nLab
fibration sequence

Idea

A fibration sequence is a “long right-exact sequence” (or left-exact, I keep mixing that up) in an (∞,1)-category.

Traditionally fibration 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.

Definition

Let $C$ be an (∞,1)-category with small limits and consider pointed objects of $C$, i.e. morphisms $*\to A$ from the terminal object $*$ (the point) to some object $A$. 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 $C$ happens to be a stable (∞,1)-category for which $*=0$ 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 $C$ is a stable $(\mathrm{\infty }\mathrm{,1})$-category. In particular, in the context of nonabelian cohomology (see gerbe and principal 2-bundle) one considers fibration sequences in non-stable $(\mathrm{\infty }\mathrm{,1})$-categories.

Now let $f:A\to B$ be a morphism in $C$.

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

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

Long fibration sequences

A crucial difference between $\mathrm{\infty }$-categorical fibration sequences and ordinary 1-categorical sequences is that the former are always long : in contrast to the ordinary kernel of a kernel, wich 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 $*\to B$ is the loop space object $\Omega B$ of $B$ (by definition):

$$\begin{array}{ccc}\Omega B& \to & *\\ \downarrow & & \downarrow \\ *& \stackrel{}{\to }& B\end{array}.$$\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 $\ker (f)\to A$ constructed above is by definition the homotopy limit in the diagram

$$\begin{array}{ccc}\ker (\ker (f))& \to & \ker (f)\\ \downarrow & & \downarrow \\ *& \to & A\end{array}$$\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 $\ker (f)$

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

Since the $(\mathrm{\infty }\mathrm{,1})$-categorical homotopy pullback comopose 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 $\Omega B$ this means that the kernel of the kernel is loops:

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

I.e. all three squares in

$$\begin{array}{ccccc}\Omega B& \to & \ker (f)& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & A& \stackrel{f}{\to }& B\end{array}$$\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, we obtain a long sequence of morphisms to the left

$$\cdots \to \Omega \Omega B\to \Omega A\stackrel{\overline{\Omega }f}{\to }\Omega B\to \ker (f)\to A\stackrel{f}{\to }B.$$\cdots \to \Omega \Omega B \to \Omega A \stackrel{\bar \Omega f}{\to} \Omega B \to ker(f) \to A \stackrel{f}{\to} B \,.

Here the $\overline{\Omega }$ indicates that the map involves reversing the direction of loops. This comes about by looking closely at the pullback diagrams that this comes from

$$\begin{array}{ccc}\Omega (A)& \to & *\\ {\downarrow }^{\overline{\Omega }f}& & \downarrow \\ \Omega B& \to & \ker (f)& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & A& \stackrel{f}{\to }& B\end{array}.$$\array{ \Omega(A) &\to& {*} \\ \downarrow^{\bar \Omega f} && \downarrow \\ \Omega B &\to& ker(f) &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ {*} &\to &A &\stackrel{f}{\to}& B } \,.

Again, all squares and all pasting squares appearing here are homtopy pullback squares. If I had labeled to two morphisms to the point out of the loop object one would see that $\overline{\Omega }f$ indeed reverses orientation of loops.

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 $X$ and $A$ objects in an (∞,1)-category $C$ that is an (∞,1)-topos, the $\mathrm{\infty }$-groupoid of $A$-valued cocycle on $X$ is just ${\mathrm{Hom}}_C(X,A)$, so that the corresponding cohomology classes are

$$H(X,A)={\Pi }_0{\mathrm{Hom}}_C(X,A)={\mathrm{Ho}}_C(X,A),$$H(X,A) = \Pi_0 Hom_C(X,A) = Ho_C(X,A) \,,

where ${\mathrm{Ho}}_C$ is the corresponding homotopy category of an (∞,1)-category.

The upshot being that in the right $(\mathrm{\infty }\mathrm{,1})$-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 $(\mathrm{\infty }\mathrm{,1})$-categories just as well as for ordinary categories. For our context this means in particular that for

$$\begin{array}{ccc}A{\times }_{K}B& \to & B\\ \downarrow & & \downarrow \\ A& \to & K\end{array}$$\array{ A \times_K B &\to& B \\ \downarrow && \downarrow \\ A &\to& K }

a homotopy pullback in $C$, for every $X\in K$ the induced diagram

$$\begin{array}{ccc}{\mathrm{Hom}}_C(X,A{\times }_{K}B)& \to & {\mathrm{Hom}}_C(X,B)\\ \downarrow & & \downarrow \\ {\mathrm{Hom}}_C(X,A)& \to & {\mathrm{Hom}}_C(X,K)\end{array}$$\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 ${\mathrm{Hom}}_C(X,A{\times }_{K}B)\to {\mathrm{Hom}}_C(X,A){\times }_{{\mathrm{Hom}}_C(X,K)}{\mathrm{Hom}}_C(X,B)$ induced by the universal property of homotopy pullback is an equivalence.

So in particular for

$$\cdots \to \Omega \Omega B\to \Omega \ker (f)\to \Omega A\stackrel{\overline{\Omega }f}{\to }\Omega B\to \ker (f)\to A\stackrel{f}{\to }B$$\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 $X$ any object, there is a fibration sequence

$$\cdots \to {\mathrm{Hom}}_C(X,\Omega \Omega B)\to {\mathrm{Hom}}_C(X,\Omega \ker (f))\to {\mathrm{Hom}}_C(X,\Omega A)\stackrel{{\mathrm{Hom}}_C(X,\overline{\Omega }f)}{\to }{\mathrm{Hom}}_C(X,\Omega B)\to {\mathrm{Hom}}_C(X,\ker (f))\to {\mathrm{Hom}}_C(X,A)\stackrel{{\mathrm{Hom}}_C(X,f)}{\to }{\mathrm{Hom}}_C(X,B)$$\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 $\mathrm{\infty }$-groupoids. By projecting everything to connected components with ${\Pi }_0$ this then yields an ordinary long exact sequence of pointed sets

$$\cdots \to {\mathrm{Ho}}_C(X,\Omega \Omega B)\to {\mathrm{Ho}}_C(X,\Omega \ker (f))\to {\mathrm{Ho}}_C(X,\Omega A)\stackrel{{\mathrm{Ho}}_C(X,\overline{\Omega }f)}{\to }{\mathrm{Ho}}_C(X,\Omega B)\to {\mathrm{Ho}}_C(X,\ker (f))\to {\mathrm{Ho}}_C(X,A)\stackrel{{\mathrm{Ho}}_C(X,f)}{\to }{\mathrm{Ho}}_C(X,B).$$\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 ${\mathrm{Ho}}_C(X,A)=:H(X,A)$, this is a “long exact sequence in cohomology”

$$\cdots \to H(X,\Omega \Omega B)\to H(X,\Omega \ker (f))\to H(X,\Omega A)\stackrel{H(X,\overline{\Omega }f)}{\to }H(X,\Omega B)\to H(X,\ker (f))\to H(X,A)\stackrel{H(X,f)}{\to }H(X,B).$$\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) \,.

Examples

Fibration sequences are familiar from the context of principal bundles.

Let $G$ be a group and let $BG$ denote the corresponding one-object groupoid (in the world of ∞-groupoids) or else the classifying space $ℬG$.

Notice that

$$G\simeq \Omega BG.$$G \simeq \Omega \mathbf{B} G \,.

Then that a $G$-principal bundle $P\to X$ is classified by morphism $X\to BG$ 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

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \to & BG\end{array}$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G }

by first forming the standard fibrant replacement of the diagram $X\to BG\leftarrow *$. That is given by the diagram

$$X\to BG\leftarrow EG,$$X \to \mathbf{B}G \leftarrow \mathbf{E}G \,,

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

$$\begin{array}{ccc}P& \to & EG\\ \downarrow & & \downarrow \\ \hat{X}& \stackrel{g}{\to }& BG\end{array},$$\array{ P &\to & \mathbf{E}G \\ \downarrow && \downarrow \\ \hat X &\stackrel{g}{\to}& \mathbf{B}G } \,,

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

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

The fact that such pullbacks really are bundles whose fiber is $G$ is the statement of the long fibration sequence induced by $g$ which says that picking any point $*\to X$ of $X$ and then pulling back $P$ to that point (i.e. restricting it to that point) yields $\Omega BG=G$:

$$\begin{array}{ccccc}G\simeq \Omega BG& \to & P& \to & EG\\ \downarrow & & \downarrow & & \downarrow \\ *& \stackrel{x}{\to }& \hat{X}& \stackrel{g}{\to }& BG\end{array},$$\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.