current

    • create page with some references

      Jalfy Jalfry

      v1, current

    • Stub, for completeness and for satisfying links

      v1, current

    • for completeness, and to satisfy links

      v1, current

    • I added a brief general idea, and remarked that whether Tarski should appear here is debatable. Not terribly satisfactory, but it’s really hard to do justice to these nebulous philosophical ideas.

      diff, v2, current

    • bare minimum, for the moment just so as to record the basic references

      v1, current

    • added a minimum sentence to the Definition-section

      diff, v2, current

    • Edit to: intersecting D-brane model by Urs Schreiber at 2018-04-01 00:55:01 UTC.

      Author comments:

      hyperlinked pointer to textbook by Ibanez-Uranga

    • starting something, not done yet

      v1, current

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