nLab
homotopy pullback

Idea

A homotopy pullback is a special kind of homotopy limit: the appropriate notion of pullback in the context of homotopy theory. Homotopy pullbacks model the quasi-category pullbacks in the (infinity,1)-category that is presented by a given homotopical category. Since pullbacks are also called fiber products, homotopy pullbacks are also called homotopy fiber products.

For more details see homotopy limit.

Definition

As with all homotopy limits, there is both a local and a global notion of homotopy pullback.

The global definition says that the homotopy pullback of a a cospan $X\to Z\leftarrow Y$ in a category with weak equivalences $C$ is its image under the right derived functor of the base change functor $\mathrm{pb}:C^{\to \leftarrow }\to C$.

The local definition says that the homotopy pullback of $X\to Z\leftarrow Y$ in a category with a notion of homotopy consists of a square

$$\begin{array}{ccc}P& \to & Y\\ \downarrow & & \downarrow \\ X& \to & Z\end{array}$$\array{ P & \to& Y \\ \downarrow && \downarrow \\ X &\to& Z }

that commutes up to homotopy, and such that for any other square

$$\begin{array}{ccc}T& \to & Y\\ \downarrow & & \downarrow \\ X& \to & Z\end{array}$$\array{ T & \to& Y \\ \downarrow && \downarrow \\ X &\to& Z }

that commutes up to homotopy, there exists a morphism $T\to P$ making the two triangles commute up to homotopy, and similarly for homotopies and higher homotopies. In other words, there is an equivalence

$$\mathrm{Map}(T,P)\simeq \mathrm{HoSq}(T,X\to Z\leftarrow Y)$$Map(T,P) \simeq HoSq(T,X\to Z\leftarrow Y)

between the space of maps $T\to P$ and the space of homotopy commutative squares with vertex $T$.

In good situations, such as when $X,Y,Z$ are fibrant? in a model category, the two constructions agree up to weak equivalence.

Note that in both cases, there is a canonical map from the actual pullback $X{\times }_{Z}Y$ to the homotopy pullback $X{\times }_Z^{h}Y$. In the global case this comes by the definition of a derived functor; in the local case it comes because a commutative square is, in particular, a homotopy commutative one.

Concrete construction

Suppose now that $Z$ has a path object $Z^I$ that represents homotopies into $Z$. (For instance, this is often the case when $C$ is a closed monoidal homotopical category with an interval object $I$.) Then we can consider the (ordinary) limit

$$\begin{array}{ccccc}X{\times }_Z^{h}Y& \to & \to & \to & Y\\ \downarrow & & & & \downarrow \\ \downarrow & & Z^I& \stackrel{d_1}{\to }& Z\\ \downarrow & & {\downarrow }^{d_0}\\ X& \to & Z\end{array}.$$\array{ X \times_{Z}^h Y &\to&\to& \to& Y \\ \downarrow &&&& \downarrow \\ \downarrow & & Z^I &\stackrel{d_1}{\to}& Z \\ \downarrow & & \downarrow^{d_0} \\ X & \to & Z } \,.

or equivalently as the ordinary pullback

$$\begin{array}{ccc}X{\times }_Z^{h}Y& \to & Z^I\\ \downarrow & & \downarrow \\ X\times Y& \to & Z\times Z.\end{array}$$\array{X\times_Z^h Y & \to & Z^I\\ \downarrow && \downarrow \\ X\times Y & \to & Z\times Z.}

In good situations, this will be a (local) homotopy pullback of $X\to Z\leftarrow Y$. This is the case when $C=$ Top with its canonical interval object $[\mathrm{0,1}]$, and also in many model categories (when $X,Y,Z$ are fibrant) and categories of fibrant objects. For details on the latter case, see section 5 of Nonabelian cocycles and their sigma model QFTs.

The canonical morphism $X{\times }_{Z}Y\to X{\times }_Z^{h}Y$ here is induced by the section $Z\to Z^I$.

The homotopy pullback constructed in this way is an example of a strict homotopy limit, as mentioned at homotopy limit. In such a case, one can say that an arbitrary homotopy-commutative square

$$\begin{array}{ccc}W& \to & Y\\ \downarrow & & \downarrow \\ Y& \to & X\end{array}$$\array{ W & \to& Y \\ \downarrow && \downarrow \\ Y &\to& X }

is a homotopy pullback square if the induced morphism from $W$ to the strict homotopy pullback is a weak equivalence.

Fibration sequences

Of particular interest are consecutive homotopy pullbacks of point inclusions. These give rise to fibration sequences and loop space objects.