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Pullback

In category theory, a pullback is a limit of a diagram consisting of two morphisms with a common codomain, yielding an object that captures the "fibered" relationship between the domain objects over the shared target, generalizing the Cartesian product in the category of sets.^[1] Formally, given morphisms f: A \to C and g: B \to C, the pullback is an object P equipped with morphisms p_1: P \to A and p_2: P \to B such that f \circ p_1 = g \circ p_2, satisfying a universal property: for any object Q with morphisms q_1: Q \to A and q_2: Q \to B where f \circ q_1 = g \circ q_2, there exists a unique morphism u: Q \to P making the diagram commute.^[2] This construction is unique up to isomorphism when it exists and is dual to the pushout.^[3] 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.^[2] In the category of sets (Set), the pullback of f: X \to Z and g: Y \to Z is explicitly the fiber product X \times_Z Y = \{(x, y) \mid f(x) = g(y)\}, with projection maps \pi_1(x, y) = x and \pi_2(x, y) = y.^[1] Categories with all pullbacks, such as the category of topological spaces or smooth manifolds, enable the formation of fiber products that preserve relevant structures, like continuity or differentiability.^[4] Beyond pure category theory, pullbacks have significant applications across mathematics. In algebraic topology, the pullback of fiber bundles along continuous maps f: X \to Y produces a new bundle f^*E over X, which is homotopy invariant when the bundle projection is a fibration.^[4] In differential geometry, the pullback operation on differential forms—defined for a smooth map \phi: M \to N as (\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 Stokes' theorem.^[4] In K-theory, pullbacks induce homomorphisms between K-groups of spaces or manifolds, again with homotopy invariance for homotopic maps.^[4] 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^*, refers to the operation of precomposition. Given functions f: X \to Y and g: Y \to Z, the pullback f^* g is defined as the composite function g \circ f: X \to Z. This operation embodies the intuitive process of substituting the output of f into g; specifically, if y = f(x), then f^* h(y) = h(f(x)) for any suitable h: Y \to Z. The defining equation is

(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.^[5] A simple example illustrates this for scalar functions. Consider g(y) = y^2 where y \in \mathbb{R}, and let f: \mathbb{R} \to \mathbb{R} be f(x) = x + 1. Then f^* g (x) = g(f(x)) = (x + 1)^2 = x^2 + 2x + 1, effectively pulling back the squaring operation along f via variable substitution.^[5] When restricted to linear maps between vector spaces, the pullback preserves linearity: if f and g are linear, then f^* 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 f and g are injective, then g \circ f is injective; similarly, if both are surjective, then g \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 cornerstone of 19th-century mathematical analysis, 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 category theory.^[6]

Cartesian Products in Sets

In the category of sets, the pullback of two functions p: X \to Z and q: Y \to Z is constructed as a subset of the Cartesian product X \times Y. Specifically, the pullback object, denoted X \times_Z Y, consists of all ordered pairs (x, y) such that p(x) = q(y), i.e.,

X \times_Z Y = \{(x, y) \in X \times Y \mid p(x) = q(y)\}.

This set is equipped with projection maps \pi_X: X \times_Z Y \to X defined by \pi_X(x, y) = x and \pi_Y: X \times_Z Y \to Y defined by \pi_Y(x, y) = y, which satisfy the commuting condition p \circ \pi_X = q \circ \pi_Y. To verify that this construction satisfies the universal property, consider any set W together with maps a: W \to X and b: W \to Y such that p \circ a = q \circ b. Define u: W \to X \times_Z Y by u(w) = (a(w), b(w)). Since p(a(w)) = q(b(w)) for all w \in W, the image of u lies in X \times_Z Y. Moreover, \pi_X \circ u = a and \pi_Y \circ u = b. Uniqueness follows because any such map must send w to the unique pair (a(w), b(w)) that satisfies the projections. Thus, X \times_Z Y is the universal object mediating maps over Z. A prominent example occurs when Z is a singleton set (a terminal object in Set), in which case p and q are the unique maps to the point, and the pullback X \times_Z Y reduces to the ordinary Cartesian product X \times Y. Another case arises when Y = Z and q is the identity map \mathrm{id}_Z; here, the pullback X \times_Z Z = \{ (x, p(x)) \mid x \in X \}, which is isomorphic to X via the first projection \pi_X, with the second projection \pi_Y = p \circ \pi_X. If p: X \to X is an endomorphism, the equalizer of p and \mathrm{id}_X (a subobject consisting of fixed points \{ x \in X \mid p(x) = x \}) can be constructed as the pullback of p and \mathrm{id}_X along \mathrm{id}_X. The universal property can be stated formally as follows: For any set W with maps a: W \to X and b: W \to Y satisfying p \circ a = q \circ b, there exists a unique map u: W \to X \times_Z Y such that

\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}

This diagram commutes, with the triangles sharing the commuting square p \circ a = q \circ b.

Categorical Framework

Definition via Universal Property

In category theory, given a category \mathcal{C} and two morphisms f: A \to C and g: B \to C with common codomain C, a pullback consists of an object P in \mathcal{C} together with morphisms p: P \to A and q: P \to B such that the diagram

\begin{CD} P @>q>> B \\ @VpVV @VVgV \\ A @>>f> C \end{CD}

commutes, meaning f \circ p = g \circ q.^[7] This square is called a pullback diagram, and the pair (P, p, q) is universal in the sense that for any other object W with morphisms u: W \to A and v: W \to B satisfying f \circ u = g \circ v, there exists a unique morphism w: W \to P such that p \circ w = u and q \circ w = v.^[7] The universality encodes the idea that P is the "most efficient" solution to the commuting condition, factoring any compatible pair of morphisms uniquely through it. This property can be formalized as a natural isomorphism of hom-sets:

\Hom(W, P) \cong \left\{ (u, v) \in \Hom(W, A) \times \Hom(W, B) \mid f \circ u = g \circ v \right\},

natural in W.^[7] In categories like \mathbf{Set}, this recovers the Cartesian product restricted to pairs mapping to the same element in C, serving as a motivating special case.^[7] Pullbacks exist in any category with all finite limits, such as \mathbf{Set}, \mathbf{[Top](/page/Top)}, \mathbf{Grp}, and \mathbf{[Ab](/page/Ab)}, where they arise as the limit of the cospan diagram A \to C \leftarrow B (a diagram shaped like two arrows sharing a codomain).^[7] More precisely, the pullback is the limit over the index category with three objects and two non-identity morphisms pointing to the codomain object, dual to the pushout as a colimit.^[7] Not all categories admit pullbacks; for instance, finite categories may lack them unless complete.^[7] A key theorem states that in categories with pullbacks, the pullback functor (reindexing along a morphism) preserves finite limits, meaning the pullback of a finite limit diagram is again a limit.^[7] Right adjoint functors also preserve all pullbacks, reflecting their limit-preserving nature.^[7] For an example, consider the category \mathbf{Grp} of groups, which has all finite limits. The pullback of group homomorphisms f: G \to K and h: H \to K is the subgroup of the direct product G \times H consisting of pairs (g, k) such that f(g) = h(k), with k \in H, equipped with componentwise group operation; the projections are the restrictions of the product projections, satisfying the universal property via the equalizer of the induced maps G \times H \rightrightarrows K.^[1]

Fiber Products as Pullbacks

In category theory, the fiber product provides a canonical instance of the pullback construction. Given a category \mathcal{C} equipped with pullbacks and two morphisms f: X \to S and g: Y \to S in \mathcal{C}, the fiber product is the object X \times_S Y together with projection morphisms p_1: X \times_S Y \to X and p_2: X \times_S Y \to Y such that the diagram

\begin{tikzcd} X \times_S Y \arrow[r, "p_1"] \arrow[d, "p_2"'] & X \arrow[d, "f"] \\ Y \arrow[r, "g"'] & S \end{tikzcd}

commutes, and this square is universal with respect to that property. The fiber product encodes the "fibers over points in S" as products of the preimages f^{-1}(s) \times g^{-1}(s) for each s \in S, when these make sense in the ambient category.^[8] Fiber products underpin the structure of fibered categories, where the base category admits a fibration with pullbacks along base morphisms inducing base change functors that preserve the fibered structure.^[9] This base change operation is functorial: for composable base morphisms, pullbacks of pullbacks can be interchanged, yielding a lax 2-functorial property that ensures stability under iterated base changes.^[10] In the category of topological spaces, the fiber product X \times_S Y is the subspace of the Cartesian product X \times Y consisting of pairs (x, y) such that f(x) = g(y), equipped with the subspace topology induced from the product topology.^[8] In the category of abelian groups, it is the kernel subgroup of the induced map f \oplus (-g): X \oplus Y \to S, equivalently the set of pairs (x, y) \in X \oplus Y with f(x) = g(y). Fiber products exist in the category of rings; for ring homomorphisms \phi: R \to S and \psi: T \to S, the fiber product ring R \times_S T is the subring of the direct product R \times T comprising pairs (r, t) such that \phi(r) = \psi(t).^[11] This construction, which generalizes to noncommutative rings via amalgamated free products in some cases, highlights the role of fiber products beyond set-based categories.^[8]

Applications in Geometry

Pullback of Differential Forms

In differential geometry, the pullback operation allows the transfer of differential forms from one manifold to another via a smooth map. Given smooth manifolds M and N, and a smooth map f: M \to N, the pullback f^*: \Omega^k(N) \to \Omega^k(M) associates to each k-form \omega on N a k-form f^* \omega on M. Specifically, for p \in M and tangent vectors v_1, \dots, v_k \in T_p M,

(f^* \omega)_p (v_1, \dots, v_k) = \omega_{f(p)} (df_p v_1, \dots, df_p v_k),

where df_p: T_p M \to T_{f(p)} N is the differential of f at p.^[12] This construction satisfies several key properties that underscore its utility. It is natural with respect to composition: for smooth maps f: M \to N and g: N \to P, (g \circ f)^* = f^* \circ g^*. Additionally, the pullback commutes with the exterior derivative, so if \omega is a closed form (i.e., d\omega = 0), then f^* \omega is also closed. For integration on oriented manifolds, if f is an orientation-preserving diffeomorphism, the change-of-variables formula holds: \int_M f^* \omega = \int_N \omega when \omega is an n-form on the n-dimensional manifold N.^[12]^[13] Examples illustrate the geometric role of pullbacks. Under a diffeomorphism f: M \to N, the pullback of a volume form \omega on N yields the corresponding volume form on M, preserving the measure of submanifolds and enabling change-of-variables in multiple integrals, such as transforming \int_N g \, \omega = \int_M (g \circ f) \cdot | \det(df) | \, f^* \omega for scalar densities.^[12] In local coordinates, suppose N has coordinates (y^1, \dots, y^n) and \omega = \sum_{i_1 < \cdots < i_k} a_{i_1 \dots i_k}(y) \, dy^{i_1} \wedge \cdots \wedge dy^{i_k}. Then, for coordinates (x^1, \dots, x^m) on M,

f^* \omega = \sum_{i_1 < \cdots < i_k} a_{i_1 \dots i_k}(f(x)) \left( \sum_{j_1, \dots, j_k} \frac{\partial f^{i_1}}{\partial x^{j_1}} \cdots \frac{\partial f^{i_k}}{\partial x^{j_k}} \, dx^{j_1} \wedge \cdots \wedge dx^{j_k} \right),

arising from the chain rule applied to the coordinate basis forms.^[12] The pullback of differential forms was developed by Élie Cartan as part of his work on exterior differential calculus in the early 20th century (1899–1926).^[14]

Pullback of Fiber Bundles

In the context of fiber bundles, the pullback construction allows one to induce a new bundle over a different base space from an existing bundle via a map between bases. Given a fiber bundle \pi: E \to B with fiber type F and a continuous map f: M \to B, the pullback bundle f^*E \to M is defined with total space \{(m, e) \in M \times E \mid \pi(e) = f(m)\} and projection map (m, e) \mapsto m.^[15] This construction ensures that f^*E is itself a fiber bundle over M with fiber type F, where the fiber over each m \in M is canonically identified with the fiber of E over f(m).^[16] The pullback operation exhibits key properties that make it a fundamental tool in bundle theory. The fibers of f^*E are isomorphic to those of E via the natural inclusion, preserving the local trivialization structure of the original bundle.^[15] Moreover, the pullback is functorial: for composable maps g: N \to M and f: M \to B, the pullback satisfies (f \circ g)^* E \cong g^* (f^* E), establishing it as a contravariant functor from the category of spaces over B to the category of bundles over those spaces.^[17] Examples illustrate the utility of pullbacks in differential geometry and physics. For an immersion i: N \hookrightarrow M, the pullback i^* TM of the tangent bundle TM \to M yields the tangent bundle TN \to N, with fibers consisting of tangent vectors to N embedded in those of M.^[15] In gauge theory, pullbacks of principal G-bundles arise naturally; for a principal bundle P \to B with structure group G and map f: M \to B, the pullback f^* P \to M inherits the right G-action, enabling the study of gauge fields and connections restricted to submanifolds or parameter spaces.^[17] A central theorem affirms that the pullback preserves the classification of bundles. If E \to B is a vector bundle (or more generally, a bundle of a specific type), then f^* E \to M is also a vector bundle of the same rank, with isomorphic classifying maps under suitable conditions on f.^[15] In the smooth category, such pullbacks exist for smooth maps f between smooth manifolds, with transition functions induced from those of E.^[17] Pullbacks extend naturally to sections of bundles. For a section s: B \to E of \pi: E \to B, the pulled-back section f^* s: M \to f^* E is given by m \mapsto (m, s(f(m))), which relates to the precomposition of s with f and preserves properties like flatness or holonomy when connections are pulled back.^[15]

Applications in Algebra

Fiber Products of Schemes

In algebraic geometry, the fiber product of schemes provides a fundamental construction for base change and families of geometric objects. Given morphisms f: X \to S and g: Y \to S of schemes, the fiber product X \times_S Y is the scheme equipped with projection morphisms p: X \times_S Y \to X and q: X \times_S Y \to Y such that the diagram

\begin{CD} X \times_S Y @>p>> X \\ @VqVV @VfVV \\ Y @>>g> S \end{CD}

commutes, and it is universal with respect to this property: for any scheme Z with morphisms a: Z \to X and b: Z \to Y such that f \circ a = g \circ b, there exists a unique morphism Z \to X \times_S Y making the triangles commute.^[18] Equivalently, X \times_S Y represents the functor on the category of S-schemes that assigns to any S-scheme T the set of pairs of T-points (x \in X(T), y \in Y(T)) compatible over S, i.e., f_*(x) = g_*(y).^[18] This functorial perspective underscores the role of fiber products as pullbacks in the category of schemes, generalizing the abstract categorical notion to the geometric setting.^[18] The explicit construction of fiber products begins in the affine case and extends to general schemes via gluing. Suppose X = \Spec A, Y = \Spec B, and S = \Spec R, with ring homomorphisms A \to R and B \to R corresponding to f and g. Then the fiber product is X \times_S Y = \Spec(A \otimes_R B), where the projections correspond to the natural ring maps A \to A \otimes_R B and B \to A \otimes_R B.^[19] More generally, for arbitrary schemes, cover S by affine open subschemes \Spec U_i, form the affine fiber products X \times_{\Spec U_i} Y over each, and glue them along the isomorphisms induced by the overlaps U_{ij} = U_i \cap U_j to obtain X \times_S Y as a scheme.^[20] This gluing process ensures the resulting object satisfies the universal property, as verified by the sheaf properties of the structure sheaves.^[21] Fiber products of schemes enjoy several key properties, established through foundational results in algebraic geometry. Their existence and representability follow from the theorem that the category of schemes admits all finite fiber products, as proven by constructing them explicitly via the above method and verifying the universal property.^[18] (EGA I, Théorème 3.2.6) A significant property is the preservation of flatness under base change: if f: X \to S is a flat morphism of schemes, then for any S-scheme S', the base-changed morphism X \times_S S' \to S' is also flat.^[22] This follows from the affine case, where flatness of A over R implies flatness of A \otimes_R R' over R', since tensoring with flat modules preserves exact sequences, and extends locally on the schemes involved.^[22] Additionally, if one morphism is an open or closed immersion, the corresponding projection in the fiber product inherits the same type of immersion.^[23] Representative examples illustrate the utility of fiber products. When S = \Spec k for a field k, and X, Y are varieties over k, the fiber product X \times_k Y recovers the usual product variety, with coordinate ring A \otimes_k B in the affine case, allowing the study of joint families over k.^[21] Another prominent application arises in the construction of Hilbert schemes: the Hilbert scheme \Hilb^d_{P^n} parametrizing subschemes of projective space P^n of degree d involves universal families formed as fiber products Z \times_{\Hilb} T over test schemes T, and in iterative constructions (e.g., building higher-point Hilbert schemes from lower ones via Grassmannians), these are iterated to ensure representability and flatness of families.^[24] (Eisenbud-Harris, Section VI.2) Over a point s \in S, the fiber of the fiber product is explicitly (X \times_S Y)_s = X_s \times_{\Spec \kappa(s)} Y_s, where \kappa(s) is the residue field at s, reflecting the local structure via the tensor product over \kappa(s).^[18] In more advanced contexts, fiber products facilitate descent theory, where effective descent of schemes along flat morphisms can be checked using the fppf (faithfully flat and locally of finite presentation) topology, ensuring that objects glued from local data over a cover recover the global scheme uniquely.^[25]

Pullbacks in Ring Theory

In commutative ring theory, the pullback provides an algebraic construction for combining two rings sharing a common quotient. Given commutative rings S and T with ring homomorphisms \phi: S \to R and \psi: T \to R, the pullback ring, also known as the fiber product, is the subring P = S \times_R T of the direct product S \times T consisting of all pairs (s, t) such that \phi(s) = \psi(t) in R. The ring operations on P are defined componentwise:

(s_1, t_1) + (s_2, t_2) = (s_1 + s_2, t_1 + t_2),

(s_1, t_1) \cdot (s_2, t_2) = (s_1 s_2, t_1 t_2),

with multiplicative identity (1_S, 1_T).^[26] If \phi and \psi are surjective, then P carries a natural R-algebra structure via the diagonal map R \to P sending r \mapsto (s, t) for any lifts s \in \phi^{-1}(r) and t \in \psi^{-1}(r), with R-action r \cdot (s, t) = (\tilde{s}, \tilde{t}) where \phi(\tilde{s}) = r = \psi(\tilde{t}).^[27] The pullback ring P inherits many structural properties from S and T. In particular, it is an [R](/page/Ring)-algebra when the maps are surjective, and the projections \pi_S: P \to S and \pi_T: P \to T are ring homomorphisms inducing a common map P \to R. Regarding ideals, the preimage under \pi_S (or \pi_T) of any ideal in S (or T) is an ideal in P; more generally, prime ideals in P correspond to pairs of prime ideals in S and T whose images under \phi and [\psi](/page/Psi) coincide, though the exact structure depends on conditions like the maps being surjective or the rings being domains.^[28] A representative example is the fiber product of two fields K and L over a base ring R, where \phi: K \to R and \psi: L \to R are the quotient maps (possible if R is a common subfield or quotient field); in such cases, P may decompose as a direct sum or exhibit reduced structure reflecting the shared base elements. This construction arises in descent theory for modules, where, under faithfully flat conditions on the maps to R, quasi-coherent modules over R correspond to modules over P equipped with descent data—specifically, isomorphisms between the restrictions along the two projections that satisfy cocycle conditions on further pullbacks.^[26] For R-modules M and N, the pullback along the structure maps corresponding to the R-algebra P is realized via the tensor product M \otimes_R N, which serves as the module over P obtained by base change; this equivalence holds in the context of descent, where compatible R-module structures lift to modules over the pullback ring.^[29]

Operator Theory

Pullback Operators on Function Spaces

In functional analysis, the pullback operator associated to a measurable map f: (X, \mathcal{A}, \mu) \to (Y, \mathcal{B}, \nu) between measure spaces is defined on the space of continuous functions C(Y) by f^* g = g \circ f for g \in C(Y), providing a linear map f^*: C(Y) \to C(X).^[30] This operator, often viewed as precomposition with f, extends naturally to the Lebesgue spaces L^p(Y, \nu) for $1 \leq p \leq \infty, yielding bounded linear operators f^*: L^p(Y, \nu) \to L^p(X, \mu) under suitable conditions on f, such as when f is nonsingular (i.e., f^{-1}(B) has measure zero whenever B does) and the pushforward measure f_* \mu \ll \nu with the Radon-Nikodym derivative d(f_* \mu)/d\nu essentially bounded.^[31] When f is measure-preserving (meaning f_* \mu = \nu), the pullback f^* is an isometry on L^p(X, \mu) for all $1 \leq p \leq \infty, satisfying \|f^* g\|_p = \|g\|_p.^[31] Moreover, f^* is continuous with respect to composition of maps: if h: Z \to X is another measurable map, then (f \circ h)^* = h^* f^*, preserving the algebraic structure of pullbacks.^[32] For invertible measure-preserving f, f^* is an isomorphism on each L^p space. A key theorem states that on L^\infty(Y, \nu), the pullback f^* induced by a measure-preserving map f is an isometry, with \|f^* g\|_\infty = \|g\|_\infty, since the essential supremum is preserved under composition with f.^[31] For non-injective f, the kernel of f^*: L^p(Y, \nu) \to L^p(X, \mu) consists of those g \in L^p(Y, \nu) such that g = 0 almost everywhere on the essential image of f (i.e., g \circ f = 0 \mu-a.e.), reflecting the information loss from non-injectivity. Representative examples include pullbacks under coordinate changes in \mathbb{R}^n, where for a diffeomorphism \phi: U \to V between open sets and Lebesgue measure m, the operator \phi^* satisfies \int_U (g \circ \phi) \, dm = \int_V g \cdot |\det(d\phi)|^{-1} \, dm for integrable g, linking to the classical change-of-variables formula.^[33] In Sobolev spaces W^{k,p}(\Omega) over domains in \mathbb{R}^n, the pullback under a C^k-diffeomorphism \psi: \Omega' \to \Omega defines an isomorphism \psi^*: W^{k,p}(\Omega) \to W^{k,p}(\Omega') by \psi^* u = u \circ \psi, preserving weak derivatives and norms up to the Jacobian factor.^[34] The pullback also connects integrals to pushforward measures: for integrable g \geq 0 on Y,

\int_X (f^* g) \, d\mu = \int_X g \circ f \, d\mu = \int_Y g \, d(f_* \mu),

where f_* \mu(B) = \mu(f^{-1}(B)) is the pushforward of \mu under f, enabling change-of-variables in measure-theoretic contexts.^[33]

Adjointness with Pushforwards

In the context of function spaces over manifolds, the pushforward operator f_*: C(X) \to C(Y) associated to a smooth map f: X \to Y acts on densities g by integrating over the preimage fibers: (f_* g)(y) = \int_{f^{-1}(y)} g \, d\mu, where \mu is a suitable measure on the fiber.^[12] This construction ensures that the pullback f^*: C(Y) \to C(X) serves as the left adjoint to f_* with respect to the L^2 inner product on these spaces, satisfying \langle f^* \phi, \psi \rangle_{L^2(X)} = \langle \phi, f_* \psi \rangle_{L^2(Y)} for suitable test functions \phi, \psi.^[35] In Hilbert spaces of sections, such as L^2 spaces of differential forms or vector fields over Riemannian manifolds, the pullback f^* acts as the formal adjoint of the pushforward f_*, preserving the sesquilinear structure induced by the manifold's metric.^[12] When f is an isometry, both operators exhibit unitarity, meaning f^* is unitary with respect to the Hilbert space inner product, as the map preserves lengths and angles in the tangent spaces.^[36] A representative example arises in quantum mechanics, where the pullback of wavefunctions under coordinate transformations maintains the unitarity of the representation on the Hilbert space L^2(\mathbb{R}^n, d^n q). For instance, under a change from position coordinates q to new coordinates via a diffeomorphism, the wavefunction \psi(q) transforms via precomposition adjusted by the Jacobian determinant to preserve the L^2 norm, effectively implementing f^* as the operator ensuring invariance of probabilities.^[37] Similarly, the Fourier transform establishes a relation between pullback and pushforward in momentum space, where the unitary Fourier operator intertwines the position and momentum representations, with pullback corresponding to multiplication in one domain and differentiation (pushforward-like) in the dual.^[37] For Riemannian manifolds (M, g) and (N, h), if f: M \to N is an isometry (satisfying f^* h = g), then the pullback f^* preserves inner products on differential forms: for k-forms \alpha, \beta on N, \langle f^* \alpha, f^* \beta \rangle_g = \langle \alpha, \beta \rangle_h, as the metric compatibility ensures the induced L^2 structure is invariant.^[36] This follows from the fact that isometries pull back the metric tensor, thereby preserving the pointwise inner products on \Lambda^k T^* N via the dual map on cotangent spaces.^[35] The adjoint relation manifests explicitly in integration: for volume forms \mathrm{vol}_X on X and \mathrm{vol}_Y on Y,

\int_X (f^* \phi) \wedge \psi \, \mathrm{vol}_X = \int_Y \phi \wedge (f_* \psi) \, \mathrm{vol}_Y,

where \phi is a form on Y and \psi on X. For volume forms specifically, if \mathrm{vol}_Y is the Riemannian volume element induced by h, the explicit computation yields f_* \mathrm{vol}_X = | \det(df) | \cdot f^{-1 *} \mathrm{vol}_Y on fibers (adjusted for orientation), ensuring the integral equality holds by change of variables and Fubini's theorem over the fibers f^{-1}(y).^[12]

Interconnections

Links Between Precomposition and Categorical Pullbacks

Precomposition, denoted f^* g = g \circ f for a morphism f: X \to Y in a category \mathcal{C} and a morphism g: Y \to Z, establishes a direct link to categorical pullbacks by recovering the universal property in specific settings. Precomposition establishes a direct link to categorical pullbacks through base-change functors in slice categories. The pullback functor f^*: \mathcal{C}/Y \to \mathcal{C}/X induced by f: X \to Y acts on representables \hom(-, A) for A \in \mathcal{C}/Y via precomposition with f, yielding \hom(-, A) \circ f \cong \hom(f(-), A).^[38] Thus, precomposition provides the concrete mechanism underlying the abstract pullback construction in functor categories.^[8]^[39] In the category \mathbf{Set} of sets, precomposition along f: X \to Y pulls back functions g: Y \to Z to functions g \circ f: X \to Z, which realizes the fiber product X \times_Y Z equipped with the projection to X. Similarly, for predicates—subsets S \subseteq Y—precomposition with the characteristic function yields the preimage f^{-1}(S) \subseteq X, isomorphic to the fiber product X \times_Y S over the inclusions S \hookrightarrow Y and X \twoheadrightarrow Y.^[8] This isomorphism highlights how precomposition captures the universal property of pullbacks in \mathbf{Set}, where the pullback object consists of pairs (x, z) such that f(x) = p_Y(z) for the structure map p_Y: Z \to Y.^[40] A key example arises in algebraic topology, where the pullback f^*: H^*(Y; G) \to H^*(X; G) in cohomology with coefficients in an abelian group G is induced by precomposition on cochains. Specifically, for a continuous map f: X \to Y, the chain map f_\#: S_*(X) \to S_*(Y) on singular chains induces the cochain map f^\#: C^*(Y; G) \to C^*(X; G) by (f^\# \phi)(\sigma) = \phi(f_\# \sigma) for a cochain \phi and simplex \sigma, which is precomposition of \phi with f_\#; this descends to cohomology since boundaries are preserved. This mechanism ensures homotopy invariance, as homotopic maps induce chain homotopic maps, yielding the same pullback in cohomology. In the broader context of toposes, pullbacks along geometric morphisms coincide with precompositional substitutions in the internal logic. A geometric morphism f: \mathcal{E} \to \mathcal{F} between toposes consists of an inverse image functor f^*: \mathcal{F} \to \mathcal{E} that preserves finite limits, including pullbacks, and a direct image f_*: \mathcal{E} \to \mathcal{F} that is right adjoint; for subobjects (predicates), f^* performs the substitution corresponding to precomposition along the unit of the adjunction. This unification shows how the concrete precompositional action in \mathbf{Set} generalizes to abstract toposes, where substitutions preserve the Heyting algebra structure of subobject lattices.^[41] The sheaf pullback f^{-1} \mathcal{F} for a morphism f: X \to Y of spaces and sheaf \mathcal{F} on Y extends precomposition to local data by sheafifying the presheaf U \mapsto \mathcal{F}(f(U)), which assigns to open sets U \subseteq X the sections of \mathcal{F} over f(U) via precomposition with f|_U; this allows gluing of locally defined sections while respecting the topology.^[42] Unlike global precomposition, which may fail to preserve sheaf conditions, f^{-1} ensures the result is a sheaf, capturing local consistency essential for cohomology and other sheaf-theoretic constructions.^[42] In fiber bundles, sections of the pullback bundle likewise arise via precomposition of sections with the base map.

Unifying Examples Across Fields

One illustrative example that bridges analysis, geometry, and function theory is the change of variables formula in multiple integrals, where the pullback of a differential form under a diffeomorphism \phi: \mathbb{R}^n \to \mathbb{R}^n transforms the integral \int_U \omega to \int_{\phi^{-1}(U)} \phi^* \omega, incorporating the Jacobian determinant via precomposition to ensure invariance.^[43] This unifies precomposition of functions, which adjusts integrands analytically, with the geometric pullback of forms that preserves orientation and volume, and the operator perspective where pullback acts as a bounded map on form spaces, linking these fields through the chain rule for exterior derivatives.^[43] A foundational instance connecting set theory, topology, and algebraic geometry is the inverse image f^{-1}(U) of a subset U under a continuous map f: X \to Y, which realizes the pullback sheaf f^{-1}\mathcal{F} for a sheaf \mathcal{F} on Y, defined as the sheafification of the presheaf V \mapsto \lim_{f(V) \to W} \mathcal{F}(W).^[44] This construction links the topological inverse image, which pulls back open sets to preserve covers, with algebraic sheaves that encode local data compatibly, demonstrating how pullbacks extend set-theoretic preimages to coherent structures across these domains.^[44] In algebraic geometry and topology, pullbacks in étale cohomology connect schemes to fiber bundles through categorical limits: for a morphism f: X \to Y of schemes and an étale sheaf \mathcal{F} on Y, the pullback f^*\mathcal{F} on X computes cohomology groups H^i(X_{\text{ét}}, f^*\mathcal{F}) via the fiber product, enabling base change isomorphisms that relate étale covers to bundle sections.^[45] This ties algebraic schemes, where pullbacks resolve intersections, to geometric bundles, where they induce sections over pulled-back bases, all unified by limits in the étale site.^[45] Across these examples, a unifying property emerges: pullbacks invert pushforwards in fibered categories, such as the category of schemes over a base, where for a cartesian square

\begin{tikzcd} X' \arrow \arrow & Y' \arrow[d, "g"] \\ X \arrow[r, "f"'] & Y \end{tikzcd}

with X' = X \times_Y Y', the base change isomorphism f_* g^* \cong g'_* (f')^* holds, illustrating how pullbacks along g recover data pushed forward by f in varieties. This inversion, rooted in the universal property of pullbacks as right adjoints to pushforwards in fibered settings, threads through all cases without repetition of core mechanisms. A modern application beyond classical scopes appears in string theory, where pullbacks of the tangent bundle over Calabi-Yau manifolds compactify extra dimensions, ensuring Ricci-flat metrics and supersymmetry in type II theories via the pullback operation on Kähler forms.^[46]

References

[1]
[PDF] limits in category theory - UChicago Math
Aug 17, 2007 · The pullback (fiber product) is the last limit I will define explicitly. ... In any category, a pullback can be constructed using a product.
[2]
[PDF] Basic Concepts in category theory
Definition 1.2. We say a category C is small if it has only a set of objects ... terminal such completion is called a pullback for the pair (f,g). If C ...
[3]
[PDF] Pushouts, Pullbacks and Their Properties - People | MIT CSAIL
This technique is theoretically based on pushouts and pullbacks, which are involved with given categories. This paper deals with the definition of pushout and ...
[4]
Pullback - an overview | ScienceDirect Topics
Pullback is defined as a construction in category theory that involves a commutative diagram of morphisms, allowing the definition of objects based on pullback ...
[5]
[PDF] MULTIVARIABLE ANALYSIS What follows are lecture notes from an ...
Lecture 24: Pullback of differential forms; forms on bases. Pullbacks. (24.1) Definition. We already defined the pullback of 0-forms (functions) and 1-forms ...<|separator|>
[6]
[PDF] The Change-of-Variables Theorem for the Lebesgue Integral
The calculus version of the change-of-variables formula has a long history and is connected with names such as L. Euler, J.-L. Lagrange, S. Laplace, C. F. Gauss ...
[7]
[PDF] maclane-categories.pdf - MIT Mathematics
... Categories for the working mathematician/Saunders Mac Lane. -. 2nd ed. p. cm. - (Graduate texts in mathematics; 5). Includes bibliographical references and ...
[8]
pullback in nLab
Sep 19, 2025 · A pullback is a limit of a diagram like this: Such a diagram is also called a pullback diagram or a cospan. If the limit exists, we obtain a commutative square.2-pullback · Homotopy pullback · Wide pullback · Pullback bundle
[9]
Section 4.33 (02XJ): Fibred categories—The Stacks project
A choice of pullbacks for p : \mathcal{S} \to \mathcal{C} is given by a choice of a strongly cartesian morphism f^\ast x \to x lying over f for any morphism f: ...
[10]
base change in nLab
Jul 12, 2022 · In a fibered category. The concept of base change generalises from this case to other fibred categories. Base change geometric morphisms.
[11]
Section 15.5 (08KG): Fibre products of rings, I—The Stacks project
Fibre products of rings have to do with pushouts of schemes. Some cases of pushouts of schemes are discussed in More on Morphisms, Section 37.14. Lemma 15.5.1.Missing: ×_S | Show results with:×_S
[12]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · One of the goals of this text on differential forms is to legitimize this interpretation of equa- tion (1) in 𝑛 dimensions and in fact, ...Missing: citation | Show results with:citation
[13]
[PDF] Differential Forms and Integration - UCLA Mathematics
The concept of a closed form corresponds to that of a conservative force in physics (and an exact form corresponds to the concept of having a potential function) ...
[14]
[PDF] Differential Topology of Fiber Bundles
Jan 22, 2024 · Definition 1.3.7. (Pullbacks of fiber bundles) If (E,B,F,q) is a fiber bundle and f : X → B a smooth map, then f∗E := {(x, e) ∈ X × E : f ...
[15]
[PDF] notes on fiber bundles - UChicago Math
Jan 23, 2019 · Definition 2.1. If π : E → X is an F bundle, and g : Y → X is a map, then there is a pullback F bundle g∗π : g∗E → Y over Y whose fiber over ...
[16]
[PDF] GAUGE THEORY FOR FIBER BUNDLES - Fakultät für Mathematik
From the Lemma itself it follows, that the pullback f∗P over a smooth mapping f : M0 → M is again a principal fiber bundle. 10.5. Homogeneous spaces. Let G be a ...
[17]
Section 26.17 (01JO): Fibre products of schemes—The Stacks project
26.17 Fibre products of schemes. Here is a review of the general definition, even though we have already shown that fibre products of schemes exist.Missing: ×_S | Show results with:×_S
[18]
https://stacks.math.columbia.edu/tag/01JO
[19]
https://stacks.math.columbia.edu/tag/01JQ
[20]
[PDF] The Geometry of Schemes
Schemes are defined as sets, topological spaces, and as schemes (structure sheaves). They can be affine or in general, and include subschemes.<|separator|>
[21]
[PDF] Math 216A. Preservation of properties of morphisms under base ...
For a scheme S, an S-map f: X → Y, and S-scheme S0, f0 = f × idS0 : X ×S S0 → Y ×S S0 is the base change of f. If f satisfies P, then f0 does too.
[22]
http://virtualmath1.stanford.edu/~conrad/216APage/handouts/basechange.pdf
[23]
Section 44.2 (0B94): Hilbert scheme of points—The Stacks project
In this section we will show that \mathrm{Hilb}^ d_{X/S} is representable by a scheme if any finite number of points in a fibre of X \to S are contained in an ...
[24]
Section 29.25 (01U2): Flat morphisms—The Stacks project
29.25 Flat morphisms. Flatness is one of the most important technical tools in algebraic geometry. In this section we introduce this notion.
[25]
[PDF] Course notes for Math 6170: Algebraic geometry
Apr 28, 2013 · For any commutative ring B define F(B) = {ϕ ∈ HomComRng(A, B) ... fiber product of rings yields the theorem. In general, we may select ...
[26]
[PDF] Algebraic Geometry - Arun Debray
May 5, 2016 · m, a commutative ring with multiplication defined pointwise ... limits, so we get a fiber product of rings. Page 47. 15 More Examples ...
[27]
Ideal Theory in Pullbacks - SpringerLink
In this article, we shall discuss pullback diagrams of the following type:
[28]
[PDF] Separated presentations of modules over pullback rings - arXiv
Dec 15, 2013 · The main result of this paper is the reduction procedure: it takes as input any separated presentation of a module M, and returns its R-diagram.
[29]
[PDF] FUNCTIONAL ANALYSIS - ETH Zürich
Jun 8, 2017 · These are notes for the lecture course “Functional Analysis I” held by the second author at ETH Zürich in the fall semester 2015.<|control11|><|separator|>
[30]
[PDF] Isolation amongst composition operators on $L^p(\mu) - Ele-Math
Let (X,3,μ) be a σ -finite measure space and ϕ : X → X be a measurable transformation which induces a composition operator Cϕ : Lp(μ) →. LP(μ). Then Cϕ is ...
[31]
[PDF] Notes on ergodic theory
Jul 5, 2017 · The identity map on any measure space is measure preserving. Example ... bT on functions is a norm-preserving linear operator of Lp, and if T is ...
[32]
[PDF] MEASURE AND INTEGRATION - ETH Zürich
This book is based on notes for the lecture course “Measure and Integration” held at ETH Zürich in the spring semester 2014. Prerequisites are the first.Missing: ∘ | Show results with:∘
[33]
[PDF] Introduction to Sobolev Spaces
Dec 23, 2018 · These are Lecture Notes 1 written for the last third of the course “MM692. Análise Real II” in 2018-2 at UNICAMP.
[34]
[PDF] INTRODUCTION TO DIFFERENTIAL GEOMETRY - ETH Zürich
Chapter 1 gives a brief historical introduction to differential geometry and explains the extrinsic versus the intrinsic viewpoint of the subject.2 This chapter ...
[35]
[PDF] Riemannian Manifolds: An Introduction to Curvature
Because parallel translation preserves inner products, it is easy to see that the vector fields {Ej} form an orthonormal frame. Since ∇XEj is linear over C.<|separator|>
[36]
[PDF] Lectures on Quantum Mechanics Leon A. Takhtajan
gendre's transformation associated with the Lagrangian L, if. θL = τ∗L(θ). In standard coordinates the Legendre's transformation is given by. τL(q, ˙q)=(p, q ...
[37]
[PDF] Category Theory - Index of /
Why write a new textbook on Category Theory, when we already have Mac. Lane's Categories for the Working Mathematician? ... Pullback is a functor. I.e. for ...
[38]
[PDF] Category theory in context Emily Riehl - Johns Hopkins University
Atiyah described mathematics as the “science of analogy”; in this vein, the purview of category theory is mathematical analogy. Specifically, category theory ...
[39]
[PDF] Notes on Category Theory - UT Math
Feb 28, 2018 · ... Categories for the Working Mathematician by. Mac Lane and Lang's Algebra are good standard resources. For a more modern take, see Emily ...
[40]
[PDF] Notes on toposes - LIX
Given a Grothendieck topos E and a ∈ E, SubE (a) is a Heyting algebra. For every morphism f : a → b, the pullback f∗ : Sub(b) → Sub(a) has left and right.Missing: precompositional | Show results with:precompositional
[41]
Section 7.45 (06UM): Pullback maps—The Stacks project
7.45 Pullback maps. It sometimes happens that a site \mathcal{C} does not have a final object. In this case we define the global section functor as follows.Missing: theory | Show results with:theory
[42]
[PDF] Notes on Differential Forms Lorenzo Sadun - arXiv
Apr 26, 2016 · This is a series of lecture notes, with embedded problems, aimed at students studying differential topology.
[43]
Section 59.36 (03PZ): Inverse image—The Stacks project
In this section we briefly discuss pullback of sheaves on the small étale sites. The precise construction of this is in Topologies, Section 34.4.
[44]
[PDF] Lectures on etale cohomology - James Milne
Behaviour of cohomology with respect to inverse limits of schemes. Let I be a directed set, and let .Xi /i2I be a projective system of schemes indexed by I.
[45]
[PDF] string theory on calabi-yau manifolds - arXiv
These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the ...