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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2012
   • (edited Jan 7th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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](http://ncatlab.org/nlab/show/higher+parallel+transport#FlatInTop) 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 Porter|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 Schreiber|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 $Cat$-bundle $Cat_* \to Cat$ along a functor $F : C \to Cat$. 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 $F : C \to Cat$ 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 [[generalized smooth space|smooth]] [[groupoid]]s (which is a [[category of fibrant objects]]), let for any manifold $X$ the groupoid $P_1(X)$ be the groupoid of smooth thin-homotopy classes of paths in $X$, let $G$ be any Lie group, $\mathbf{B} G$ the corresponding one-object Lie groupoid and consider the _universal fibration _ $\mathbf{E} G \to \mathbf{B}G$ -- the groupoid incarnation of the universal $G$-bundle as described at [[generalized universal bundle]]. Then Theorem: $G$-bundles with connection on $X$ are equivalent to functors $tra : \widehat{P_1(X)} \to \mathbf{B}G$ out of acyclic fibrations $\widehat{P_1(X)} \to P_1(X)$ over $P_1(X)$ (i.e. smooth [[anafunctor]]s $P_1(X) \to \mathbf{B}G$). These functors are literally the corresponding "parallel transport": indeed, evaluated on a path $\gamma$ in $X$ there is locally a 1-form $A \in \Omega^1(X, Lie(G))$ such that the group element $tra(\gamma)$ is the traditional parallel transport of that 1-form, $tra(\gamma) = P exp(\int_\gamma A)$. Now, we can form the fibration which is associated with this parallel transport, namely the pullback $$ \array{ tra^* \mathbf{E} G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \widehat{P_2(X)} &\stackrel{tra}{\to}& \mathbf{B}G \\ \downarrow \\ P_2(X) } \,. $$ This fibration $tra^* \mathbf{E}G \to \widehat{P_2(X)}$ is what is properly speaking the [[action groupoid]] of $tra$ acting on the fibers of the principal $G$-bundle. _[[Mike Shulman|Mike]]_: Can you clarify the distinction between "lifting" and "transport"? In what way does the lifting of a path $f$ starting at a point $e$ _not_ transport $e$ along $f$? 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

   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 $Cat$-bundle $Cat_* \to Cat$ along a functor $F : C \to Cat$.

   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 $F : C \to Cat$ 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 $X$ the groupoid $P_1(X)$ be the groupoid of smooth thin-homotopy classes of paths in $X$, let $G$ be any Lie group, $\mathbf{B} G$ the corresponding one-object Lie groupoid and consider the _universal fibration _ $\mathbf{E} G \to \mathbf{B}G$ – the groupoid incarnation of the universal $G$-bundle as described at generalized universal bundle. Then

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

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

   $$\array{ tra^* \mathbf{E} G &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ \widehat{P_2(X)} &\stackrel{tra}{\to}& \mathbf{B}G \\ \downarrow \\ P_2(X) } \,.$$

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

   Mike: Can you clarify the distinction between “lifting” and “transport”? In what way does the lifting of a path $f$ starting at a point $e$ not transport $e$ along $f$? 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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexcontinuation of forwarded discussion *** _[[Mike Shulman|Mike]] adds_: In fact, any topological fibration over $X$ (not just a bundle) induces a "transport" map from (the fundamental $\infty$-groupoid of) $X$ to the $(\infty,1)$-category $Top$, taking each point to its fiber. Putting extra structure on the fibration, such as making it a $G$-bundle, corresponds to restricting the codomain of the transport functor to land in some smaller subcategory, such as $\mathbf{B}G$. A reference in classical homotopy theory is May's "Classifying spaces and fibrations." I agree with what Urs said that the "transport" properly refers to the corresponding functor into $Top$, $Cat$, or $\mathbf{B}G$, with the fibration being the "action object." Then the property of "being a fibration" means "can be equipped with a transport" or "can be obtained from a transport." These are all important points to be added, but I think that what I wrote is still correct. -[[Tim Porter|Tim]] The transport aspect of course is there, but as the lifting is not unique the idea of the transport being 'up to coherent homotopy' or '$\infty$-homotopy is much more sophisticated than the simple point of the existence of a 'lifting'. My point is that although correct your definition may be accused of being 'mathematics made difficult'. To have a definition of fibration that somehow requires one to have understood $(\infty,1)$-categories and the fundamental $\infty$-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable). That process is only recently understood and is still only understood by a handful of people who work with fibrations. If you look at the classical treatment of fibre spaces (and I found a copy of Grothendieck's 1950s notes on them in a pile of stuff today), there is no thought of transport, just of lifting. Serre, who revolutionised the algebraic topology of fibre spaces and fibrations is using lifting, not transport. The link between the transport functor and the notion of fibration emerged via the 'Grothendieck construction' (due to Ehresmann as well.. and there transport was part of the story that he wanted) but was part of the image of fibre spaces\fibrations as being like semi-direct products and that took a long time to get to the present state. _[[Mike Shulman|Mike]]_: Well, I think a lot of what we do here could be accused of being "mathematics made difficult." Cf. the first sentence of [[group]]. But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice. For instance, I believe that the "real" definition of a [[monoidal category]] requires that "all diagrams of constraints commute," while the fact that it's enough to check two particular diagrams is a happy accident. Presenting the pentagon and unit axioms as the God-given definition of a monoidal category, without mentioning that the only reason this is an okay definition is because you can prove a coherence theorem from it, leads the student quite naturally to wonder why God likes pentagons. Likewise, I believe that the only reason the definition of a fibration via lifting properties is a _correct_ definition is that it _does_ lead to the complete functoriality up to higher homotopies. (One doesn't need complicated notions of $\infty$-categories to get an intuitive feel for how this works: it's enough to observe that lifting of 2-cells gives you homotopy-uniqueness for lifts of paths, lifting of 3-cells gives you homotopy-uniqueness for lifts of 2-cells, etc.) The fact that it took many years for this to be understood is not, to me, an argument against its truth, or against its expository helpfulness. Also, I wrote this page not to be just about the topological notion of fibration, but to include categorical (Grothendieck) fibrations as well and show the commonality between them. And for that I think the notion of transport is indispensable, since categorical fibrations can't be defined by a simple lifting property. *** fowarded discussion to be continued in the next comment

   continuation of forwarded discussion

   ---

   Mike adds: In fact, any topological fibration over $X$ (not just a bundle) induces a “transport” map from (the fundamental $\infty$-groupoid of) $X$ to the $(\infty,1)$-category $Top$, taking each point to its fiber. Putting extra structure on the fibration, such as making it a $G$-bundle, corresponds to restricting the codomain of the transport functor to land in some smaller subcategory, such as $\mathbf{B}G$. A reference in classical homotopy theory is May’s “Classifying spaces and fibrations.”

   I agree with what Urs said that the “transport” properly refers to the corresponding functor into $Top$, $Cat$, or $\mathbf{B}G$, with the fibration being the “action object.” Then the property of “being a fibration” means “can be equipped with a transport” or “can be obtained from a transport.” These are all important points to be added, but I think that what I wrote is still correct.

   -Tim The transport aspect of course is there, but as the lifting is not unique the idea of the transport being ’up to coherent homotopy’ or ’$\infty$-homotopy is much more sophisticated than the simple point of the existence of a ’lifting’. My point is that although correct your definition may be accused of being ’mathematics made difficult’. To have a definition of fibration that somehow requires one to have understood $(\infty,1)$-categories and the fundamental $\infty$-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable). That process is only recently understood and is still only understood by a handful of people who work with fibrations.

   If you look at the classical treatment of fibre spaces (and I found a copy of Grothendieck’s 1950s notes on them in a pile of stuff today), there is no thought of transport, just of lifting. Serre, who revolutionised the algebraic topology of fibre spaces and fibrations is using lifting, not transport. The link between the transport functor and the notion of fibration emerged via the ’Grothendieck construction’ (due to Ehresmann as well.. and there transport was part of the story that he wanted) but was part of the image of fibre spaces\fibrations as being like semi-direct products and that took a long time to get to the present state.

   Mike: Well, I think a lot of what we do here could be accused of being “mathematics made difficult.” Cf. the first sentence of group. But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice.

   For instance, I believe that the “real” definition of a monoidal category requires that “all diagrams of constraints commute,” while the fact that it’s enough to check two particular diagrams is a happy accident. Presenting the pentagon and unit axioms as the God-given definition of a monoidal category, without mentioning that the only reason this is an okay definition is because you can prove a coherence theorem from it, leads the student quite naturally to wonder why God likes pentagons.

   Likewise, I believe that the only reason the definition of a fibration via lifting properties is a correct definition is that it does lead to the complete functoriality up to higher homotopies. (One doesn’t need complicated notions of $\infty$-categories to get an intuitive feel for how this works: it’s enough to observe that lifting of 2-cells gives you homotopy-uniqueness for lifts of paths, lifting of 3-cells gives you homotopy-uniqueness for lifts of 2-cells, etc.) The fact that it took many years for this to be understood is not, to me, an argument against its truth, or against its expository helpfulness.

   Also, I wrote this page not to be just about the topological notion of fibration, but to include categorical (Grothendieck) fibrations as well and show the commonality between them. And for that I think the notion of transport is indispensable, since categorical fibrations can’t be defined by a simple lifting property.

   ---

   fowarded discussion to be continued in the next comment

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexcontinuation of forwarded discussion *** _Mike, again_: Okay, I basically rewrote the whole page, trying to take your comments into account. What do you think of it now? [[Tim Porter|Tim]] : It looks good. My objection was, sort of, pedagogical as I did not want to put off the reader who might be wanting to understand fibrations from the infinity cat point of view. You nicely ease the way in, suggesting how the understanding of the basic classical notion evolves. I like it. Thanks. PS. Wikipedia entries for fibration and fibred category exist and could be used if you think they are consistent enough with you wishes for this. [[Urs Schreiber|Urs]]: thanks, nice entry. Eventually it would be nice if we could say more about _which fibrations are fibrations_: which notion of fibred ($n$-)categories/groupoids are fibrations with respect to which model structure. I once tried to collect a bit of data on this, but I need to dig out my old notes, and they were likely very incomplete anyway. [[Ronnie Brown|Ronnie]]: I made a few minor changes, but I also find it odd to define a fibration in terms of a hypothetical fundamental $\infty$-groupoid of $B$. What is this animal? With regard to transport I like the old papers of Philip R. Heath on operations and fibrations- Heath, Philip R. Groupoid operations and fibre-homotopy equivalence. I. Math. Z. 130 (1973), 207--233. (the first of 3 papers) If $p:E \to B$ is a fibration then there is an operation of the paths on the base on the fibres the laws of an operation holding up to homotopy. At Hull, Phil and I were amused by the fact that the unit interval $I=[0,1]$ is a cogroupoid up to homotopy where homotopy is defined using $I$!. This led to his thesis work on fibrations, getting away from the base point approach of the loops on the base operates on the fibre. Maybe effort should be put in to defining and constructing an $\infty$-cogroupoid (up to homotopy)! =-- *** end of forwarded discussion

   continuation of forwarded discussion

   ---

   Mike, again: Okay, I basically rewrote the whole page, trying to take your comments into account. What do you think of it now?

   Tim : It looks good. My objection was, sort of, pedagogical as I did not want to put off the reader who might be wanting to understand fibrations from the infinity cat point of view. You nicely ease the way in, suggesting how the understanding of the basic classical notion evolves. I like it. Thanks.

   PS. Wikipedia entries for fibration and fibred category exist and could be used if you think they are consistent enough with you wishes for this.

   Urs: thanks, nice entry. Eventually it would be nice if we could say more about which fibrations are fibrations: which notion of fibred ($n$-)categories/groupoids are fibrations with respect to which model structure. I once tried to collect a bit of data on this, but I need to dig out my old notes, and they were likely very incomplete anyway.

   Ronnie: I made a few minor changes, but I also find it odd to define a fibration in terms of a hypothetical fundamental $\infty$-groupoid of $B$. What is this animal?

   With regard to transport I like the old papers of Philip R. Heath on operations and fibrations-

   Heath, Philip R. Groupoid operations and fibre-homotopy equivalence. I. Math. Z. 130 (1973), 207–233. (the first of 3 papers)

   If $p:E \to B$ is a fibration then there is an operation of the paths on the base on the fibres the laws of an operation holding up to homotopy.

   At Hull, Phil and I were amused by the fact that the unit interval $I=[0,1]$ is a cogroupoid up to homotopy where homotopy is defined using $I$!. This led to his thesis work on fibrations, getting away from the base point approach of the loops on the base operates on the fibre. Maybe effort should be put in to defining and constructing an $\infty$-cogroupoid (up to homotopy)!

   =–

   ---

   end of forwarded discussion

4. • CommentRowNumber4.
   • CommentAuthorjim_stasheff
   • CommentTimeJan 7th 2012
   • PermaLink
   Author: jim_stasheff
   Format: TextI beg to differ with some of this. @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. Yes, but that is usually a theorem not a definition. A fibration is (almost always) defined as a map such that... @Urs it is the classifying functor F:C→Cat which should be addressed as the "(parallel) transport", why confuse related ideas by naming them all with the same `address' cf the damage created by multiple equivalent defintiions of `connection' 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. which is why to be careful about names These functors are literally NOT literally the corresponding "parallel transport": indeed, evaluated on a path γ in X there is locally a 1-form A∈Ω 1(X,Lie(G)) Notice you have now abandoned topolgoical fibrations by switching to 1-forms such that the group element tra(γ) is the traditional parallel transport of that 1-form, tra(γ)=Pexp(∫ γA). 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). Can do, but why invoke the universal as oppose to working directly over X? @Mike: Can you clarify the distinction between "lifting" and "transport"? In what way does the lifting of a path f starting at a point e not transport e along f? Certainly in geometric situations to get a parallel notion of transport, you need a connection, but I see that as a stronger requirement. Transport as a verb - you are right but as a noun it (usually) refers to a lifting with certain properties. cf. "can be equipped with a transport" or "can be obtained from a transport." Yes, geometry needs more than topology. @Tim: My point is that although correct your definition may be accused of being 'mathematics made difficult'. To have a definition of fibration that somehow requires one to have understood (∞,1)-categories and the fundamental ∞-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable). Amen! @Mike But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice. Yes, we need to walk on 2 legs

   I beg to differ with some of this.
   
   @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.
   
   Yes, but that is usually a theorem not a definition. A fibration is (almost always) defined as a map such that...
   
   
   @Urs it is the classifying functor
   F:C→Cat which should be addressed as the "(parallel) transport",
   
   why confuse related ideas by naming them all with the same `address'
   cf the damage created by multiple equivalent defintiions of `connection'
   
   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.
   
   which is why to be careful about names
   
   
   These functors are literally
   
   NOT literally
   
   the corresponding "parallel transport": indeed, evaluated on a path
   γ in
   X there is locally a 1-form
   A∈Ω 1(X,Lie(G))
   
   
   
   Notice you have now abandoned topolgoical fibrations by switching to 1-forms such that the group element
   tra(γ) is the traditional parallel transport of that 1-form,
   tra(γ)=Pexp(∫ γA).
   
   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).
   
   Can do, but why invoke the universal as oppose to working directly over X?
   
   @Mike: Can you clarify the distinction between "lifting" and "transport"? In what way does the lifting of a path
   f starting at a point e not transport e along f? Certainly in geometric situations to get a parallel notion of transport,
   you need a connection, but I see that as a stronger requirement.
   
   Transport as a verb - you are right but as a noun it (usually) refers to a lifting with certain properties.
   cf. "can be equipped with a transport" or "can be obtained from a transport."
   Yes, geometry needs more than topology.
   
   @Tim: My point is that although correct your definition may be accused of being 'mathematics made difficult'. To have a definition of fibration that somehow requires one to have understood
   (∞,1)-categories and the fundamental
   ∞-groupoid of a space is, to me, the wrong way around. The simply stated lifting property leads eventually naturally to the beautiful transport description (and is more general, more classically based, and probably more approachable).
   
   Amen!
   
   @Mike But I believe that it is useful and important to have a correct conceptual understanding, in addition to any technically simpler definition that one works with in practice.
   
   Yes, we need to walk on 2 legs
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexJim., this is some very old discussion that I moved over here only because we agree'd to do so, instead of deleting it. Is there anything in the current entry _[[fibration]]_ that you feel needs to be changed? Let's talk about that.

   Jim., this is some very old discussion that I moved over here only because we agree’d to do so, instead of deleting it.

   Is there anything in the current entry fibration that you feel needs to be changed? Let’s talk about that.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 20th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexLooking at this page I was surprised to see there's no mention of type theory, so I added a section to connect to dependent types. There's no doubt loads more tying in one could do. <a href="https://ncatlab.org/nlab/revision/diff/fibration/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/fibration/30">v30</a>, <a href="https://ncatlab.org/nlab/show/fibration">current</a>

   Looking at this page I was surprised to see there’s no mention of type theory, so I added a section to connect to dependent types.

   There’s no doubt loads more tying in one could do.

   diff, v30, current

7. • CommentRowNumber7.
   • CommentAuthorjesuslop
   • CommentTimeJan 11th 2020
   • PermaLink
   Author: jesuslop
   Format: MarkdownItexIn section 4, "Fibrations in Category Theory", Grothendiek and Cartesian fibrations are treated on the same footing saying that the fibration functor $p:E \to B$ produces a pseudofunctor $B \to Cat$, but I think that in the Cartesian case the functors should be $(\infty, 1)$.

   In section 4, “Fibrations in Category Theory”, Grothendiek and Cartesian fibrations are treated on the same footing saying that the fibration functor $p:E \to B$ produces a pseudofunctor $B \to Cat$, but I think that in the Cartesian case the functors should be $(\infty, 1)$.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 12th 2020
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think that whoever wrote this paragraph was using all four terms to mean the same thing. It's somewhat of an accident of history that our page [[Grothendieck fibration]] is about the 2-categorical version and our page [[Cartesian fibration]] is about the $(\infty,1)$-categorical one; there's nothing intrinsically more "Grothendieck" about the 2-categorical version or "Cartesian" about the $(\infty,1)$-categorical one. It might be better if the page currently called "Cartesian fibration" were renamed something like "Cartesian fibration of quasi-categories". But I agree, because of that accident, the current phrasing on the page [[fibration]] is potentially confusing.

   I think that whoever wrote this paragraph was using all four terms to mean the same thing. It’s somewhat of an accident of history that our page Grothendieck fibration is about the 2-categorical version and our page Cartesian fibration is about the $(\infty,1)$-categorical one; there’s nothing intrinsically more “Grothendieck” about the 2-categorical version or “Cartesian” about the $(\infty,1)$-categorical one. It might be better if the page currently called “Cartesian fibration” were renamed something like “Cartesian fibration of quasi-categories”. But I agree, because of that accident, the current phrasing on the page fibration is potentially confusing.