Contents

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.

A homotopy pushout is the dual concept.

For more details see homotopy limit.

In the context of homotopy type theory a homotopy pullback is the interpretation of the space of solutions to an equation.

Definition

In category theory

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 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

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

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

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.

In homotopy type theory

In homotopy type theory the homotopy pullback of a term of function type

along a term of function type

is given formally by precisely the same formula that would also define the ordinary fiber product of functions of sets, namely by

Spelled out, this is the dependent sum over the dependent identity type over the evaluation of $f$ and $g$.

What in classical logic is interpreted as the set of pairs $(a,b)$ such that $f(a)$ and $g(b)$ are equal here becomes the restriction of a mapping cocylinder.

Formal proof that this is the homotopy pullback in homotopy type theory is in (Brunerie). Proof in the categorical semantics of homotopy type theory is below.

Concrete constructions

We discuss various concrete constructions by ordinary pullbacks and ordinary limits such that under some sufficient conditions these compute homotopy pullbacks, up to weak equivalence.

General

Both statements are classical. They are reviewed for instance as Lurie, prop. A.2.4.4. The proof of the second statement is spelled out here.

Due to prop. 1 one typically computes homotopy pullbacks of a diagram by first forming a resolution of one of the two morphisms by a fibration and then forming an ordinary pullback.

If all objects involved are already fibrant, then such a resolution is provided by the factorization lemma. This says that a fibrant resoltuion of $B\to C$ is given by the total composite vertical morphism in

where $C\stackrel{\simeq }{\to }{C}^I\to C\times C$ is a path object for the fibrant object $C$ (For instance, when $C$ is a closed monoidal homotopical category then this can be taken to be the internal hom with an interval object $I$.)

Then by prop. 1 we have

The canonical morphism $A{\times }_{C}B\to A{\times }_C^{h}B$ here is induced by the canonical section $s:C\to C^I$ into the path space object, hence by the commutativity of the diagram

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

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

A useful class of example of this is implied by the following:

In homotopy type theory

If we unwind the categorical semantics of the above definition

of the homotopy pullback in homotopy type theory, we re-obtain the above prescription for how to construct homotopy pullbacks.

So let the ambient category be a suitable type-theoretic model category.

The type $a:A,b:B⊢(f(a)=g(b))$ is obtained by substitution from the identity type of $C$. By the discussion there, the categorical semantics of substitution is given by pullback of the fibrations that interpret the dependent types, and so this is interpreted as the pullback $[a:A,b:B⊢(f(a)=g(b))]≔(f,g{)}^{*}{C}^I$ of the path space object of $C$:

Forming the dependent sum over $a:A,b:B$ is simply interpreted as regarding the resulting object $(f,g{)}^{*}{C}^I$ as an object in $𝒞\simeq {𝒞}_{/*}$ instead of as an object in the slice category ${𝒞}_{/A\times B}$.

Since by assumption on the categorical interpretation of a type, all objects here are fibrant, this coincides with the expression of the homotopy pullback from corollary 1 above.

Examples

Fiber sequences

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

Related concepts

• equation

• homotopy limit

• (∞,1)-pullback, (∞,1)-limit

References

General

See the references at homotopy limit .

A fairly comprehensive resource is the appendix of

• Jacob Lurie, Higher Topos Theory

In terms of homotopy type theory

A proposal for a formalization of homotopy pushouts by higher inductive types in homotopy type theory has been given in

• Mike Shulman, Homotopy Type Theory VI (blog post)

A HoTT-Coq-coding of homotopy pullbacks and pushouts is in

• Guillaume Brunerie, HoTT/Coq/Limits/Pullbacks.v