fiber sequence




Special and general types

Special notions


Extra structure



Limits and colimits



A homotopy fiber sequence is a “long left-exact sequence” in an (∞,1)-category. (The dual concept is that of cofiber sequence.)

Traditionally fiber sequences have been considered in the context of homotopical categories such as model categories and Brown category of fibrant objects which present the (∞,1)-category in question. In particular, classically this was considered for Top itself. In these cases they are obtained in terms of homotopy pullbacks. Since, as discussed there, the homotopy fiber of a morphism may be computed as the ordinary 1-categorical fiber of any fibration resolution of this morphism, one often also speaks of fibration sequences.


In (,1)(\infty,1)-category theory

Let CC be an (∞,1)-category with small limits and consider pointed objects of CC, i.e. morphisms *A{*} \to A from the terminal object *{*} (the point) to some object AA. All unlabeled morphisms from the point in the following are these chosen ones and all other morphisms are taken with respect to these points.

Notice that in the case that CC happens to be a stable (∞,1)-category for which *=0{*} = 0 all objects are canonically pointed and the notions of left- and right-exact fibration sequences coincide.

But for the notion of fibration sequence to make sense, we do not need to assume that CC is a stable (,1)(\infty,1)-category. In particular, in the context of nonabelian cohomology (see gerbe and principal 2-bundle) one considers fibration sequences in non-stable (,1)(\infty,1)-categories.

Now let f:ABf : A \to B be a morphism in CC.

The homotopy fiber or homotopy kernel or mapping cocone of ff is the pullback (which in our (,1)(\infty,1)-categorical context means homotopy pullback) of the point along ff:

ker(f) * A f B. \array{ ker(f) &\to& {*} \\ \downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \,.

In categories of fibant objects

under construction

In (Quillen 67, section I.3) it was shown how the theory of fiber sequences and cofiber sequences arises in the abstract homotopy theory of model categories. Focusing on the fiber sequences, this perspective depends only on the category of fibrant objects inside the model category, and in fact makes sense generally in this context. This was spelled out in (Brown 73, section 4), which we review here.


In pointed objects 𝒞 f */\mathcal{C}_f^{\ast/} of a category of fibrant objects 𝒞 f\mathcal{C}_f, def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}, consider a morphism of fiber-diagrams, hence a commuting diagram of the form

fib(f 1) X 1 Fibf 1 Y 1 fib(f 2) X 2 Fibf 2 Y 2. \array{ fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{}} && \downarrow && \downarrow \\ fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2 } \,.

If the two vertical morphisms on the right are weak equivalences, then so is the vertical morphism in the left

(Brown 73, section 4, lemma 3)


Factor the diagram in question

fib(f 1) X 1 Fibf 1 Y 1 f fib(f 2) X 2 Fibf 2 Y 2 \array{ fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{}} && \downarrow && \downarrow^{\mathrlap{f}} \\ fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2 }

through the pullback of the bottom horizontal line:

fib(f 1) X 1 Fibf 1 Y 1 W id fib(ϕ) f *X 2 Fibϕ Y 1 W W f fib(f 2) X 2 Fibf 2 Y 2 \array{ fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{id}} \\ fib(\phi) &\longrightarrow& f^\ast X_2 &\underoverset{\in Fib}{\phi}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2 }

Here f *X 2X 2f^\ast X_2 \to X_2 is a weak equivalence by lemma \ref{InCfPullbackAlongFibrationPreservesWeakEquivalences} and with this X 1f *X 2X_1 \to f^\ast X_2 is a weak equivalence by assumption and two-out-of-three.

Moreover, this diagram exhibits fib(f 1)fib(ϕ)fib(f_1)\to fib(\phi) as the base change (along *Y 2\ast \to Y_2) of X 1f *X 2X_1 \to f^\ast X_2.

Hence it is now sufficient to observe that in category of fibrant objects, base change preserves weak equivalences (…).

Hence we say:


Let 𝒞\mathcal{C} be a model category. For f:XYf \colon X \longrightarrow Y any morphism, then its homotopy fiber

hofib(f)X hofib(f)\longrightarrow X

is the morphism in the homotopy category Ho(𝒞)Ho(\mathcal{C}), def. \ref{HomotopyCategoryOfAModelCategory}, which is represented by the fiber, def. \ref{FiberAndCofiberInPointedObjects}, of any fibration resolution of ff.

We may now state the abstract version of the statement of prop. \ref{SerreFibrationGivesExactSequenceOfHomotopyGroups}:


Let 𝒞\mathcal{C} be a model category. For f:XYf \colon X \to Y any morphism of pointed objects, and for AA a pointed object, def. \ref{CategoryOfPointedObjects}, then the sequence

[A,hofib(f)] *i *[A,X] *f *[A,Y] * [A,hofib(f)]_\ast \overset{i_\ast}{\longrightarrow} [A,X]_\ast \overset{f_\ast}{\longrightarrow} [A,Y]_{\ast}

is exact (the sequence being the image of the homotopy fiber sequence of def. 1 under the hom-functor of the pointed homotopy category of a model category

[A,] *:Ho(𝒞 /*)Set */. [A,-]_\ast \;\colon\; Ho(\mathcal{C}^{/\ast}) \longrightarrow Set^{\ast/} \,.

We may choose representatives such that AA is cofibrant, and ff is a fibration. Then we are faced with an ordinary pullback diagram

hofib(f) i X p * Y \array{ hofib(f) &\overset{i}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \ast &\longrightarrow& Y }

and the hom-classes are represented by genuine morphisms in 𝒞\mathcal{C}. From this it follows immediately that ker(p *)ker(p_\ast) includes im(i *)im(i_\ast). Hence it remains to show that every element in ker(p *)ker(p_\ast) indeed comes from im(i *)im(i_\ast).

But an element in ker(p *)ker(p_\ast) is represented by a morphism α:AX\alpha \colon A \to X such that there is a left homotopy as in the following diagram

A α X i 0 η˜ p A i 1 Cyl(A) η Y = * Y. \array{ && A &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{i_0}}\downarrow &{}^{\tilde \eta}\nearrow& \downarrow^{\mathrlap{p}} \\ A &\overset{i_1}{\longrightarrow} & Cyl(A) &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow^{\mathrlap{=}} \\ \ast && \longrightarrow && Y } \,.

Now by lemma \ref{ComponentMapsOfCylinderAndPathSpaceInGoodSituation} the square here has a lift η˜\tilde \eta, as shown. This means that i 1η˜i_1 \circ\tilde \eta is left homotopic to α\alpha. But by the universal property of the fiber, i 1η˜i_1 \circ \tilde \eta factors through i:hofib(f)Xi \colon hofib(f) \to X.

In homotopy type theory

In homotopy type theory the homotopy fiber of a function term f:ABf : A \to B over a function term pt B:*Bpt_B : * \to B is the type

a:A(f(a)=pt B), \sum_{a : A} (f(a) = pt_B) \,,

hence the dependent sum over AA of the identity type on BB with f(a)f(a) and pt Bpt_B substituted. (A special case of the discussion at homotopy pullback)

For the corresponding Coq code see

Long fibration sequences

A crucial difference between \infty-categorical fibration sequences and ordinary 1-categorical sequences is that the former are always long : in contrast to the ordinary kernel of a kernel, which is necessarily trivial, the homotopy kernel of a homotopy kernel is typically far from trivial, but is a loop space object. Due to that, each fibration sequence extend to the left by as many steps (times 3) as the objects involved have nontrivial homotopy groups.

Kernel of a kernel: loop objects

In particular the homotopy fiber of the point *B{*} \to B is the loop space object ΩB\Omega B of BB (by definition):

ΩB * * B. \array{ \Omega B &\to& {*} \\ \downarrow && \downarrow \\ {*} &\stackrel{}{\to}& B } \,.

Notice that the ordinary 1-categorical pullback of a point to itself is necessarily just the point again. Much of what makes (∞,1)-category-theory richer than ordinary category theory is this fact that the kernel of the point is not trivial, but loops. This implies in particular that the kernel of the kernel is in general nontrivial.

Namely the homotopy kernel of the morphism ker(f)Aker(f) \to A constructed above is by definition the homotopy limit in the diagram

ker(ker(f)) ker(f) * A \array{ ker(ker(f)) &\to& ker(f) \\ \downarrow && \downarrow \\ {*} &\to & A }

This is the same kind of diagram as before, just depicted after taking its mirror image along a diagonal. The point of drawing it this way is that this suggests to form the pasting diagram with the one that defines ker(f)ker(f)

ker(ker(f)) ker(f) * * A f B. \array{ ker(ker(f)) &\to& ker(f) &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& A &\stackrel{f}{\to}& B } \,.

Since the (,1)(\infty,1)-categorical pullback satisfies the pasting law just as ordinary pullback diagrams do, it follows that the total outer square obtained this way is itself a homotopy pullback. But by definition of te loop space object ΩB\Omega B this means that the kernel of the kernel is loops:

ker(ker(f))ΩB. ker(ker(f)) \simeq \Omega B \,.

I.e. all three squares in

ΩB ker(f) * * A f B \array{ \Omega B &\to& ker(f) &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& A &\stackrel{f}{\to}& B }

are (homotopy) pullback squares.

Long fibration sequences

Continuing this way to the left with the pasting law, we obtain a long fiber sequence of morphisms to the left of the form

Ωker(f)ΩAΩfΩBker(f)gAfB. \cdots \to \Omega ker(f) \longrightarrow \Omega A \stackrel{\Omega f}{\longrightarrow} \Omega B \longrightarrow ker(f) \stackrel{g}{\longrightarrow} A \stackrel{f}{\longrightarrow} B \,.

A subtlety to be aware of here is that ΩB\Omega B is not quite ker(ker(f))ker(ker(f)), but the latter instead is Ω¯f\bar \Omega f, where Ω¯\bar \Omega denotes loops with reversed orientation.

A classical discussion of this in terms of computing homotopy fibers via path object fibrant replacements is e.g. in (Switzer 75, around 2.57). But let’s see it just diagrammatically:

First observe that it is indeed Ωf\Omega f and not Ω¯f\bar \Omega f that appears in the above: by “bending around” the bottom left “*\ast \to ” we get

ΩA * Ωf ΩB ker(f) * g * A f B(* * ker(f) * g A f B * *)lim(ΩAΩfΩB). \array{ \Omega A &\longrightarrow& \ast \\ \downarrow^{\mathrlap{\Omega f}} && \downarrow \\ \Omega B & \longrightarrow & ker(f) &\longrightarrow& \ast \\ \downarrow && \downarrow^{\mathrlap{g}} && \downarrow \\ \ast &\longrightarrow & A &\stackrel{f}{\longrightarrow}& B } \;\;\;\;\; \simeq \;\;\;\;\; \left( \array{ \ast &\longrightarrow& \ast \\ \downarrow && \downarrow \\ ker(f) &\longrightarrow& \ast \\ \downarrow^{\mathrlap{g}} && \downarrow \\ A &\stackrel{f}{\longrightarrow}& B \\ \uparrow && \uparrow \\ \ast &\longrightarrow& \ast } \right) \stackrel{\underset{\longleftarrow}{\lim}}{\mapsto} \left( \Omega A \stackrel{\Omega f}{\longrightarrow} \Omega B \right) \,.

On the other hand, if we define the homotopy fiber of any morphism ϕ\phi by the diagram

ker(ϕ) * ϕ \array{ ker(\phi) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ & \stackrel{\phi}{\longrightarrow} & }

then ker(g)ker(g) is given by the diagram

ker(g) * ker(f) g A \array{ ker(g) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ ker(f) &\stackrel{g}{\longrightarrow}& A }

but what appears in the above pasting diagram is instead this diagram “reflected at the diagonal axis”

ker(g) * ker(f) g AΩB * ker(f) g A(ΩB ker(f) * A) 1 \array{ ker(g) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ ker(f) &\stackrel{g}{\longrightarrow}& A } \;\;\; \simeq \;\;\; \array{ \Omega B &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ ker(f) &\stackrel{g}{\longrightarrow}& A } \;\;\;\; \simeq \;\;\;\; \left( \array{ \Omega B &\longrightarrow& ker(f) \\ \downarrow &\swArrow& \downarrow \\ \ast &\longrightarrow& A } \right)^{-1}

Here “() 1(-)^{-1}” denotes the inverse of the 2-morphism (homotopies). Since it is these 2-morphisms/homotopies that become the loops in the loop space, it is here that loop reversal appears in translating between the naive iterated homotopy fiber to the construction that actually appears in the above pasting composite.

Long exact sequences in cohomology

Usually, when looking at fibration sequences in 1-categorical contexts of the homotopy category of an (∞,1)-category, one doesn’t see these long fibration squences directly, but only “in cohomology”.

This can be usefully understood as follows:

recall from cohomology that for XX and AA objects in an (∞,1)-category CC that is an (∞,1)-topos, the \infty-groupoid of AA-valued cocycle on XX is just Hom C(X,A)Hom_C(X,A), so that the corresponding cohomology classes are

H(X,A)=Π 0Hom C(X,A)=Ho C(X,A), H(X,A) = \Pi_0 Hom_C(X,A) = Ho_C(X,A) \,,

where Ho CHo_C is the corresponding homotopy category of an (∞,1)-category.

The upshot being that in the right (,1)(\infty,1)-context cohomology is just the hom-object.

But the hom-functor has the crucial property that it is an exact functor in both arguments. This holds for (,1)(\infty,1)-categories just as well as for ordinary categories. For our context this means in particular that for

A× KB B A K \array{ A \times_K B &\to& B \\ \downarrow && \downarrow \\ A &\to& K }

a homotopy pullback in CC, for every XKX \in K the induced diagram

Hom C(X,A× KB) Hom C(X,B) Hom C(X,A) Hom C(X,K) \array{ Hom_C(X,A \times_K B) &\to& Hom_C(X,B) \\ \downarrow && \downarrow \\ Hom_C(X,A) &\to& Hom_C(X,K) }

is again homotopy pullback diagram (of ∞-groupoids); in particular the morphism Hom C(X,A× KB)Hom C(X,A)× Hom C(X,K)Hom C(X,B)Hom_C(X,A \times_K B)\to Hom_C(X,A) \times_{Hom_{C}(X,K)} Hom_C(X,B) induced by the universal property of homotopy pullback is an equivalence.

So in particular for

ΩΩBΩker(f)ΩAΩ¯fΩBker(f)AfB \cdots \to \Omega \Omega B \to \Omega ker(f) \to \Omega A \stackrel{\bar \Omega f}{\to} \Omega B \to ker(f) \to A \stackrel{f}{\to} B

a fibration sequence and for XX any object, there is a fibration sequence

Hom C(X,ΩΩB)Hom C(X,Ωker(f))Hom C(X,ΩA)Hom C(X,Ω¯f)Hom C(X,ΩB)Hom C(X,ker(f))Hom C(X,A)Hom C(X,f)Hom C(X,B) \cdots \to Hom_C(X,\Omega \Omega B) \to Hom_C(X,\Omega ker(f)) \to Hom_C(X,\Omega A) \stackrel{Hom_C(X,\bar \Omega f)}{\to} Hom_C(X,\Omega B) \to Hom_C(X,ker(f)) \to Hom_C(X,A) \stackrel{Hom_C(X,f)}{\to} Hom_C(X,B)

is again a fibration sequence, now of \infty-groupoids. By projecting everything to connected components with Π 0\Pi_0 this then yields an ordinary long exact sequence of pointed sets

Ho C(X,ΩΩB)Ho C(X,Ωker(f))Ho C(X,ΩA)Ho C(X,Ω¯f)Ho C(X,ΩB)Ho C(X,ker(f))Ho C(X,A)Ho C(X,f)Ho C(X,B). \cdots \to Ho_C(X,\Omega \Omega B) \to Ho_C(X,\Omega ker(f)) \to Ho_C(X,\Omega A) \stackrel{Ho_C(X,\bar \Omega f)}{\to} Ho_C(X,\Omega B) \to Ho_C(X,ker(f)) \to Ho_C(X,A) \stackrel{Ho_C(X,f)}{\to} Ho_C(X,B) \,.

Due to the identitfication of cohomology with these homotopy hom-sets via Ho C(X,A)=:H(X,A)Ho_C(X,A) =: H(X,A), this is a “long exact sequence in cohomology”

H(X,ΩΩB)H(X,Ωker(f))H(X,ΩA)H(X,Ω¯f)H(X,ΩB)H(X,ker(f))H(X,A)H(X,f)H(X,B). \cdots \to H(X,\Omega \Omega B) \to H(X,\Omega ker(f)) \to H(X,\Omega A) \stackrel{H(X,\bar \Omega f)}{\to} H(X,\Omega B) \to H(X,ker(f)) \to H(X,A) \stackrel{H(X,f)}{\to} H(X,B) \,.

Long exact sequences of homotopy groups

See long exact sequence of homotopy groups.


Characterization of equivalences

We may read off from the non-triviality of the homotopy fiber AA of a morphism f:BCf : B \to C to which extent ff fails to be an equivalence.


A morphism f:BCf : B \to C in ∞Grpd is an equivalence precisely if all its homotopy fibers over every point of CC are contractible, i.e. are equivalent to **.

More generally, a morphism f:BCf : B \to C in any (∞,1)-category 𝒞\mathcal{C} is an equivalence if for all objects X𝒞X \in \mathcal{C} all homotopy fibers of the morphism 𝒞(X,f):𝒞(X,B)𝒞(X,C)\mathcal{C}(X,f) : \mathcal{C}(X,B) \to \mathcal{C}(X,C) are contractible.

For more on this see n-truncated object of an (∞,1)-category. Also HTT, around example For more on homotopy fibers of hom-spaces see the section below.

Fiber sequences of (,1)(\infty,1)-functor categories

We have seen that for ABfCA \to B \stackrel{f}{\to} C a fiber sequence in an (,1)(\infty,1)-category 𝒞\mathcal{C}, then for any other object XX we obtain a fiber sequence

Hom 𝒞(X,A)Hom 𝒞(X,B)f *Hom 𝒞(X,C) Hom_{\mathcal{C}}(X,A) \to Hom_{\mathcal{C}}(X,B) \stackrel{f_*}{\to} Hom_{\mathcal{C}}(X,C)

in ∞Grpd, where the point of Hom 𝒞(X,C)Hom_{\mathcal{C}}(X,C) is X*CX \to * \to C with *C* \to C the point of CC, so that Hom 𝒞(X,A)Hom_{\mathcal{C}}(X,A) is the homotopy fiber over this point of the morphism given by postcomposition with BCB \to C.

Often it is important to know the homotopy fibers of f *f_* also over other objects in Hom 𝒞(X,C)Hom_{\mathcal{C}}(X,C). This is notably the case when considering twisted cohomology with coefficients in AA.


The homotopy fiber of Hom 𝒞(X,B)f *Hom 𝒞(X,C)Hom_{\mathcal{C}}(X,B) \stackrel{f_*}{\to} Hom_{\mathcal{C}}(X,C) over a morphism c:XCc : X \to C may be identified with the hom-object

Hom 𝒞 /C(c,f) Hom_{\mathcal{C}_{/C}}(c,f)

in the over (∞,1)-category 𝒞 /C\mathcal{C}_{/C}.

This is HTT, prop.


Model all (∞,1)-categories as quasi-categories.

Using the discussion at hom-object in a quasi-category, we observe that

Hom 𝒞(X,C)𝒞 /C× 𝒞{X} Hom_{\mathcal{C}}(X,C) \simeq \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\}


Hom 𝒞(X,B)𝒞 /B× 𝒞{X}𝒞 /f× 𝒞{X} Hom_{\mathcal{C}}(X,B) \simeq \mathcal{C}_{/B} \times_{\mathcal{C}} \{X\} \simeq \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\}

under which identification the map f *f_* is induced by the canonical

ϕ:𝒞 /f𝒞 /C. \phi'' : \mathcal{C}_{/f} \to \mathcal{C}_{/C} \,.

So we are done if we can show that the ordinary pullback diagram

𝒞 /f× 𝒞 /C{c} 𝒞 /f× 𝒞{X} ϕ ϕ {c} 𝒞 /C× 𝒞{X} \array{ \mathcal{C}_{/f} \times_{\mathcal{C}_{/C}} \{c\} &\to& \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\} \\ \downarrow^{\mathrlap{\phi}} && \downarrow^{\mathrlap{\phi'}} \\ \{c\} &\to& \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\} }

is a homotopy pullback square, because we have an isomorphism of simplicial sets

𝒞 /f× 𝒞 /C{c}Hom 𝒞 /X R(c,f), \mathcal{C}_{/f} \times_{\mathcal{C}_{/C}} \{c\} \simeq Hom^R_{\mathcal{C}_{/X}}(c,f) \,,

as one checks.

Since in the above diagram all objects are Kan complexes, for the diagram to be a homotopy pullback it is sufficient that ϕ\phi' is a Kan fibration for which in turn it is sufficient that it is a left fibration.

That follows by noticing that the right vertical morphism fits into the pullback diagram

𝒞 /f× 𝒞{X} 𝒞 /f ϕ ϕ 𝒞 /C× 𝒞{X} 𝒞 /C. \array{ \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\} &\to& \mathcal{C}_{/f} \\ \downarrow^{\mathrlap{\phi'}} && \downarrow^{\mathrlap{\phi''}} \\ \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\} &\to& \mathcal{C}_{/C} } \,.

But by general properties of left fibrations, the right vertical map is a left fibration. And since these are stable under pullbacks, so is ϕ\phi'.

Principal \infty-bundles

Fibration sequences are familiar from the context of principal bundles.

Let GG be a group and let BG\mathbf{B}G denote the corresponding one-object groupoid (in the world of ∞-groupoids) or else the classifying space G\mathcal{B}G.

Notice that

GΩBG. G \simeq \Omega \mathbf{B} G \,.

Then that a GG-principal bundle PXP \to X is classified by morphism XBGX \to \mathbf{B}G means that it is the homotopy fiber of this morphism.

Indeed, as indicated at generalized universal bundle and at homotopy limit, we may compute the homotopy pullback

P * X BG \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G }

by first forming the standard fibrant replacement of the diagram XBG*X \to \mathbf{B}G \leftarrow {*}. That is given by the diagram

XBGEG, X \to \mathbf{B}G \leftarrow \mathbf{E}G \,,

where EG*\mathbf{E}G \simeq {*} is the total “space” (or 2-groupoid) of the universal GG-bundle. Once we have done this weakly equivalent replacement, the homotopy pullback may be computed as the ordinary pullback

P EG X^ g BG, \array{ P &\to & \mathbf{E}G \\ \downarrow && \downarrow \\ \hat X &\stackrel{g}{\to}& \mathbf{B}G } \,,

in the ordinary 1-category of nn-groupoids or spaces, using a replacement X^X\hat X \stackrel{\simeq}{\twoheadrightarrow} X of XX by an acyclic fibration (called “hypercover” in this context) (for instance the Čech groupoid associated with an open cover of XX).

One recognizes the usual statement that principal GG-bundles all arise as pullbacks of the universal GG-principal bundle.

The fact that such pullbacks really are bundles whose fiber is GG is the statement of the long fibration sequence induced by gg which says that picking any point *X{*} \to X of XX and then pulling back PP to that point (i.e. restricting it to that point) yields ΩBG=G\Omega \mathbf{B}G = G:

GΩBG P EG * x X^ g BG, \array{ G \simeq \Omega \mathbf{B}G &\to& P &\to & \mathbf{E}G \\ \downarrow && \downarrow && \downarrow \\ {*}&\stackrel{x}{\to}& \hat X &\stackrel{g}{\to}& \mathbf{B}G } \,,

The same logic – even the same diagrams – work for principal 2-bundles and generally for principal ∞-bundles.

Action groupoids

Let GG be an ∞-group in that BG\mathbf{B}G is an ∞-groupoid with a single object. An action of GG on an (∞,1)-category is an (∞,1)-functor

ρ:BG(,1)Cat \rho : \mathbf{B}G \to (\infty,1)Cat

to (∞,1)Cat. This takes the single object of BG\mathbf{B}G to some (,1)(\infty,1)-category VV.

The action groupoid V//GV//G is the (∞,1)-categorical colimit over the action:

C//G:=lim ρ. C//G := \lim_\to \rho \,.

By the result described here this is, equivalent to the pullback of the “universal (,1)Cat(\infty,1)Cat-bundle” Z(,1)CatZ \to (\infty,1)Cat, namely to the coCartesian fibration

V//G Z BG ρ (,1)Grpd \array{ V//G &\to& Z \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Grpd }

classified by ρ\rho under the (∞,1)-Grothendieck construction. We obtain a fiber sequence to the left by adjoining the (,1)(\infty,1)-categorical pullback along the point inclusion *BG* \to \mathbf{B}G

V V//G Z * BG ρ (,1)Grpd. \array{ V&\to& V//G &\to& Z \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Grpd } \,.

The resulting total (,1)(\infty,1)-pullback rectangle is the fiber of Z(,1)CatZ \to (\infty,1)Cat over the (,1)(\infty,1)-category CC, which is VV itself, as indicated.

Notice that every fibration sequence VV//GBGV \to V//G \to \mathbf{B}G with V//GBGV//G \to \mathbf{B}G a coCartesian fibration arises this way, up to equivalence.

Integral versus real cohomology

One of the most basic fibration sequences that appears all over the place in practice is the sequence of Eilenberg-MacLane objects

B n1/B nB nB n/B n+1 \cdots \to \mathbf{B}^{n-1} \mathbb{R}/\mathbb{Z} \to \mathbf{B}^n \mathbb{Z} \to \mathbf{B}^n \mathbb{R} \to \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{B}^{n+1} \mathbb{Z} \to \cdots

in H=\mathbf{H} = ∞Grpd/Top, where \mathbb{R} is the abelian group (under addition) of real number, \mathbb{Z} the abelian group of integers, and where /=S 1=U(1)\mathbb{R}/\mathbb{Z} = S^1 = U(1) is their quotient group, the circle group or unitary group U(n)U(n) for n=1n=1.

A quick way to see that this is indeed a fibration sequence is to realize that /\mathbb{R}/\mathbb{Z} is equivalent to the weak quotient action groupoid //\mathbb{R}//\mathbb{Z}. Since everything here is in the image of the Dold-Kan correspondence

GrpdKanCplxsSetFUsAbN ΞCh + \infty Grpd \simeq KanCplx \hookrightarrow sSet \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}} sAb \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N^\bullet}{\to}} Ch^+

it is useful to model this fiber sequence as a sequence of chain complexes

(B nB n//B n+1B)=UΞ([ 0 ][ ][ 0 ]). ( \mathbf{B}^n\mathbb{R} \to \mathbf{B}^n \mathbb{R}//\mathbb{Z} \to \mathbf{B}^{n+1}\mathbf{B} \mathbb{Z} ) \;\;\; = \;\;\; U \Xi \left( \left[ \array{ \vdots \\ \downarrow \\ 0 \\ \downarrow \\ \mathbb{R} \\ \downarrow \\ \vdots } \right] \;\;\; \to \;\;\; \left[ \array{ \vdots \\ \downarrow \\ \mathbb{Z} \\ \downarrow \\ \mathbb{R} \\ \downarrow \\ \vdots } \right] \;\;\; \to \;\;\; \left[ \array{ \vdots \\ \downarrow \\ \mathbb{Z} \\ \downarrow \\ 0 \\ \downarrow \\ \vdots } \right] \right) \,.

Written this way the second morphism is evidently a degreewise surjection, hence is a fibration in the model structure on chain complexes. Therefore this being a fibration sequence is equivalent (as described at homotopy pullback) to the first morphism being equivalent to the ordinary kernel of the second, which clearly it is.

From this fiber sequence we obtain long exact sequences in cohomology, for instance in singular cohomology: let XX be a topological space and for AA an abelian group let

H n(X,A):=π 0Grpd(Π(X),B nA) H^n(X,A) := \pi_0 \infty Grpd( \Pi(X), \mathbf{B}^n A)

be its cohomology with coefficients in AA, computed in ∞Grpd which we may present by sSet, where the fundamental ∞-groupoid Π(X)\Pi(X) is the singular simplicial complex SingXSing X and we have

H n(X,A)=π 0sSet(SingX,UΞA[n]). H^n(X,A) = \pi_0 sSet(Sing X, U \Xi A[n]) \,.

Then, as discussed above, the fiber sequence of coefficients B n1/B nB nB n/B n+1 \cdots \to \mathbf{B}^{n-1} \mathbb{R}/\mathbb{Z} \to \mathbf{B}^n \mathbb{Z} \to \mathbf{B}^n \mathbb{R} \to \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{B}^{n+1} \mathbb{Z} \to \cdots yields the long exact sequence in cohomology

H n1(X,/)H n(X,)H n(X,)H n(X,/)H n+1(X,). \cdots H^{n-1}(X,\mathbb{R}/\mathbb{Z}) \to H^n(X,\mathbb{Z}) \to H^n(X,\mathbb{R}) \to H^n(X,\mathbb{R}/\mathbb{Z}) \to H^{n+1}(X,\mathbb{Z}) \to \cdots \,.

In applications it often happens that one has a situation where the real cohomology is trivial, i.e H n(X,)=0H^n(X,\mathbb{R}) = 0 in some degree nn. In that case the exactness of the sequence

H n(X,)0H n(X,/)H n+1()0 \cdots \to H^n(X,\mathbb{Z}) \to 0 \to H^n(X,\mathbb{R}/\mathbb{Z}) \to H^{n+1}(\mathbb{Z}) \to 0 \to \cdots

implies that in this case we have an isomorphism

H n(X,/)H n+1(). H^n(X,\mathbb{R}/\mathbb{Z}) \stackrel{\simeq}{\to} H^{n+1}(\mathbb{Z}) \,.

Mayer-Vietoris sequences

A Mayer-Vietoris sequence is a fiber sequence obtained from an (,1)(\infty,1)-pullback diagram of pointed objects:


A× CB B g A f C \array{ A \times_C B &\to& B \\ \downarrow && \downarrow^{\mathrlap{g}} \\ A &\stackrel{f}{\to}& C }

is an infinity-pullback diagram in H\mathbf{H}, then it naturally induces a fiber sequence that starts out as

ΩCA× CBA×B. \Omega C \to A \times_C B \to A \times B \,.

This – or its associated long exact sequence of homotopy groups – is called the Mayer-Vietoris sequence of the pullback. See there for details.

Of a cover

The original example of Mayer-Vietoris sequences is obtained from the situtation where a homotopy pushout diagram

UV U V X \array{ U \cap V &\hookrightarrow& U \\ \downarrow && \downarrow \\ V &\to& X }

in H=\mathbf{H} = ∞Grpd/Top is given (which modeled in sSet or in terms of CW-complexes in Top may be modeled by an ordinary pushout), and where AGrpdA \in \infty Grpd is some coefficient group object. Then applying the (,1)(\infty,1)-categorical hom H(,A):H opGrpd\mathbf{H}(-, A) : \mathbf{H}^{op} \to \infty Grpd yields the \infty-pullback diagram

H(X,A) H(U,A) H(V,A) H(UV,A). \array{ \mathbf{H}(X, A) &\to& \mathbf{H}(U,A) \\ \downarrow && \downarrow \\ \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A) } \,.

By the above this is equivalent to

H(X,A)H(U,A)×H(V,A)H(UV,A) \mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A) \to \mathbf{H}(U \cap V, A)

being a fibration sequence. The corresponding long exact sequence in cohomology (as discussed above) is what is traditionally called the Mayer-Vietoris sequence of the cover of XX by UU and VV in AA-cohomology.


The observation of long exact sequences of homotopy groups for homotopy fiber sequences originates (according to Switzer 75, p. 35) in

  • M. G. Barratt, Track groups I, II. Proc. London Math. Soc. 5, 71-106, 285-329 (1955).

The first exhaustive study of these is due to

  • Dieter Puppe, Homotopiemengen und ihre induzierten Abbildungen I, Math. Z. 69, 299-344 (1958).

whence the terminology Puppe sequences.

Classical textbook accounts include

The construction in the axiomatic homotopy theory of model categories is due to

  • Daniel Quillen, chapter I.3 of Axiomatic homotopy theory in Homotopical algebra, Lecture Notes in Mathematics, No. 43 43, Berlin (1967)

and in the context of categories of fibrant objects due to

A discussion of fiber sequences in terms of associated ∞-bundles is in

Related discussion on the behaviour of fiber sequences under left Bousfield localization of model categories is in

Revised on April 27, 2016 15:28:26 by Urs Schreiber (