Subcategory

In category theory, a subcategory of a category C is a category D whose objects form a subclass of those of C and whose morphisms form a subcollection of those of C, such that the identity morphisms and composition operations in D coincide with those in C, thereby forming a category in their own right.^[1] This structure allows D to inherit the foundational properties of C while focusing on a restricted portion of its elements.^[2] Subcategories play a central role in category theory by enabling the study of specialized structures within broader categorical frameworks, often preserving key universal properties and facilitating abstractions across mathematical disciplines.^[3] They are classified into types based on the extent to which they retain objects and morphisms from the parent category; for instance, a full subcategory includes all objects from a selected subclass of C along with the complete hom-sets between them, meaning no morphisms are omitted.^[3] In contrast, a subcategory may be non-full if it selects only a proper subset of the morphisms between its objects, as seen in the category of rings (Ring) as a subcategory of the category of rngs (Rng), where certain zero morphisms present in Rng are excluded from Ring.^[3] Notable examples illustrate the versatility of subcategories: the category of abelian groups (Ab) forms a full subcategory of the category of groups (Grp), retaining all group homomorphisms between abelian groups.^[3] Similarly, the category of finite-dimensional vector spaces over a field F (FVect_F) is a full subcategory of the category of all vector spaces (Vect_F).^[3] The category of topological spaces (Top) serves as a subcategory of the category of sets (Set), with continuous functions as morphisms, while the category of groups (Grp) is itself a subcategory of Set with group homomorphisms. These constructions underpin advanced concepts like reflective subcategories and equivalences, highlighting subcategories' importance in unifying diverse mathematical objects through functorial relationships.^[3]

Definition and Basics

Formal Definition

In category theory, a subcategory S of a category C is defined by selecting a subclass of objects \mathrm{Ob}(S) \subseteq \mathrm{Ob}(C) and, for each pair of objects A, B \in \mathrm{Ob}(S), a subclass of morphisms \mathrm{Hom}_S(A, B) \subseteq \mathrm{Hom}_C(A, B).^[2] This selection must satisfy three key conditions to ensure S itself forms a category: first, for every object A \in \mathrm{Ob}(S), the identity morphism \mathrm{id}_A \in \mathrm{Hom}_C(A, A) must belong to \mathrm{Hom}_S(A, A); second, the hom-sets \mathrm{Hom}_S(A, B) are non-empty only when defined for objects in S, meaning that if a morphism f: A \to B is in S, then both A and B are in \mathrm{Ob}(S); and third, composition is preserved, so that if f: A \to B and g: B \to D are in \mathrm{Hom}_S, then their composite g \circ f: A \to D is also in \mathrm{Hom}_S(A, D).^[2] These conditions guarantee that the inclusion mapping from S to C defines a functor, inheriting the categorical structure while restricting to a subset.^[2] Unlike a mere subset of objects and arrows from C, a subcategory S must independently satisfy the axioms of a category—identities, composition, and domain-codomain typing—drawing directly from C's operations without alteration. This formal structure, as originally articulated in foundational texts, underpins the hierarchical organization of categories and enables the study of specialized substructures within broader categorical frameworks.

Full Subcategories

A full subcategory of a category \mathcal{C} is a subcategory \mathcal{S} whose class of objects \mathrm{Ob}(\mathcal{S}) is a subclass of \mathrm{Ob}(\mathcal{C}), and whose morphisms consist of all arrows in \mathcal{C} between objects in \mathcal{S}; formally, for all A, B \in \mathrm{Ob}(\mathcal{S}), the hom-set satisfies \mathcal{S}(A, B) = \mathcal{C}(A, B).^[4] This condition ensures that \mathcal{S} inherits the complete relational structure between its objects directly from \mathcal{C}, distinguishing it as a refinement of the general subcategory notion where only some morphisms need be included. The inclusion of all relevant morphisms in a full subcategory automatically preserves the identity arrows and composition operations required for \mathcal{S} to form a category, as these are subsets of the morphisms in \mathcal{C}.^[4] Consequently, full subcategories are particularly useful for isolating a subclass of objects while retaining the full morphism data between them, thereby maintaining the structural integrity of interactions without alteration or loss. This property makes full subcategories a common tool in categorical constructions where the focus is on object selection rather than morphism restriction. In notation, a full subcategory is often specified simply by its class of objects, denoted as the full subcategory of \mathcal{C} on a given subclass \mathcal{X} \subseteq \mathrm{Ob}(\mathcal{C}), with the understanding that all morphisms from \mathcal{C} between elements of \mathcal{X} are included.^[4] This concise designation underscores how the subcategory is uniquely determined by the choice of objects alone.

Properties and Embeddings

Inclusion Functors

In category theory, the inclusion functor associated to a subcategory \mathcal{S} of a category \mathcal{C} is the functor I: \mathcal{S} \to \mathcal{C} that maps every object and morphism of \mathcal{S} to the corresponding object and morphism in \mathcal{C}, thereby preserving the category structure by construction.^[4] This functor is canonical, as it embeds \mathcal{S} into \mathcal{C} without altering any elements, ensuring that identities and compositions in \mathcal{S} coincide with those in \mathcal{C}.^[5] The inclusion functor I is always faithful, meaning that for any objects A, B \in \mathcal{S}, the induced map I: \mathcal{S}(A, B) \to \mathcal{C}(I A, I B) on hom-sets is injective, since morphisms in \mathcal{S} are a subset of those in \mathcal{C} and are mapped identically.^[4] It is full precisely when \mathcal{S} is a full subcategory of \mathcal{C}, in which case the map on hom-sets is also surjective, capturing all morphisms in \mathcal{C} between objects of \mathcal{S}.^[5] However, the inclusion functor I is not necessarily essentially surjective, since its image on objects is exactly the subclass of objects of \mathcal{S}; it is essentially surjective if and only if every object of \mathcal{C} is isomorphic to some object of \mathcal{S}.^[4] This functor plays a central role in category theory by providing the standard mechanism to regard \mathcal{S} as embedded within \mathcal{C}, facilitating the transfer of structures such as limits, colimits, and adjoints from \mathcal{C} to \mathcal{S} when applicable.^[5] For instance, it induces functors on functor categories or enables the study of reflective subcategories where I admits a left adjoint.^[4] When \mathcal{S} is full, the inclusion further ensures that \mathcal{S} inherits the hom-set structure directly from \mathcal{C}.^[5]

Faithfulness and Fullness

In category theory, the inclusion functor I: S \to C of a subcategory S into a category C is always faithful, meaning that for every pair of objects A, B in S, the induced map I: \hom_S(A, B) \to \hom_C(I(A), I(B)) is injective.^[4] This injectivity holds because the morphisms in S form a subset of those in C, so distinct morphisms in S remain distinct under inclusion without any collapse.^[2] Faithfulness ensures that the subcategory preserves the distinctness of arrows between its objects, providing a faithful representation of S's internal structure within C.^[4] A functor is full if the map on hom-sets is surjective, so that every morphism in C between images of objects from S arises from a morphism in S.^[4] For the inclusion I: S \to C, this surjectivity occurs precisely when S is a full subcategory of C, meaning S contains all morphisms from C between its objects. In this case, I is both full and faithful, often called fully faithful, and the map \hom_S(A, B) \to \hom_C(I(A), I(B)) is bijective for all A, B in S.^[6] Examples include the subcategory of finite sets as a full subcategory of the category of sets, where all functions between finite sets are included.^[4] When the inclusion I: S \to C is fully faithful, S is said to be embedded in C, and S is isomorphic to its image under I in C.^[6] This embedding property implies that S can be regarded as a "copy" of itself inside C without losing or gaining morphisms between its objects, facilitating the study of S via properties inherited from C.^[4] If the inclusion is additionally injective on objects, it constitutes a full embedding, preserving the object's identity strictly. Such embeddings are foundational for constructing reflective subcategories and analyzing functorial properties.^[4]

Types of Subcategories

Wide Subcategories

A wide subcategory, also known as a lluf subcategory, of a category \mathcal{C} is a subcategory \mathcal{S} whose class of objects coincides with that of \mathcal{C}, so \mathrm{Ob}(\mathcal{S}) = \mathrm{Ob}(\mathcal{C}), while for all objects A, B \in \mathrm{Ob}(\mathcal{C}), the hom-set satisfies \mathcal{S}(A, B) \subseteq \mathcal{C}(A, B); moreover, \mathcal{S} includes all identity morphisms and is closed under composition.^[7] The inclusion functor \mathcal{S} \hookrightarrow \mathcal{C} is always faithful, since each map \mathcal{S}(A, B) \to \mathcal{C}(A, B) is the injective inclusion of subsets, but it need not be full unless \mathcal{S} = \mathcal{C}.^[2] Wide subcategories thus provide a means to impose quotient-like structures on the morphisms of \mathcal{C} without altering its objects, facilitating the study of relational properties among objects via restricted arrows.^[8] In contrast to full subcategories, which preserve all morphisms between their (possibly restricted) objects, wide subcategories retain every object of \mathcal{C} but selectively omit morphisms.^[7] This distinction proves motivationally useful in homological algebra, where wide subcategories enable the analysis of derived functors by ignoring maps that do not preserve exactness or other chain conditions.^[9]

Replete Subcategories

A replete subcategory of a category \mathcal{C} is a full subcategory \mathcal{S} that is closed under isomorphisms: if an object A of \mathcal{C} is isomorphic to an object B of \mathcal{S}, then A belongs to \mathcal{S}.^[10] This closure ensures that \mathcal{S} includes all objects from the isomorphism classes represented in its object set, while retaining every morphism of \mathcal{C} between its objects.^[11] Equivalently, the inclusion functor \mathcal{S} \hookrightarrow \mathcal{C} is fully faithful and an isofibration, meaning it preserves and reflects isomorphisms.^[11] The property of repleteness guarantees that \mathcal{S} captures entire isomorphism classes without omission, making the inclusion functor not only full and faithful but also equivalent to the subcategory formed by quotienting \mathcal{C} by isomorphisms.^[10] For any full subcategory, its repletion—the smallest replete subcategory containing it—is obtained by adjoining all isomorphic objects, a process that preserves fullness and yields an isomorphism-closed structure.^[10] This makes replete subcategories particularly useful in contexts requiring invariance under equivalence, such as localizing categories or studying essential images of functors.^[11] Replete subcategories are also termed isomorphism-closed subcategories, emphasizing their invariance with respect to isomorphisms in the ambient category.^[10] In contrast, skeletal subcategories select a single representative object from each isomorphism class, thereby reducing redundancy but excluding isomorphic copies.^[11]

Serre Subcategories

In an abelian category \mathcal{C}, a Serre subcategory \mathcal{S} is a nonempty full subcategory that is closed under taking subobjects, quotient objects, and extensions.^[12] That is, if A, C \in \mathcal{S} and there exists an exact sequence $0 \to A \to B \to C \to 0 in \mathcal{C}, then B \in \mathcal{S}; equivalently, \mathcal{S} is closed under extensions in the sense that for any exact sequence A \to B \to C with A, C \in \mathcal{S}, it follows that B \in \mathcal{S}.^[12] This closure ensures that \mathcal{S} inherits the exact structure necessary for homological applications. The inclusion functor i: \mathcal{S} \to \mathcal{C} is exact, meaning it preserves and reflects exact sequences, and \mathcal{S} itself forms an abelian category with kernels and cokernels computed as in \mathcal{C}.^[12] These properties make Serre subcategories particularly useful for constructing quotient categories \mathcal{C}/\mathcal{S}, where objects are those of \mathcal{C} and morphisms are taken modulo those factoring through \mathcal{S}, often serving as localizations in homological algebra.^[13] The notion was introduced by Jean-Pierre Serre in his seminal work on coherent sheaves, where it played a central role in generalizing classical torsion theory to algebraic geometry; it has since become foundational in the study of derived categories, enabling the localization of triangulated categories at acyclic complexes concentrated in \mathcal{S}.^[14]^[13]

Examples and Applications

In the Category of Sets

In the category of sets, denoted Set, subcategories provide concrete illustrations of how restrictions on objects and morphisms preserve the categorical structure while highlighting different aspects of set-theoretic relationships. A basic example is the discrete subcategory generated by a fixed collection of sets, such as the singleton sets {a}, {b}, and {c} for distinct elements a, b, c; here, the only morphisms are the identity functions on each singleton, ensuring the subcategory consists solely of these identities and satisfies the category axioms with trivial composition.^[15] This setup emphasizes the minimal structure needed for a subcategory, where no non-trivial functions exist between distinct objects, akin to isolated points in a graph with no edges connecting them. A prominent full subcategory of Set is FinSet, comprising all finite sets as objects and all functions between them as morphisms; for instance, the sets ∅, {1}, and {1,2} are objects, with functions like the constant map sending both elements of {1,2} to 1 in {1} serving as morphisms.^[16] This subcategory inherits the full hom-sets from Set, meaning every possible function between finite sets is included, and it is both full and replete, closed under isomorphisms such as bijections between finite sets of the same cardinality. Everyday applications appear in combinatorics, where FinSet models counting problems on bounded collections, like permutations of a fixed number of items. For a wide subcategory, consider the category Inj with all sets as objects but only injective functions as morphisms; an example morphism is the inclusion {1,2} ↪ {1,2,3}, which embeds the smaller set without repetition, while non-injective functions like the constant map {1,2} → {a} (sending both to a) are excluded.^[15] This restriction preserves composition, as the composite of injections remains injective, and identities are injective, illustrating how Set's morphisms allow selective narrowing to emphasize one-to-one correspondences, such as in database relations where unique mappings prevent data duplication. In these examples, subcategory embeddings into Set are concrete inclusions, faithfully representing the restricted structures within the ambient category without altering the underlying sets or functions; this underscores Set's flexibility in permitting varied subcategory types through simple object or morphism subsets, facilitating intuitive explorations of categorical concepts like faithfulness in everyday set operations.^[17]

In Abelian Categories

In the category of abelian groups, denoted \mathrm{Ab}(\mathbb{Z}), the full subcategory consisting of torsion abelian groups forms a Serre subcategory.^[12] This subcategory is closed under the formation of subgroups, quotient groups, and extensions, as the torsion property is preserved by these operations.^[12] The quotient category \mathrm{Ab}(\mathbb{Z}) / \mathcal{T}, where \mathcal{T} is the subcategory of torsion groups, is equivalent to the category of \mathbb{Q}-vector spaces.^[18] In the category of R-modules, denoted \mathrm{Mod}_R, for a commutative ring R, the full subcategory of modules of finite length also constitutes a Serre subcategory.^[19] Submodules, quotient modules, and extensions of finite length modules remain of finite length, ensuring the required closure properties.^[19] This example highlights how finite length conditions capture essential algebraic structures in module theory. Serre subcategories play a key role in localizing abelian categories via quotient constructions, which facilitate the study of homological properties by isolating torsion or finite phenomena.^[12] In derived category theory, they enable the computation of derived functors by identifying kernels of exact functors and supporting Verdier localizations.^[13] These applications are fundamental in algebraic geometry and representation theory for analyzing sheaf cohomology and module resolutions.^[13]