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Inverse image functor

The inverse image functor, commonly denoted f^{-1} or f^*, is a contravariant functor in sheaf theory that, given a continuous morphism f: X \to Y between topological spaces and a sheaf \mathcal{F} on Y, produces a sheaf f^{-1} \mathcal{F} (or f^* \mathcal{F}) on X by pulling back the sections of \mathcal{F} along f.^[1] This construction is essential for transferring sheaf data from the target space Y to the source space X, facilitating the analysis of geometric and algebraic structures in a localized manner.^[2] The explicit construction of the inverse image begins with a presheaf on X, where for an open set U \subseteq X, the sections are given by the colimit (f^{-1} \mathcal{F})(U) = \varinjlim_{V \supseteq f(U)} \mathcal{F}(V), taken over all open sets V in Y containing the image f(U); the sheaf f^{-1} \mathcal{F} is then obtained by sheafifying this presheaf.^[1] In the more structured setting of ringed spaces, such as schemes in algebraic geometry, for a sheaf of \mathcal{O}_Y-modules \mathcal{G} on Y, the inverse image f^* \mathcal{G} is defined as f^{-1} \mathcal{G} \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X, where f^{-1} first produces a sheaf of f^{-1} \mathcal{O}_Y-modules, and the tensor product equips it with an \mathcal{O}_X-module structure using the natural ring homomorphism induced by f.^[3] A defining feature of the inverse image functor is its role as the left adjoint to the direct image functor f_*, satisfying \Hom_{\Sh(X)}(f^{-1} \mathcal{G}, \mathcal{F}) \cong \Hom_{\Sh(Y)}(\mathcal{G}, f_* \mathcal{F}) for sheaves \mathcal{G} on Y and \mathcal{F} on X, which ensures compatibility with limits and colimits in the category of sheaves.^[2] It is exact, preserving finite limits and thus short exact sequences, and commutes with stalks: the stalk of f^{-1} \mathcal{G} at a point x \in X equals the stalk of \mathcal{G} at f(x).^[2] These properties make f^* indispensable for computations in sheaf cohomology, where it allows restriction of cohomological data along morphisms, and in defining operations like tensor products and Hom sheaves on schemes.^[3]

Background Concepts

Topological Spaces and Continuous Maps

A topological space is a pair (X, \tau), where X is a set and \tau is a collection of subsets of X, called open sets, satisfying the following axioms: (1) the empty set \emptyset and X itself are in \tau; (2) the union of any arbitrary collection of sets in \tau is in \tau; and (3) the intersection of any finite collection of sets in \tau is in \tau.^[4] These axioms ensure that the open sets form a stable structure under the operations relevant to notions of continuity and proximity, generalizing the familiar open intervals in the real line.^[4] Given topological spaces (X, \tau_X) and (Y, \tau_Y), a function f: X \to Y is continuous if, for every open set U \in \tau_Y, the inverse image f^{-1}(U) is an open set in \tau_X.^[4] The inverse image, or preimage, of a subset V \subseteq Y under f is defined as f^{-1}(V) = \{x \in X \mid f(x) \in V\}, which induces a map f^{-1}: \mathcal{P}(Y) \to \mathcal{P}(X) from the power set of Y to the power set of X.^[4] This preimage operation is fundamental to the topological definition of continuity, as it captures how the function interacts with the open structures of the spaces without relying on a metric.^[5] The notion of continuous functions originated in the context of metric spaces but was generalized to abstract topological spaces by Maurice Fréchet in his 1906 doctoral thesis Sur quelques points du calcul fonctionnel, where he introduced the idea of abstract spaces to extend concepts like limits and continuity beyond Euclidean settings.^[6] This development was formalized and expanded by Felix Hausdorff in his 1914 monograph Grundzüge der Mengenlehre, which axiomatized topology using open sets and established the preimage criterion for continuity as a cornerstone of the theory.^[7]

Functors and Contravariant Functors

In category theory, a category \mathcal{C} consists of a class of objects \mathrm{Ob}(\mathcal{C}), a class of morphisms (or arrows) between objects, a composition operation that associates to each pair of morphisms f: A \to B and g: B \to C a composite morphism g \circ f: A \to C, and for each object A an identity morphism \mathrm{id}_A: A \to A, satisfying the axioms of associativity of composition and the identity acting as a unit for composition.^[8]^[9] A functor F: \mathcal{C} \to \mathcal{D} between categories \mathcal{C} and \mathcal{D} is a mapping that sends objects of \mathcal{C} to objects of \mathcal{D} and morphisms f: A \to B in \mathcal{C} to morphisms F(f): F(A) \to F(B) in \mathcal{D}, such that F preserves identities, F(\mathrm{id}_A) = \mathrm{id}_{F(A)}, and preserves composition, F(g \circ f) = F(g) \circ F(f).^[10] Such a functor is called covariant, as it maintains the direction of arrows.^[11] A contravariant functor F: \mathcal{C} \to \mathcal{D} similarly maps objects to objects but reverses the direction of morphisms, sending f: A \to B to F(f): F(B) \to F(A), while preserving identities, F(\mathrm{id}_A) = \mathrm{id}_{F(A)}, and satisfying the reversed composition rule F(g \circ f) = F(f) \circ F(g).^[12]^[13] Equivalently, a contravariant functor from \mathcal{C} to \mathcal{D} is a covariant functor from \mathcal{C} to the opposite category \mathcal{D}^{\mathrm{op}}, where all arrows in \mathcal{D} are reversed.^[13] An example of a covariant functor is the forgetful functor U: \mathbf{Grp} \to \mathbf{Set} from the category of groups to the category of sets, which maps a group to its underlying set and a group homomorphism to its restriction as a set function, preserving the structure without regard to the group operations.^[14]^[11] In contrast, the power set functor P: \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set} (or contravariantly P: \mathbf{Set} \to \mathbf{Set}^{\mathrm{op}}) sends a set X to its power set \mathcal{P}(X) and a function f: X \to Y to the preimage map f^{-1}: \mathcal{P}(Y) \to \mathcal{P}(X), reversing the arrow direction.^[15] Contravariant functors are often denoted using superscript notation, such as f^* for a morphism f, to indicate the reversal of direction, as seen in pullback constructions.^[16]

Formal Definition

Definition in the Category of Topological Spaces

In the category \mathbf{[Top](/page/Top)}, whose objects are topological spaces and whose morphisms are continuous functions between them, the inverse image functor arises from a fixed continuous map f: X \to Y. It is contravariant in f and initially defined by its action on the lattice of subsets or open sets of the spaces. Specifically, f induces a function f^*: \mathcal{P}(Y) \to \mathcal{P}(X) on power sets given by

f^*(A) = f^{-1}(A) = \{ x \in X \mid f(x) \in A \}

for any A \subseteq Y. This map preserves the Boolean lattice operations: f^{-1}(\bigcup_i A_i) = \bigcup_i f^{-1}(A_i), f^{-1}(\bigcap_i A_i) = \bigcap_i f^{-1}(A_i), and f^{-1}(Y \setminus A) = X \setminus f^{-1}(A). Restricting to open subsets yields f^*: \mathcal{O}(Y) \to \mathcal{O}(X), where \mathcal{O}(Z) is the collection of open sets in Z, preserving arbitrary unions and finite intersections (hence a frame homomorphism \mathcal{O}(Y)^{op} \to \mathcal{O}(X)).^[17] This construction extends to a functor between slice categories by pulling back topological structures along f. The slice category \mathbf{Top}/Y has objects continuous maps p: Z \to Y (spaces over Y) and morphisms continuous maps h: Z \to Z' such that p' \circ h = p; \mathbf{Top}/X is defined analogously. The inverse image functor is then f^*: \mathbf{Top}/Y \to \mathbf{Top}/X, acting on objects by sending p: Z \to Y to the projection \mathrm{pr}_X: Z \times_Y X \to X, where Z \times_Y X = \{ (z, x) \in Z \times X \mid p(z) = f(x) \} carries the subspace topology induced from the product topology on Z \times X. On morphisms, if k: (Z \to Y) \to (Z' \to Y) is a morphism over Y, then f^*(k): (Z \times_Y X \to X) \to (Z' \times_Y X \to X) is the continuous map (z, x) \mapsto (k(z), x). This defines f^* fully as a functor in \mathbf{Top}, with the preimage on subsets recovering the action on the structure of open sets in the pullback space.^[18]

Generalization to Other Categories

The inverse image functor extends naturally to the category of locales, where objects are frames (complete Heyting algebras satisfying the infinite distributive law) and morphisms are frame homomorphisms. For a locale morphism f: X \to Y, the inverse image f^*: O(Y) \to O(X) is a frame homomorphism that preserves finite meets and arbitrary joins, mapping open sets in Y to open sets in X while respecting the lattice structure of the frames O(Y) and O(X).^[19] This construction generalizes the topological case by replacing open sets with the more abstract frame elements, allowing for point-free topology where locales capture spatial properties without requiring points.^[19] In the category of measurable spaces, denoted Meas, objects consist of sets equipped with \sigma-algebras, and morphisms are measurable functions f: (X, \Sigma_X) \to (Y, \Sigma_Y). The inverse image functor f^*: \Sigma_Y \to \Sigma_X is defined by f^*(B) = f^{-1}(B) for B \in \Sigma_Y, producing a sub-\sigma-algebra of \Sigma_X that preserves countable unions, intersections, and complements. This functor ensures that measurability is preserved under preimages, facilitating the study of measure-theoretic constructions like integration and probability in a categorical framework.^[20] In algebraic geometry, for a morphism of schemes f: X \to Y, the inverse image functor f^* acts on the category of quasi-coherent sheaves, pulling back a sheaf \mathcal{G} on Y to f^*\mathcal{G} on X via tensor product with the structure sheaf on affine opens: if U = \operatorname{Spec} A and f(U) \subset \operatorname{Spec} B with \mathcal{G}|_{\operatorname{Spec} B} = \tilde{N}, then \Gamma(U, f^*\mathcal{G}) = A \otimes_B N.^[21] The functor is exact if f is flat, preserving exact sequences of sheaves, and it is left adjoint to the direct image functor f_* for quasicompact quasiseparated morphisms.^[21]^[2] Within site theory, particularly the étale site of schemes, the inverse image functor f^{-1}: \operatorname{Sh}(Y_{\acute{e}tale}) \to \operatorname{Sh}(X_{\acute{e}tale}) for a scheme morphism f: X \to Y is the left adjoint to the pushforward f_*, defined via colimits over the fiber product: for an étale U \to X, sections are \operatorname{colim}_{(V/Y, \varphi: U \to X \times_Y V)} \mathcal{G}(V/Y).^[2] This preserves stalks and representables, with f^{-1} h_V = h_{X \times_Y V} for étale V \to Y, and relates to flatness conditions for descent in the étale topology.^[2] In general categories equipped with pullbacks, the inverse image functor arises via representable functors or comma categories, where for a morphism f: X \to Y, f^* precomposes contravariant functors on Y with f, often realized as the left adjoint in a geometric morphism when the category is a topos.

Key Properties

Preservation of Topological Structures

The inverse image functor f^*, induced by a continuous map f: X \to Y between topological spaces, preserves the openness and closedness of sets. Specifically, for any open set U \subseteq Y, the preimage f^{-1}(U) is open in X, and for any closed set V \subseteq Y, the preimage f^{-1}(V) is closed in X. This property is equivalent to the definition of continuity for f.^[22] The preimage operation also preserves arbitrary unions and arbitrary intersections set-theoretically, ensuring that these operations remain compatible with the topological structure when the sets are open or closed. In particular,

f^{-1}\left( \bigcup_{i \in I} U_i \right) = \bigcup_{i \in I} f^{-1}(U_i)

for any index set I and family of subsets \{U_i\}_{i \in I} \subseteq Y, and

f^{-1}\left( \bigcap_{i \in I} U_i \right) = \bigcap_{i \in I} f^{-1}(U_i)

for arbitrary intersections (including finite ones). These equalities hold for any subsets and underscore the functor's role in pulling back the lattice of open sets as a frame homomorphism, preserving finite meets and arbitrary joins.^[23] The inverse image functor preserves connectedness and path-connectedness of subsets. If C \subseteq Y is connected, then f^{-1}(C) is connected in X, with the subspace topology inherited from X. A similar preservation holds for path-connected subsets, where preimages of path-connected sets are path-connected under continuous f.^[23] For compactness, the preimage of a compact subset K \subseteq Y is compact in X when the map f is proper (i.e., preimages of compact sets are compact by definition). In Hausdorff spaces, compact sets are closed, so f^{-1}(K) is closed, but compactness requires the properness condition to ensure every open cover of f^{-1}(K) has a finite subcover via pullback from K.^[24] Regarding separation axioms, the inverse image does not preserve them in general. If Y satisfies a separation axiom T_n (such as T_2, Hausdorffness), X does not necessarily inherit this property via f^* unless f satisfies additional conditions, such as being an embedding (continuous and injective with the property that f maps open sets in X to open sets in the image subspace of Y). For open maps, inheritance can occur in specific cases like quotient maps where the topology on X is defined to reflect Y's separation, but counterexamples exist where open continuous maps from non-Hausdorff X to Hausdorff Y fail to preserve the axiom. For instance, consider the constant map f: X \to Y where X is the real line with the indiscrete topology (non-Hausdorff) and Y = \mathbb{R} (Hausdorff); here, f^{-1}(Y) = X retains the non-Hausdorff structure despite Y's properties.^[25]

Adjunction with the Direct Image Functor

The inverse image functor f^* : \Top / Y \to \Top / X, induced by a continuous map f : X \to Y, sends an object (Z \to Y) to the pullback object (Z \times_Y X \to X ). This functor is left adjoint to the direct image functor f_* : \Top / X \to \Top / Y, which sends (W \to X) to the composite (W \to X \to Y), and can be constructed via Kan extension along the base change induced by f.^[18] The adjunction f^* \dashv f_* is characterized by the natural bijection of hom-sets

\Hom_{\Top / X}(f^* B, A) \cong \Hom_{\Top / Y}(B, f_* A),

where B is an object in \Top / Y and A is an object in \Top / X. This bijection is natural in B and A, arising from the universal property of the pullback defining f_. A similar adjunction holds in the category of sheaves on topological spaces, where f^ : Sh(Y) → Sh(X) is left adjoint to f_* : Sh(X) → Sh(Y), with the same form of the hom-isomorphism for sheaves \mathcal{G} on Y and \mathcal{F} on X.^[26]^[27] As the left adjoint, f^* preserves all colimits, including coproducts and coequalizers. As the right adjoint, f_* preserves all limits, including products and equalizers; in particular, f_* is left exact when restricted to appropriate subcategories such as sheaves of abelian groups.^[26] The unit of the adjunction is the natural transformation \eta : \id_{\Top / Y} \to f_* f^*, and the counit is \varepsilon : f^* f_* \to \id_{\Top / X}. Restricting to the poset of open sets (viewed as representable objects in the slice categories via inclusions), for an open U \subset Y regarded as the object (U \to Y) in Top/Y, the unit \eta_U : (U \to Y) \to f_* f^* (U \to Y) is the canonical map from U to the pushforward of its pullback along f. The corresponding counit component provides the natural map from the pullback-pushforward to the identity on opens in Top/X. This Galois connection on opens specializes the categorical adjunction.^[27] This adjunction is monadic in certain settings, meaning the comparison functor from \Top / Y to the Eilenberg-Moore category of algebras for the monad f_* f^* on \Top / X is an equivalence; this equivalence encodes descent data along f and underpins descent theory for topological structures.

Examples and Applications

Basic Examples in Topology

In the category of topological spaces, the inverse image functor, often denoted f^*, associates to a continuous map f: X \to Y the contravariant operation that pulls back open sets in Y to open sets in X via preimages, preserving the topological structure on the lattice of open sets. This functorial behavior is fundamental to understanding continuity and is best illustrated through simple examples in Euclidean spaces. Consider the inclusion map i: \{0\} \to \mathbb{R}, where \{0\} is equipped with the subspace topology induced from the standard topology on \mathbb{R}. For any open interval (-\epsilon, \epsilon) in \mathbb{R} containing 0, with \epsilon > 0, the inverse image i^{-1}((-\epsilon, \epsilon)) = \{0\}, which is open in the subspace topology on \{0\} since the entire space is always open.^[28] This example demonstrates how the inverse image functor restricts open neighborhoods to the whole subspace, reflecting the embedding of a point. Another basic illustration is the projection map \pi: \mathbb{R}^2 \to \mathbb{R} defined by \pi(x, y) = x, which is continuous in the standard topologies. The inverse image of an open interval (a, b) \subset \mathbb{R} is the horizontal strip \{(x, y) \in \mathbb{R}^2 \mid a < x < b\}, an open set in \mathbb{R}^2 as it is the product of (a, b) with \mathbb{R}.^[29] Thus, the functor pulls back one-dimensional opens to cylindrical regions in the plane, preserving openness. For the constant map c: X \to \{\mathrm{pt}\}, where \{\mathrm{pt}\} is the singleton space with the unique topology \{\emptyset, \{\mathrm{pt}\}\}, the inverse image c^{-1}(\{\mathrm{pt}\}) = X and c^{-1}(\emptyset) = \emptyset, both open in X.^[30] This shows the functor mapping the nontrivial open in the codomain to the entire domain, highlighting its contravariant nature on constant morphisms. The inverse image operation is only functorial for continuous maps, as discontinuous functions may fail to send open sets to open sets. For instance, the step function f: \mathbb{R} \to \mathbb{R} with f(x) = 0 if x < 0 and f(x) = 1 if x \geq 0 is discontinuous, and f^{-1}((0.5, 1.5)) = [0, \infty), which is not open in \mathbb{R}.^[5] Continuity ensures the preservation of open sets under inverse images, a key property briefly referenced here.^[31] A concrete computation for the continuous map f: \mathbb{R} \to \mathbb{R} given by f(x) = x^2 yields f^{-1}((-1, 1)) = (-1, 1), which is open, confirming preservation of openness. However, f^{-1}(\{0\}) = \{0\}, a closed set in \mathbb{R}, illustrating that the functor also preserves closed sets under continuous maps.^[31]

Role in Sheaf Theory and Cohomology

In sheaf theory, the inverse image functor enables the transfer of sheaf data along continuous maps between topological spaces, facilitating the study of local-to-global properties. For a continuous map f: X \to Y and a sheaf \mathcal{F} on Y, the pullback sheaf f^* \mathcal{F} (also denoted f^{-1} \mathcal{F}) on X is defined such that sections over an open subset V \subset X satisfy (f^* \mathcal{F})(V) = \lim_{U \supset f(V)} \mathcal{F}(U), where the direct limit is over all open sets U in Y containing the image f(V). This construction ensures that f^* \mathcal{F} captures sections of \mathcal{F} that are compatible with the geometry induced by f, and it extends naturally to presheaves via sheafification followed by the limit. The functor f^* forms the left adjoint to the direct image functor f_* in the category of sheaves, establishing a fundamental duality in sheaf operations.^[32]^[2] A significant property of f^* is its exactness under suitable conditions, which preserves the homological structure of sheaf complexes. In the category of sheaves of abelian groups on topological spaces, f^* is exact, transforming short exact sequences of sheaves on Y into short exact sequences on X. In the more refined setting of algebraic geometry over schemes, the pullback preserves exactness for quasi-coherent sheaves precisely when f is a flat morphism, as flatness ensures that the associated tensor product f^{-1} \mathcal{F} \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X remains exact; this extends to étale morphisms, which are locally of finite presentation and flat, allowing faithful transfer of exactness in étale cohomology contexts. This preservation is crucial for maintaining exactness in derived categories and for applications in intersection theory and deformation theory.^[33]^[2]^[34] In sheaf cohomology, the inverse image functor induces canonical maps f^*: H^*(Y, \mathcal{F}) \to H^*(X, f^* \mathcal{F}) on cohomology groups, which are natural transformations compatible with the cup product structure, thereby preserving the ring structure of cohomology when H^* carries a graded-commutative algebra. For instance, in Čech cohomology computed with respect to an open cover, the pullback f^* acts by restricting cocycles from covers on Y to the preimage covers on X, enabling explicit computations of restrictions and refinements in topological invariants like characteristic classes. A pivotal application arises in the Leray spectral sequence, which decomposes the cohomology of the pullback as E_2^{p,q} = H^p(Y, R^q f_* f^* \mathcal{F}) \Rightarrow H^{p+q}(X, f^* \mathcal{F}), relating global cohomology on X to sheaf cohomology on Y via higher direct images and facilitating reductions in computational complexity for fibrations and coverings.^[35]^[33]^[36] The inverse image functor's integration into sheaf cohomology was revolutionized by Alexander Grothendieck in his 1957 Tohoku paper, where it underpinned the development of derived functors in abelian categories, unifying module cohomology with sheaf cohomology and establishing f^* as essential for abstract homological algebra on topoi.^[37]

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