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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 16th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI would like to express my annoyance that while a "bundle" and a "fibration" are very similar things, and the words are sometimes used almost synonymously, a "trivial bundle" and a "trivial fibration" are very *different* things. That is all.

   I would like to express my annoyance that while a “bundle” and a “fibration” are very similar things, and the words are sometimes used almost synonymously, a “trivial bundle” and a “trivial fibration” are very different things.

   That is all.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2012
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   Author: Urs
   Format: MarkdownItexFor that reason I have always preferred "acyclic fibration" over "trivial fibration". The former is nicely descriptive. $\,$ (Different can of worms below.) $\,$ $\,$ $\,$ P.S. But I am also wary of thinking of "bundle" and "fibration" as very similar. To a large extent that apparent similarity is a coincidence due to the fact that topological spaces happen to support a model structure for discrete $\infty$-groupoids. So while up to weak homotopy equivalence every map of topological spaces is locally trivial and is a fibration, not every map of topological spaces is a bundle or even locally trivial, up to homeomorphism. By the above coincicence it just so happens that every bundle of discrete $\infty$-groupoids has a presentation by a map of topological spaces. But topological bundles with their structure-preserving morphisms are something very different from discrete $\infty$-bundles with their structure-preserving morphisms.

   For that reason I have always preferred “acyclic fibration” over “trivial fibration”. The former is nicely descriptive.

   $\,$

   (Different can of worms below.)

   $\,$

   $\,$

   $\,$

   P.S. But I am also wary of thinking of “bundle” and “fibration” as very similar. To a large extent that apparent similarity is a coincidence due to the fact that topological spaces happen to support a model structure for discrete $\infty$-groupoids. So while up to weak homotopy equivalence every map of topological spaces is locally trivial and is a fibration, not every map of topological spaces is a bundle or even locally trivial, up to homeomorphism. By the above coincicence it just so happens that every bundle of discrete $\infty$-groupoids has a presentation by a map of topological spaces. But topological bundles with their structure-preserving morphisms are something very different from discrete $\infty$-bundles with their structure-preserving morphisms.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJan 16th 2012
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   Author: Mike Shulman
   Format: MarkdownItex> For that reason I have always preferred "acyclic fibration" over "trivial fibration". Yeah, me too. But sometimes that seems to be a losing battle. > topological bundles with their structure-preserving morphisms are something very different from discrete ∞-bundles with their structure-preserving morphisms. Really? Doesn't it just depend on the structure ($\infty$-)group you choose? That is, a topological bundle has a structure group like Homeo(F) or O(V), while an $\infty$-bundle has a structure group like hAut(F). There's something else to be said about the notions of "local triviality" but I thought that was encapsulated in the fact that the way (nice) topological spaces model $\infty$-groupoids is via their topos-theoretic $\Pi_\infty$.

   For that reason I have always preferred “acyclic fibration” over “trivial fibration”.

   Yeah, me too. But sometimes that seems to be a losing battle.

   topological bundles with their structure-preserving morphisms are something very different from discrete ∞-bundles with their structure-preserving morphisms.

   Really? Doesn’t it just depend on the structure ($\infty$-)group you choose? That is, a topological bundle has a structure group like Homeo(F) or O(V), while an $\infty$-bundle has a structure group like hAut(F). There’s something else to be said about the notions of “local triviality” but I thought that was encapsulated in the fact that the way (nice) topological spaces model $\infty$-groupoids is via their topos-theoretic $\Pi_\infty$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 16th 2012
   • (edited Jan 16th 2012)
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   Author: Urs
   Format: MarkdownItex> Really? The thing is that $Top$ models _disrcrete $\infty$-groupoids_. Intrinsically this has nothing to do with topology, as you know. Nevertheless, by the fact that it is a model, every single discrete $\infty$-bundle has a "presentation" by a topological one. I often see that people use this coincidence to confuse their readers (or themselves) about what they are doing. I won't name example now. Though later I might come up with some that I can mention without upsetting too many potential lurkers. ;-)

   Really?

   The thing is that $Top$ models disrcrete $\infty$-groupoids. Intrinsically this has nothing to do with topology, as you know. Nevertheless, by the fact that it is a model, every single discrete $\infty$-bundle has a “presentation” by a topological one. I often see that people use this coincidence to confuse their readers (or themselves) about what they are doing. I won’t name example now. Though later I might come up with some that I can mention without upsetting too many potential lurkers. ;-)

5. • CommentRowNumber5.
   • CommentAuthorStephan A Spahn
   • CommentTimeJan 16th 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexThere is the [[Milnor slide trick]] (at least in Top) but I'm afraid it can't prevent that ''fiber bundle'' and ''fibration'' are not literally the same.

   There is the Milnor slide trick (at least in Top) but I’m afraid it can’t prevent that ”fiber bundle” and ”fibration” are not literally the same.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 17th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI'm afraid I still don't understand what you're getting at. What I'm saying is that we can either 1. Regard a topological space as a presentation of a discrete $\infty$-groupoid, in which case we have a notion of discrete $\infty$-bundle over it (with fibers given by other $\infty$-groupoids, perhaps presented by topological spaces as well), or 1. Regard a topological space as a "cohesive" space, paying attention to the details of the topology, in which case we have a notion of (locally trivial) $\infty$-bundle over it (with fibers given by certain discrete $\infty$-groupoids) and that these are actually the same (over nice topological spaces). Because a locally trivial $\infty$-bundle (with cohesively-discrete fibers) over a cohesive object $X$ is classified by maps $\Pi_\infty(X) \to \infty Gpd$, and when a topological space $X$ is regarded cohesively, then $\Pi_\infty(X)$ is the $\infty$-groupoid that it would present if we regarded it in the other way.

   I’m afraid I still don’t understand what you’re getting at. What I’m saying is that we can either

   1. Regard a topological space as a presentation of a discrete $\infty$-groupoid, in which case we have a notion of discrete $\infty$-bundle over it (with fibers given by other $\infty$-groupoids, perhaps presented by topological spaces as well), or

   2. Regard a topological space as a “cohesive” space, paying attention to the details of the topology, in which case we have a notion of (locally trivial) $\infty$-bundle over it (with fibers given by certain discrete $\infty$-groupoids)

   and that these are actually the same (over nice topological spaces). Because a locally trivial $\infty$-bundle (with cohesively-discrete fibers) over a cohesive object $X$ is classified by maps $\Pi_\infty(X) \to \infty Gpd$, and when a topological space $X$ is regarded cohesively, then $\Pi_\infty(X)$ is the $\infty$-groupoid that it would present if we regarded it in the other way.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeJan 17th 2012
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   Author: TobyBartels
   Format: MarkdownItexHow is it not true that every map of a topological spaces is a [[bundle]]? (Certainly they are not all locally trivial, nor are they even all fibre bundles.)

   How is it not true that every map of a topological spaces is a bundle? (Certainly they are not all locally trivial, nor are they even all fibre bundles.)