equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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.
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 $pb: C^{\to \leftarrow} \to C$.
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 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.
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.
Let $A \to C \leftarrow B$ be a diagram in some model category. Then sufficient conditions for the ordinary (1-categorical) pullback $A \times_C B$ to present the homotopy pullback of the diagram are
one of the two morphisms is a fibration and all three objects are fibrant objects;
one of the two morphisms is a fibration and the model category is right proper.
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.
Notice that a fibrant resolution of the diagram in the injective model structure on functors has both morphisms be a fibration. So the first point in prop. says that (in the special case of pullbacks) something weaker than this is sufficient for computing the homotopy limit of the diagram.
This can be explained in model-categorical terms by the fact that the category of cospans also has a Reedy model structure in which the fibrant objects are precisely those considered in the first point above, and that homotopy limits can equally well be computed using this model structure (specifically, the adjunction $Const \dashv Lim$ is Quillen with respect to it).
In this spirit one may ask for the largest class of morphisms such that their ordinary pullbacks are already homotopy pullbacks. This is related to the concept of sharp morphisms.
Due to prop. 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 in $A \stackrel{f}{\to} C \stackrel{g}{\leftarrow} B$ all three objects are fibrant objects, then the homotopy pullback of this diagram is presented by the ordinary pullback
or, equivalently up to isomorphism, as the ordinary pullback
See also at Mayer-Vietoris sequence this proposition.
Since the objects are already fibrant, prop. implies that it is sufficient to replace one of the morphisms by a fibrant resolution. Such a resolution is provided by the factorization lemma: by Lemma 3, $B \to C$ admits a canonical fibrant resolution
where $C \stackrel{\simeq}{\to} C^I \to C \times C$ is a path space object for $C$ (for instance, when $C$ is a closed monoidal homotopical category then this can be taken to be the internal hom with an interval object $I$).
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 examples of this is implied by the following:
Let $\mathcal{C}$ be a category of simplicial presheaves over some site equipped with a local injective model structure on simplicial presheaves with respect to that site.
Then an ordinary pullback of $A \to C \leftarrow B$ in $\mathcal{C}$ is a homotopy pullback already when one of the two morphisms is an objectwise Kan fibration.
The global projective model structure on simplicial presheaves is right proper. So by prop. the ordinary pullback in question presents the homotopy pullback in the global structure. By the discussion at homotopy limit and Bousfield localization of model categories, this presents the (∞,1)-pullback of the diagram of (∞,1)-presheaves, and the fibrant replacement of that pullback in the local model structure presents the (∞,1)-sheafification of this (∞,1)-presheaf. This is (essentially by definition, see (∞,1)-topos) a left exact (∞,1)-functor and hence preserves finite (∞,1)-limits.
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 $a : A, b : B \vdash (f(a) = g(b))$ 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 $[a : A, b : B \vdash (f(a) = g(b))] \coloneqq (f,g)^* C^I$ 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 $\mathcal{C} \simeq \mathcal{C}_{/*}$ instead of as an object in the slice category $\mathcal{C}_{/ A \times B}$.
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 above.
Specifically, let $f \colon A \longrightarrow B$ be a function, then the categorical semantics for the expression
is the canonical fibration replacement of $f$ as it appears notably in the factorization lemma
In plain homotopy types (i.e. in ∞-groupoids, in the classical model structure on simplicial sets etc.) the following holds:
a diagram
is a homotopy pullback diagram precisely if it induces a weak equivalence on all homotopy fibers
for all elements $b \in B$.
(e.g. CPS 05, 5.2, MO)
For the analog of prop. to hold in (∞,1)-categories more general than ∞-groupoids one would need either “enough global elements” of the object $B$ to detect all homotopy fibers, or else one would need a suitable “internal” formulation of the statement.
On the other hand in stable homotopy theory the statement holds generally:
For $\mathcal{C}$ a stable model category, then for a diagram
the following are equivalent
the square is a homotopy pullback square;
the square is a homotopy pushout squre;
the induced morphism on the homotopy fiber over the zero object 0 is a weak equivalence
A proof in terms of stable model categories including the third item is spelled out for instance in (Hovey 07, remark 7.1.12). (Beware: this is not in the original volume from 1999, but is in the version “reprinted with corrections” from 2007.) A proof in terms of stable (∞,1)-categories of the first two items is in (Lurie HA, prop. 1.1.3.4 (2)).
Of particular interest are consecutive homotopy pullbacks of point inclusions. These give rise to fiber sequences and loop space objects.
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)
See the references at homotopy limit .
Fairly comprehensive resources are
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol 63, 1999,reprinted 2007
Jacob Lurie, appendix of Higher Topos Theory
See also
A proposal for a formalization of homotopy pushouts by higher inductive types in homotopy type theory has been given in