Subobject

In category theory, a subobject of an object Y in a category \mathcal{C} is defined as an equivalence class of monomorphisms with codomain Y, where two monomorphisms f: X \to Y and g: Z \to Y are equivalent if there exists an isomorphism h: X \to Z such that g \circ h = f.^[1]^[2] This structure generalizes the notion of a subset or substructure in concrete categories, such as subsets in the category of sets (where monomorphisms are injective functions) or subgroups in the category of groups.^[1]^[2] Subobjects capture "inclusions" or "parts" of an object in a way that respects the categorical framework, where direct reference to elements is avoided in favor of morphisms.^[1] They form a partially ordered set (poset) under inclusion, defined by the existence of a monomorphism between representatives of two subobjects, allowing for operations like intersections (via limits of the relevant diagram) and unions in certain categories.^[1]^[2] In the category of sets, every subobject corresponds uniquely to a subset via its inclusion map, and the poset of subobjects is isomorphic to the power set ordered by inclusion.^[2] A central role for subobjects arises in the study of toposes, where the subobject classifier \Omega—an object that classifies all subobjects via characteristic morphisms—enables the internal logic of the category, analogous to truth values in set theory.^[1]^[2] For instance, in the category of sets, \Omega is the two-element set {true, false}, and subobjects biject with functions to \Omega. Subobjects are also stable under pullbacks in many categories, preserving their structure under limits, and they dualize to quotient objects, highlighting category theory's duality principles.^[1]^[2] These properties make subobjects foundational for advanced topics, including exact categories, sheaf theory, and the semantics of logic within categories.^[1]

Core Concepts

Definition

In category theory, a subobject of an object C in a category \mathcal{C} is formally defined as an equivalence class of monomorphisms with codomain C. Specifically, given monomorphisms m: A \to C and m': A' \to C, they represent the same subobject if there exists an isomorphism i: A \to A' such that m' \circ i = m. This equivalence relation identifies monomorphisms that are "essentially the same" up to isomorphism in their domains, allowing subobjects to capture injective embeddings in a category-independent manner.^[3] A monomorphism m: A \to C is a morphism that is left-cancellative: for any object X and any pair of morphisms f, g: X \to A, if m \circ f = m \circ g, then f = g. This property ensures that m embeds A into C without "collapsing" distinct elements from A, analogous to injectivity in concrete categories. Monomorphisms form the basis for subobjects by providing the arrows whose equivalence classes define them.^[3] Subobjects are commonly denoted using the equivalence class notation [m: A \hookrightarrow C], where the hook arrow \hookrightarrow indicates that m is a monomorphism. This notation highlights the representative monic morphism and its domain while abstracting away isomorphic variants.^[4]

Equivalence Relation on Monomorphisms

In category theory, the notion of a subobject is refined by imposing an equivalence relation on the class of monomorphisms with a fixed codomain. Specifically, two monomorphisms m: A \to C and m': A' \to C are deemed equivalent—and thus represent the same subobject—if there exists an isomorphism i: A \to A' such that m = m' \circ i. This relation partitions the monomorphisms into equivalence classes, each class identifying a unique subobject up to isomorphism.^[5] This equivalence relation ensures that subobjects are intrinsically defined, independent of the particular choice of representative object or embedding morphism, thereby emphasizing the categorical "image" or embedded structure rather than any specific realization. By quotienting the monomorphisms in this manner, the construction captures the essential notion of a substructure in a way that is robust across isomorphic variants, aligning with the abstract goals of category theory. Monomorphisms here function as the categorical analogue of injective functions, generalizing the idea of embeddings without presupposing underlying elements.^[5]

Interpretations and Motivations

Relation to Classical Substructures

In the category of sets, denoted Set, subobjects of an object C correspond precisely to the subsets of C. Specifically, each subset A \subseteq C determines a subobject via the inclusion monomorphism i: A \hookrightarrow C, and two such monomorphisms represent the same subobject if they are related by an isomorphism making the appropriate triangle commute, thereby identifying isomorphic copies of the same subset.^[6] This notion extends naturally to other classical structures. In the category of groups, Grp, subobjects of a group G are the subgroups of G, realized through inclusion monomorphisms that embed a subgroup injectively into G. Similarly, in the category of vector spaces over a field k, denoted Vect_k, subobjects of a vector space V correspond to its subspaces, via monomorphisms that are linear injections preserving the vector space structure.^[6]^[4] Subobjects thus offer a unified categorical framework for conceptualizing "substructures" or "parts" of objects across diverse mathematical domains, independent of any underlying set of elements. This abstraction is particularly valuable in categories lacking a global choice of elements or where direct element-wise descriptions are unavailable, allowing for consistent treatment via morphisms alone.^[6]

Categorical Perspective

In category theory, subobjects formalize the notion of "parts" of an object in a manner that is invariant under isomorphism, defined as equivalence classes of monomorphisms with a common codomain. This abstraction allows for the categorical construction of quotients and exact sequences via universal properties, particularly through kernels and cokernels. In pointed categories, the kernel of a morphism f: A \to B is the subobject \ker f \hookrightarrow A that equalizes f and the zero morphism, satisfying the universal property that any other such equalizer factors uniquely through it.^[5] Similarly, the cokernel provides the dual universal quotient. In abelian categories, this enables the definition of exactness as the coincidence of an image subobject with a kernel subobject.^[5] The non-elementary nature of subobjects distinguishes them from the set-theoretic case, where inclusions suffice; in general categories, monomorphisms may represent more structured embeddings, such as regular monomorphisms that preserve additional categorical features without relying on global elements or pointwise inclusions.^[4] This generality ensures that subobjects adapt to the ambient category's logic, avoiding dependence on extraneous set-like operations.^[7] In homological algebra, subobjects are indispensable for classifying extensions and short exact sequences, as in $0 \to M \to E \to N \to 0, where the subobject M \hookrightarrow E identifies the kernel, and equivalence classes of such extensions correspond bijectively to elements in Ext groups, providing a measure of how N extends M.^[8] This framework underpins derived functors and homological invariants, treating subobjects as the building blocks for sequence exactness and resolution theory.^[5] Philosophically, subobjects underscore category theory's morphism-centric paradigm, prioritizing relational structures defined by arrows over intrinsic object properties, thereby unifying diverse mathematical contexts through diagrammatic universality rather than elemental inspection.^[5]

Examples

In the Category of Sets

In the category of sets, denoted \mathbf{Set}, subobjects of an object C are defined as isomorphism classes of monomorphisms into C. A monomorphism in \mathbf{Set} is an injective function, and thus every subobject corresponds to the image of such an injection, which is a subset of C. Specifically, for any subset A \subseteq C, the inclusion map i_A: A \to C defined by i_A(a) = a for all a \in A is a monomorphism, providing a canonical representative for the subobject associated to A.^[4]^[7] Conversely, every monomorphism m: B \to C in \mathbf{Set} is isomorphic to the inclusion map of its image \operatorname{im}(m) = \{m(b) \mid b \in B\} \subseteq C. Here, the isomorphism arises from the fact that m factors uniquely as B \xrightarrow{\sim} \operatorname{im}(m) \hookrightarrow C, where the first map is bijective and the second is the inclusion. Two monomorphisms m_1: B_1 \to C and m_2: B_2 \to C represent the same subobject if there exists an isomorphism \phi: B_1 \to B_2 such that m_2 \circ \phi = m_1, which occurs precisely when \operatorname{im}(m_1) = \operatorname{im}(m_2) as subsets of C. This establishes a bijection between the subobjects of C and the power set \mathcal{P}(C).^[4]^[9]^[10] The empty subset \emptyset \subseteq C corresponds to the unique subobject given by the empty function \emptyset \to C, which is the only monomorphism from the empty set. For a concrete illustration, consider C = \{1, 2, 3\}. The subobjects are in one-to-one correspondence with the eight subsets of C: the empty set (unique empty inclusion), the three singletons \{1\}, \{2\}, \{3\} (via inclusions like \{1\} \to C sending 1 to 1), the three doubletons \{1,2\}, \{1,3\}, \{2,3\}, and the full set C (via the identity map \operatorname{id}_C: C \to C). Each such inclusion is monic, and no two distinct subsets yield isomorphic monomorphisms into C.^[4]^[7] This correspondence aligns with the classical notion of subsets as substructures in set theory, providing a foundational example of subobjects in an unstructured category.^[4]

In Algebraic Categories

In algebraic categories, subobjects are defined as equivalence classes of monomorphisms into a given object, where the category's structure imposes additional constraints beyond mere set-theoretic inclusion. These categories, such as the category of groups (Grp), rings (Ring), and modules over a ring R (Mod_R), arise from varieties of universal algebras, and their subobjects correspond to substructures that preserve the algebraic operations. Monomorphisms in these categories are precisely the injective homomorphisms, ensuring that subobjects respect the category's morphisms.^[4] In the category Grp, subobjects of a group G correspond to its subgroups H \leq G, represented by the inclusion monomorphisms i: H \hookrightarrow G. Two such inclusions represent equivalent subobjects if and only if there exists an isomorphism \phi: H \to H' such that the diagram with inclusions into G commutes, which holds precisely when H = H' as subsets of G; non-trivial equivalences are thus rare and typically absent in concrete realizations. For example, cyclic subgroups generated by elements of G form distinct subobjects unless they coincide. This identification aligns with the fact that Grp is a regular category where kernels classify monomorphisms.^[11]^[4] Similarly, in the category Ring of rings (with unital homomorphisms), subobjects of a ring R are subrings S \leq R, via injective ring homomorphisms that preserve both addition and multiplication. The inclusion j: S \hookrightarrow R serves as the canonical representative, and equivalence classes match the distinct subrings, as isomorphic inclusions over R require identical underlying subsets. Subrings must contain the multiplicative identity of R under standard conventions, distinguishing them further from mere additive subgroups. For instance, the prime subring generated by 1 in an integral domain is a canonical subobject.^[12]^[4] In the category Mod_R of left R-modules, subobjects of a module M are precisely its submodules N \leq M, consisting of subsets closed under addition and scalar multiplication by elements of R, represented by inclusions k: N \hookrightarrow M. Equivalence again identifies subobjects with distinct submodules, as commuting isomorphisms over M imply equality of subsets. Submodules form abelian subgroups invariant under the R-action, and examples include ideals of R viewed as R-modules. Unlike in Grp or Ring, Mod_R is abelian, allowing subobjects to interact via exact sequences.^[13]^[4] A key distinction from the category of sets is that not every subset qualifies as a subobject in these algebraic categories; only those preserving the operations do so. For example, in Grp, the trivial subgroup \{e\} (where e is the identity) is always a subobject, but an arbitrary subset, such as the positive integers in \mathbb{Z} under addition (which fails closure under inverses), is not. This structural restriction ensures subobjects capture algebraic invariants rather than arbitrary inclusions.^[4]

Properties and Structures

Partial Order of Subobjects

In category theory, the collection of subobjects of a fixed object C in a category \mathcal{C} is equipped with a natural partial order. For subobjects represented by monomorphisms m: A \hookrightarrow C and m': A' \hookrightarrow C, the order is defined by \leq [m'] if there exists a morphism f: A \to A' such that the following commutative diagram holds:

\begin{tikzcd} A \arrow[r, "f"] \arrow[d, "m"'] & A' \arrow[d, "m'"] \\ C \arrow[r, equal] & C \end{tikzcd}

That is, m' \circ f = m. This relation is independent of the choice of representatives for the equivalence classes of monomorphisms and induces a partially ordered set (poset) structure on \mathrm{Sub}(C).^[5] This partial order corresponds to the intuitive notion of inclusion or factorization, where $$ is "contained in" [m'] if the image of m factors through the image of m'. In categories where monomorphisms are stable under pullback, such as regular categories, this order aligns with the existence of a unique such f that is itself a monomorphism.^[5] The partial order exhibits monotonicity with respect to pullbacks. Specifically, for any morphism g: X \to C, the induced pullback functor g^*: \mathrm{Sub}(C) \to \mathrm{Sub}(X), which sends a subobject [m: A \hookrightarrow C] to [g^* m: g^* A \hookrightarrow X], preserves the order: if \leq [m'], then [g^* m] \leq [g^* m']. This follows from the universal property of pullbacks, ensuring that the factorization diagram pulls back coherently.^[4] In finitely complete categories (i.e., those with all finite limits), the poset \mathrm{Sub}(C) has all finite meets, constructed via pullbacks of finite families; under additional assumptions like the existence of images, it forms a complete lattice.^[5]

Subobject Lattices

In category theory, the poset of subobjects of a fixed object C forms a lattice provided that every pair of subobjects admits both a meet (greatest lower bound) and a join (least upper bound) in the poset.^[5] This structure arises in categories equipped with sufficient limits and colimits, such as regular categories, where the operations align with categorical constructions. The existence of such a lattice depends on the category's properties; for instance, meets are generally available via pullbacks, while joins require pushouts and may not always yield subobjects of C.^[14] The meet of two subobjects A \hookrightarrow C and A' \hookrightarrow C is explicitly the pullback

\begin{CD} A \times_C A' @>>> A \\ @VVV @VVV \\ A' @>>> C, \end{CD}

which factors as a monomorphism A \times_C A' \hookrightarrow C. This construction, preserved under base change in categories with pullbacks, corresponds to the intersection in concrete settings like sets.^[5]^[14] Joins present more challenges, as they exist in categories with pushouts but the resulting object may not embed as a subobject of C. The join of A \hookrightarrow C and A' \hookrightarrow C, when it exists, is the least upper bound in the poset, often obtained as the image of the pushout of the inclusions along the common codomain C. However, in general categories, this pushout may lie outside the subobject class, requiring additional conditions like coherence for the join to remain a subobject. For example, in the category of sets (\mathbf{Set}), the join is the set-theoretic union, which embeds monotonically into C.^[5]^[14] In \mathbf{Set}, the subobject lattice of an object C is precisely the power set lattice \mathcal{P}(C), a complete Boolean lattice where meets are intersections and joins are unions.^[5] Similarly, in the category of groups (\mathbf{Grp}), the subobjects are subgroups, forming the subgroup lattice with meets as intersections H \cap K and joins as the subgroup generated by the union \langle H \cup K \rangle. This lattice is modular but not necessarily distributive.^[15]

Advanced Developments

Subobject Classifiers

In category theory, the subobject classifier provides a universal means to parameterize all subobjects of objects within a category equipped with pullbacks. Specifically, in a category \mathcal{C} with pullbacks, an object \Omega is a subobject classifier if, for every monomorphism m: A \hookrightarrow C, there exists a unique characteristic morphism \chi_m: C \to \Omega such that m is the pullback of a distinguished morphism \mathrm{true}: 1 \to \Omega along \chi_m, where $1 denotes the terminal object. This construction ensures that subobjects can be "classified" externally via morphisms into \Omega, mirroring the role of the power set in the category of sets. The distinguished morphism \mathrm{true}: 1 \to \Omega serves as the characteristic morphism for the full subobject C \hookrightarrow C, the maximal subobject of C. It acts as a global element that identifies the "true" or total inclusion, forming the basis for the pullback operation that recovers any subobject from its characteristic morphism.^[16] This setup induces a bijection between the subobjects of C and the morphisms from C to \Omega: for any morphism \chi: C \to \Omega, the corresponding subobject is the pullback of \mathrm{true} along \chi, and every subobject arises uniquely in this way. Recall that subobjects are equivalence classes of monomorphisms with the same image. This correspondence equips the category with a powerful representational tool for substructure.^[17] A key consequence of the existence of a subobject classifier \Omega is that, when the category is finitely complete, it qualifies as an elementary topos, thereby inheriting rich logical and geometric structures such as internal Heyting algebras on \Omega.^[18]

Limits and Colimits Involving Subobjects

In categories with pullbacks, subobjects of an object C can be pulled back along any morphism f: D \to C to yield subobjects of D, defining a contravariant functor f^*: \mathrm{Sub}(C) \to \mathrm{Sub}(D) that sends a monomorphism m: S \hookrightarrow C to the pullback monomorphism f^*(m): S' \hookrightarrow D, where S' is the pullback of m along f. This functor preserves the partial order on subobjects, as the pullback operation respects inclusions.^[4] In abelian categories, the kernel of a morphism f: A \to B, defined as the equalizer of f and the zero morphism, is a subobject of A via a monomorphism \ker f \hookrightarrow A, and every subobject arises as the kernel of some morphism. This identification aligns kernels with the categorical notion of equalizer, facilitating homological computations.^[19] Colimits of subobjects do not generally yield subobjects of the colimit of the ambient objects; for instance, the coproduct of monomorphisms i: A \hookrightarrow X and j: B \hookrightarrow X is the monomorphism i + j: A \coprod B \hookrightarrow X \coprod X, which targets the coproduct rather than X itself, and may not factor through a subobject of X. Such discrepancies highlight that while limits of subobjects often remain within the subobject poset, colimits require additional structure to do so.^[4] In exact categories, subobjects correspond to normal monomorphisms, which are kernels of cokernels, and every morphism factors uniquely as a normal epimorphism followed by a normal monomorphism, ensuring that images of monomorphisms are well-defined subobjects preserved under these factorizations. This structure underpins the exactness of short sequences involving subobjects.^[20] In topoi, the contravariant subobject functor \mathrm{Sub}: \mathcal{E}^\mathrm{op} \to \mathbf{Poset} is logical, meaning it preserves finite limits, so that the poset of subobjects of a limit object is the limit of the posets of subobjects, computed via pullbacks for intersections. This property ensures that subobject lattices inherit the topos's limit structure coherently.^[21]