nLab obstruction

Idea

General

Given a fiber sequence $F \to A \stackrel{\mathbf{c}}{\to} B$ of classifying spaces/moduli stacks, hence $[\mathbf{c}]$ a universal characteristic class, and given an “$A$-structure” in the form of a morphism (cocycle) $f : X \to A$, then a lift $\hat f$ through $F \to A$ to an “$F$-structure” exists precisely if the induced $B$-structure $\mathbf{c}(f) : X \to B$ is trivializable in $B$-cohomology. One says that $[\mathbf{c}(f)]$ it is the obstruction to lifting the $A$-structure to an $F$-structure.

$$\array{ && F &\to& * \\ & {}^{\mathllap{\hat f}}\nearrow & \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{pt_B}} \\ X &\stackrel{f}{\to}& A &\stackrel{\mathbf{c}}{\to}& B }$$

Twisted cohomology

Conversely, by the universal property of fiber sequences, $A$-cocycles are equivalent to $B$-cocycles whose obstruction class under $f$ is trivial.

Therefore it makes sense to ask for the infinity-groupoid of $B$-cocycles whose class under $f$ has some other fixed value $\chi$. This gives $\chi$-_twisted cohomology_ with coefficients in $A$.

In terms of homotopy type theory

Formulated in homotopy type theory, obstruction theory reduces to a rather simple statement about factorization, or not, of functions through kernels of other functions. We spell out some details.

Let $\mathbf{c} : A \to B$ be a term of function type and let $pt_B : * \to B$ be a global point of $B$. The fiber of $\mathbf{c}$ over $pt_B$

$$F \coloneqq \sum_{a : A} (\mathbf{c}(a) = pt_B)$$

comes with a canonical “inclusion” function

$$i : F \to A$$

given by

$$i : (a, \mathbf{c}(a) \stackrel{\simeq}{\to} pt_B) \mapsto a \,.$$

Now let

$$f : X \to A$$

be any other function. We are asking for the obstruction to lift it to a function $\hat f : X \to F$ such that

$$i \circ \hat f \simeq f \,.$$

This exists precisely if there is an equivalence

$$\phi : \mathbf{c}\circ f \stackrel{\simeq}{\to} pt_B$$

hence if the obstruction class

$$f^* \mathbf{c} \coloneqq \mathbf{c} \circ f$$

is trivial.

Examples

Lift through Postnikov stages

If $F \to A$ in the above is a stage $\tau_{\leq n+1} B \to \tau_{\leq n}B$ in the Postnikov tower of an object $B$, then the lifting problem is that of lifting through the Postnikov tower of $A$ and the universal obstruction class is that which classified $\tau_{\leq n+1} B \to \tau_{\leq n}B$ as a $\pi_{n+1} B$-principal infinity-bundle.

Obstruction to extension

The formal dual of the lift obstruction problem discussed above is the following extension problem:

we start with a universal characteristic map

$$\mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A$$

representing a class $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ in the $A$-cohomology of $\mathbf{B}G$. Then given a morphism $\phi : \mathbf{B}G \to \mathbf{B}H$ we may ask for the obstruction to extending $\mathbf{c}$ along it.

Now the statement is: if $\phi$ is a homotopy cofiber, then there is a good obstruction theory to answer this question. Namely in that situation we are looking at a diagram of the form

$$\array{ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A \\ \downarrow && \downarrow^{\mathrlap{\phi}} & \nearrow_{\mathrlap{\hat \mathbf{c}}} \\ * &\to& \mathbf{B}H }$$

where the left square is an homotopy pushout. By its universal property, the extension $\hat {\mathbf{c}}$ of $\mathbf{c}$ exists as indicated precisely if the class

$$[f^* \mathbf{c}] \in H^n(\mathbf{B}Q, A)$$

is trivial.

One class of examples for this sort of situation is where one considers refined Lie group cohomology on simply connected Lie groups and is asking for ways to push it down to discrete quotients, hence to non-simply connected Lie groups integrating the same Lie algebra. This is often phrased in terms of “multiplicative bundle gerbes” over these Lie groups, but that is just another way of talking about the corresponding cohomology of the smooth moduli stack $\mathbf{B}G$.

Obstruction to quantization: Quantum anomaly

There are various formalizations of the notion of quantization in physics, or at least various aspects of that formalization. This involves various steps, some of which may have obstructions to being carried out. In physics such an obstruction in the process of quantization is often called a quantum anomaly.

For instance for many theories in physics the action functional is a priori not a function on the fields but a section of a circle-principal bundle. For this to qualify as an action functional therefore one needs a trivialization of that bundle and so the Chern class of the bundle is the obstruction and hence an anomaly of the system. See there for more.

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