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In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.

Precomposition

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Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function of a variable where itself is a function of another variable may be written as a function of This is the pullback of by the function

It is such a fundamental process that it is often passed over without mention.

However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as differential forms and their cohomology classes; see

Fiber-product

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Main article: Pullback bundle

The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the resulting new pullback bundle looks locally like a Cartesian product of the new base space, and the (unchanged) fiber. The pullback bundle then has two projections: one to the base space, the other to the fiber; the product of the two becomes coherent when treated as a fiber product.

Generalizations and category theory

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The notion of pullback as a fiber-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases: inverse image (and pullback) sheaves in algebraic geometry, and pullback bundles in algebraic topology and differential geometry.

See also:

Functional analysis

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See also: Transpose of a linear map

When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the transpose or composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator.

Relationship

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The relation between the two notions of pullback can perhaps best be illustrated by sections of fiber bundles: if is a section of a fiber bundle over and then the pullback (precomposition) of s with is a section of the pullback (fiber-product) bundle over

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Pullback

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In , a is a limit of a consisting of two s with a common , yielding an object that captures the "fibered" relationship between the domain objects over the shared target, generalizing the in the . Formally, given s f:A→Cf: A \to C and g:B→Cg: B \to C, the pullback is an object PP equipped with s p1:P→Ap_1: P \to A and p2:P→Bp_2: P \to B such that f∘p1=g∘p2f \circ p_1 = g \circ p_2, satisfying a : for any object QQ with s q1:Q→Aq_1: Q \to A and q2:Q→Bq_2: Q \to B where f∘q1=g∘q2f \circ q_1 = g \circ q_2, there exists a unique u:Q→Pu: Q \to P making the commute. This construction is unique up to when it exists and is dual to the pushout. Pullbacks play a foundational role in categorical limits, as every finite limit in a category can be constructed from pullbacks along with a terminal object. In the (Set), the pullback of f:X→Zf: X \to Z and g:Y→Zg: Y \to Z is explicitly the fiber product X×ZY={(x,y)∣f(x)=g(y)}X \times_Z Y = \{(x, y) \mid f(x) = g(y)\}, with projection maps π1(x,y)=x\pi_1(x, y) = x and π2(x,y)=y\pi_2(x, y) = y. Categories with all pullbacks, such as the or smooth manifolds, enable the formation of fiber products that preserve relevant structures, like continuity or differentiability. Beyond pure , pullbacks have significant applications across mathematics. In , the pullback of fiber bundles along continuous maps f:X→Yf: X \to Y produces a new bundle f∗Ef^*E over XX, which is homotopy invariant when the bundle projection is a . In , the pullback operation on differential forms—defined for a smooth ϕ:M→N\phi: M \to N as (ϕ∗ω)(p)(v1,…,vk)=ω(ϕ(p))(dϕp(v1),…,dϕp(vk))(\phi^* \omega)(p)(v_1, \dots, v_k) = \omega(\phi(p))(d\phi_p(v_1), \dots, d\phi_p(v_k))—preserves multilinearity, skew-symmetry, and smoothness, facilitating integration and . In , pullbacks induce homomorphisms between K-groups of spaces or manifolds, again with invariance for homotopic maps. These constructions underscore the pullback's utility as a tool for transferring structure and data across categorical diagrams.

Foundational Concepts

Precomposition of Functions

In the context of functions between sets, the pullback, often denoted f∗f^*, refers to the operation of precomposition. Given functions f:X→Yf: X \to Y and g:Y→Zg: Y \to Z, the pullback f∗gf^* g is defined as the composite function g∘f:X→Zg \circ f: X \to Z. This operation embodies the intuitive process of substituting the output of ff into gg; specifically, if y=f(x)y = f(x), then f∗h(y)=h(f(x))f^* h(y) = h(f(x)) for any suitable h:Y→Zh: Y \to Z. The defining equation is (f∗g)(x)=g(f(x)),(f^* g)(x) = g(f(x)), which follows directly from the standard definition of function composition as substitution of the inner function's expression into the outer one. A simple example illustrates this for scalar functions. Consider g(y)=y2g(y) = y^2 where y∈Ry \in \mathbb{R}, and let f:R→Rf: \mathbb{R} \to \mathbb{R} be f(x)=x+1f(x) = x + 1. Then f∗g(x)=g(f(x))=(x+1)2=x2+2x+1f^* g (x) = g(f(x)) = (x + 1)^2 = x^2 + 2x + 1, effectively pulling back the squaring operation along ff via variable substitution. When restricted to linear maps between vector spaces, the pullback preserves : if ff and gg are linear, then f∗g=g∘ff^* g = g \circ f is linear, as composition of linear transformations yields another linear transformation. Regarding properties under composition, the pullback preserves injectivity and surjectivity: if both ff and gg are injective, then g∘fg \circ f is injective; similarly, if both are surjective, then g∘fg \circ f is surjective. These follow from the basic axioms of set functions and mappings. The underlying concept of precomposition via variable substitution was a of 19th-century , appearing in foundational results like change-of-variables theorems for integrals, developed by figures such as Lagrange and Gauss, well before its abstraction in 20th-century .

Cartesian Products in Sets

In the , the pullback of two functions p:X→Zp: X \to Z and q:Y→Zq: Y \to Z is constructed as a of the Cartesian product X×YX \times Y. Specifically, the pullback object, denoted X×ZYX \times_Z Y, consists of all ordered pairs (x,y)(x, y) such that p(x)=q(y)p(x) = q(y), i.e., X×ZY={(x,y)∈X×Y∣p(x)=q(y)}.X \times_Z Y = \{(x, y) \in X \times Y \mid p(x) = q(y)\}. This set is equipped with projection maps πX:X×ZY→X\pi_X: X \times_Z Y \to X defined by πX(x,y)=x\pi_X(x, y) = x and πY:X×ZY→Y\pi_Y: X \times_Z Y \to Y defined by πY(x,y)=y\pi_Y(x, y) = y, which satisfy the commuting condition p∘πX=q∘πYp \circ \pi_X = q \circ \pi_Y. To verify that this construction satisfies the universal property, consider any set WW together with maps a:W→Xa: W \to X and b:W→Yb: W \to Y such that p∘a=q∘bp \circ a = q \circ b. Define u:W→X×ZYu: W \to X \times_Z Y by u(w)=(a(w),b(w))u(w) = (a(w), b(w)). Since p(a(w))=q(b(w))p(a(w)) = q(b(w)) for all w∈Ww \in W, the image of uu lies in X×ZYX \times_Z Y. Moreover, πX∘u=a\pi_X \circ u = a and πY∘u=b\pi_Y \circ u = b. Uniqueness follows because any such map must send ww to the unique pair (a(w),b(w))(a(w), b(w)) that satisfies the projections. Thus, X×ZYX \times_Z Y is the universal object mediating maps over ZZ. A prominent example occurs when ZZ is a singleton set (a terminal object in Set), in which case pp and qq are the unique maps to the point, and the pullback X×ZYX \times_Z Y reduces to the ordinary Cartesian product X×YX \times Y. Another case arises when Y=ZY = Z and qq is the identity map idZ\mathrm{id}_Z; here, the pullback X×ZZ={(x,p(x))∣x∈X}X \times_Z Z = \{ (x, p(x)) \mid x \in X \}, which is isomorphic to XX via the first projection πX\pi_X, with the second projection πY=p∘πX\pi_Y = p \circ \pi_X. If p:X→Xp: X \to X is an endomorphism, the equalizer of pp and idX\mathrm{id}_X (a subobject consisting of fixed points {x∈X∣p(x)=x}\{ x \in X \mid p(x) = x \}) can be constructed as the pullback of pp and idX\mathrm{id}_X along idX\mathrm{id}_X. The universal property can be stated formally as follows: For any set WW with maps a:W→Xa: W \to X and b:W→Yb: W \to Y satisfying p∘a=q∘bp \circ a = q \circ b, there exists a unique map u:W→X×ZYu: W \to X \times_Z Y such that W→uX×ZYa↓πX↓X→pZW→uX×ZYb↓πY↓Y→qZ.\begin{CD} W @>u>> X \times_Z Y \\ @V a V V @V \pi_X V V \\ X @>p>> Z \end{CD} \qquad \begin{CD} W @>u>> X \times_Z Y \\ @V b V V @V \pi_Y V V \\ Y @>q>> Z. \end{CD}
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