Coproduct

In category theory, a coproduct is a colimit that combines a family of objects in a category into a single object equipped with morphisms from each original object, satisfying a universal property: for any object receiving compatible morphisms from the family, there exists a unique morphism from the coproduct making the diagram commute.^[1] This construction, dual to the categorical product, generalizes intuitive notions of summation or union while abstracting away specific implementations to focus on relational structure.^[2] The universal property of the coproduct ensures its uniqueness up to isomorphism, meaning any two coproducts of the same family are canonically equivalent.^[1] For two objects A and B in a category \mathcal{C}, the coproduct A \coprod B comes with injections i_A: A \to A \coprod B and i_B: B \to A \coprod B, such that for any object X and morphisms f: A \to X, g: B \to X, there is a unique h: A \coprod B \to X with h \circ i_A = f and h \circ i_B = g.^[3] This property extends to indexed families, where the coproduct \coprod_{i \in I} X_i serves as the "least general" extension incorporating all summands.^[1] Coproducts manifest differently across categories, reflecting the ambient structure. In the category of sets (Set), the coproduct is the disjoint union, where elements are tagged by their origin to preserve distinguishability.^[1] In the category of groups (Grp), it is the free product, consisting of words formed by alternating elements from the factors under the group operation.^[3] For abelian groups or vector spaces (Ab or Vect), the coproduct coincides with the direct sum, allowing finite support combinations.^[2] In topological spaces (Top), it is again the disjoint union topology.^[3] The empty coproduct is the initial object of the category, such as the empty set in Set.^[1] As a core colimit, coproducts underpin broader constructions like pushouts and coequalizers, facilitating proofs by abstraction in algebra, topology, and beyond.^[1] Category theory, in which coproducts were formalized, emerged from work by Samuel Eilenberg and Saunders Mac Lane in the 1940s, evolving into a foundational language for unifying mathematical disciplines.^[1]

Definition in Category Theory

Universal Property

In category theory, the coproduct of two objects A and B in a category \mathcal{C} is defined by its universal property. Specifically, the coproduct A \sqcup B is an object equipped with morphisms i_A: A \to A \sqcup B and i_B: B \to A \sqcup B, called the inclusion maps, such that for every object X in \mathcal{C} and every pair of morphisms f: A \to X, g: B \to X, there exists a unique morphism h: A \sqcup B \to X making the following diagrams commute: h \circ i_A = f and h \circ i_B = g. This universal property can be illustrated by the commutative diagram below, where the solid arrows represent the given morphisms and the dashed arrow denotes the unique induced morphism h:

       i_A          h
A ──────→ A ⊔ B ────→ X
  ↘             ↗
   f           g
     ↘       ↗
       B ──→
      i_B
       i_A          h
A ──────→ A ⊔ B ────→ X
  ↘             ↗
   f           g
     ↘       ↗
       B ──→
      i_B

The uniqueness clause in the property—that h is the only such morphism—ensures that if A' \sqcup B' with inclusions i_A' and i_B' also satisfies the same universal mapping property relative to A and B, then there exists a unique isomorphism \phi: A \sqcup B \to A' \sqcup B' such that \phi \circ i_A = i_A' and \phi \circ i_B = i_B', and similarly for the inverse. Thus, coproducts, when they exist, are unique up to unique isomorphism. The coproduct is the dual concept to the categorical product, obtained by reversing all arrows in the category.

Notation and Variations

In category theory, the coproduct of two objects A and B in a category \mathcal{C} is commonly denoted by A + B or A \sqcup B, with inclusion morphisms i_A: A \to A + B and i_B: B \to A + B.^[1] Alternative symbols include \amalg for the disjoint union in the category of sets, while \oplus is often used in additive categories to denote the direct sum, which coincides with the coproduct.^[1] For infinite or indexed coproducts over a set I, the notation \bigsqcup_{i \in I} A_i or \coprod_{i \in I} A_i is standard, emphasizing the colimit over a discrete diagram.^[4] The term "coproduct" is the primary designation, but it is also referred to as the "categorical sum" to distinguish it from non-categorical notions of sum, such as the union of sets without ensuring disjointness.^[4] More generally, an n-ary coproduct can be viewed as the colimit of a diagram over a discrete category with n objects and no non-identity morphisms, aligning with the universal property for families of morphisms from the indexed objects.^[4] While the binary coproduct is defined for two objects via the universal property, it extends naturally to finite indexed families \{A_i\}_{i \in I} where |I| < \infty, with the coproduct \bigsqcup_{i \in I} A_i mediating unique morphisms for any compatible family \{f_i: A_i \to C\}_{i \in I}.^[1] In pointed categories, where every object has a distinguished zero morphism, the coproduct often coincides with the wedge sum, particularly in the category of pointed topological spaces, denoted X \vee Y or (X \sqcup Y)/(x_0 = y_0), formed by identifying basepoints after the disjoint union.

Coproducts in Common Categories

In Sets

In the category of sets, denoted Set, the coproduct of two sets A and B is their disjoint union, often denoted A \sqcup B or A + B.^[5]^[1] This construction ensures that elements from A and B are distinguishable, even if the sets overlap. Explicitly, A \sqcup B is formed as the set of pairs (a, 0) for a \in A unioned with (b, 1) for b \in B, that is,

A \sqcup B = (A \times \{0\}) \cup (B \times \{1\}).

The inclusion maps are the injections i_A: A \to A \sqcup B defined by i_A(a) = (a, 0) and i_B: B \to A \sqcup B defined by i_B(b) = (b, 1).^[5]^[1] This disjoint union satisfies the universal property of the coproduct: for any set X and functions f: A \to X, g: B \to X, there exists a unique function h: A \sqcup B \to X such that h \circ i_A = f and h \circ i_B = g. This h is explicitly given by h(a, 0) = f(a) and h(b, 1) = g(b).^[5]^[1] The coproduct is unique up to canonical isomorphism.^[1] The cardinality of the coproduct is the sum of the cardinalities of the component sets: |A \sqcup B| = |A| + |B|, holding even if A and B are not disjoint, due to the tagging that prevents overlap in the union.^[5] For an infinite family of sets \{A_i\}_{i \in I} indexed by a set I, the coproduct in Set is the disjoint union \bigsqcup_{i \in I} A_i, constructed as \bigcup_{i \in I} (A_i \times \{i\}), with inclusions i_j: A_j \to \bigsqcup_{i \in I} A_i given by i_j(a) = (a, j) for each j \in I. This satisfies the universal property for families of functions from the A_i to any set X.^[5]

In Groups

In the category of groups, the coproduct of two groups G and H is given by their free product G * H, which is the group freely generated by the elements of G and H subject only to the relations that hold within G and within H separately, with no additional relations imposed between elements from G and H.^[6]^[7] This construction ensures that elements of G * H can be represented as reduced words alternating between nontrivial elements from G and H, with the group operation defined by concatenation followed by reduction to eliminate identities or consecutive terms from the same factor.^[7] The free product admits a presentation G * H = \langle X_G \cup X_H \mid R_G \cup R_H \rangle, where X_G (resp., X_H) is a set of generators for G (resp., H) and R_G (resp., R_H) is the set of relations defining G (resp., H).^[6] There are natural inclusion homomorphisms i_G: G \to G * H and i_H: H \to G * H, which embed G and H as subgroups generated by their respective elements, preserving the group structures without introducing cross-interactions.^[7] These inclusions satisfy the universal property of the coproduct: for any group K and group homomorphisms \phi: G \to K, \psi: H \to K, there exists a unique group homomorphism \tilde{\phi}: G * H \to K such that \tilde{\phi} \circ i_G = \phi and \tilde{\phi} \circ i_H = \psi.^[6]^[8] This uniqueness arises because any such \tilde{\phi} is determined by freely extending \phi and \psi to the reduced words in G * H, respecting the relations only within each factor.^[7] A representative example is the free product \mathbb{Z} * \mathbb{Z}, which is the free group on two generators, consisting of all reduced words in two indeterminates a and b (corresponding to the generators of each \mathbb{Z}) with no relations other than the group axioms.^[7]

In Abelian Groups and Modules

In the category of abelian groups, denoted Ab, the coproduct of two objects A and B is their direct sum A \oplus B, which consists of ordered pairs (a, b) with a \in A and b \in B, equipped with componentwise addition (a, b) + (a', b') = (a + a', b + b').^[9] The canonical inclusions are the morphisms i_A: A \to A \oplus B defined by i_A(a) = (a, 0) and i_B: B \to A \oplus B defined by i_B(b) = (0, b).^[9] This construction satisfies the universal property of the coproduct: for any abelian group X and group homomorphisms f: A \to X, g: B \to X, there exists a unique homomorphism h: A \oplus B \to X such that the diagrams

\begin{CD} A @>f>> X \\ @Vi_AVV @| \\ A \oplus B @>h>> X \end{CD} \quad \text{and} \quad \begin{CD} B @>g>> X \\ @Vi_BVV @| \\ A \oplus B @>h>> X \end{CD}

commute, explicitly given by h(a, b) = f(a) + g(b).^[9] In additive categories like Ab, the direct sum is in fact a biproduct, serving simultaneously as both the categorical product and coproduct for finite families due to the zero object and abelian structure.^[9] This construction generalizes directly to the category of modules over a commutative ring R, denoted R-Mod. The coproduct of R-modules M and N is their direct sum M \oplus N, with the same componentwise R-linear structure and inclusions i_M(m) = (m, 0), i_N(n) = (0, n).^[10] The universal property holds analogously: for any R-module P and R-module homomorphisms \phi: M \to P, \psi: N \to P, there is a unique R-module homomorphism \theta: M \oplus N \to P such that \theta \circ i_M = \phi and \theta \circ i_N = \psi, defined by \theta(m, n) = \phi(m) + \psi(n).^[10] For a family of R-modules \{M_i\}_{i \in I}, the coproduct is the direct sum \bigoplus_{i \in I} M_i, comprising tuples (m_i)_{i \in I} where each m_i \in M_i and only finitely many m_i are nonzero (elements of finite support), with componentwise scalar multiplication and addition restricted to this subspace.^[10] In contrast to the coproduct, the categorical product in Ab and R-Mod is the direct product, which for infinite families \{A_i\}_{i \in I} consists of all tuples (a_i)_{i \in I} without the finite support restriction.^[9] For finite index sets, the direct sum and direct product coincide, but for infinite I, the direct sum is a proper subgroup (or submodule) of the direct product, reflecting the colimit nature of coproducts versus the limit nature of products.^[9] This distinction underscores the role of direct sums as the universal "least upper bound" under inclusions in these abelian categories.^[10]

In Topological Spaces

In the category of topological spaces, denoted Top, the coproduct of two topological spaces X and Y is given by their disjoint union X \sqcup Y, equipped with the disjoint union topology. The underlying set of X \sqcup Y is the disjoint union of the underlying sets of X and Y, and the open sets are those of the form U \sqcup V, where U is open in X and V is open in Y. The inclusion maps i_X: X \to X \sqcup Y and i_Y: Y \to X \sqcup Y are homeomorphisms onto their images, which are the connected components of X \sqcup Y. This construction satisfies the universal property of the coproduct: for any topological space Z and continuous maps f: X \to Z, g: Y \to Z, there exists a unique continuous map h: X \sqcup Y \to Z such that h \circ i_X = f and h \circ i_Y = g, defined by h on the image of i_X as f and on the image of i_Y as g. In the subcategory of pointed topological spaces, Top^*, where objects are pairs (X, x_0) with a distinguished basepoint x_0 \in X and morphisms preserve basepoints, the coproduct differs from the disjoint union. For pointed spaces (X, x_0) and (Y, y_0), the coproduct is the wedge sum X \vee Y, formed by taking the disjoint union X \sqcup Y and quotienting by the equivalence relation that identifies x_0 with y_0, yielding the quotient space (X \sqcup Y) / \{x_0 \sim y_0\} with the quotient topology. The basepoint of X \vee Y is the equivalence class of x_0 and y_0. The wedge sum satisfies the universal property in Top^*: for pointed spaces (Z, z_0) and basepoint-preserving continuous maps f: (X, x_0) \to (Z, z_0), g: (Y, y_0) \to (Z, z_0), there is a unique basepoint-preserving continuous map h: (X \vee Y, [x_0]) \to (Z, z_0) such that h \circ i_X = f and h \circ i_Y = g, where i_X and i_Y are the basepoint-preserving inclusions into the wedge sum. A representative example is the wedge sum of two circles, S^1 \vee S^1, where each S^1 is pointed at (1,0); this space is homeomorphic to the figure-eight curve, consisting of two circles joined at a single point. In general, the topologies on both the disjoint union and wedge sum arise from quotient constructions, and while they behave well for Hausdorff spaces, non-Hausdorff examples may require additional verification of continuity for the induced maps.

Properties and Constructions

Existence and Uniqueness

In any category where coproducts exist, they are unique up to unique isomorphism. Specifically, suppose A \sqcup B and A \sqcup' B are two coproducts of objects A and B, equipped with inclusion morphisms i: A \to A \sqcup B, j: B \to A \sqcup B, and similarly i': A \to A \sqcup' B, j': B \to A \sqcup' B. Then there exists a unique isomorphism \phi: A \sqcup B \to A \sqcup' B such that \phi \circ i = i' and \phi \circ j = j'.^[11] To see this, apply the universal property of the coproduct A \sqcup' B to the pair of morphisms i' and j' from A and B, yielding a unique morphism \phi: A \sqcup B \to A \sqcup' B that commutes with the inclusions. Dually, the universal property of A \sqcup B applied to i' and j' produces a unique morphism \psi: A \sqcup' B \to A \sqcup B that commutes. The uniqueness clause in each universal property then implies that \psi \circ \phi = \mathrm{id}_{A \sqcup B} and \phi \circ \psi = \mathrm{id}_{A \sqcup' B}, establishing \phi as an isomorphism with the required compatibility.^[11] Coproducts do not exist in every category. For instance, when a poset is viewed as a thin category (where morphisms are unique between comparable elements), the coproduct of two elements is their join, which may not exist for arbitrary pairs. In contrast, the category of sets (Set) admits all finite coproducts, realized as disjoint unions; the category of groups (Grp) has all finite coproducts as free products; and the category of abelian groups (Ab) has all finite coproducts as direct sums. These categories are in fact cocomplete, possessing coproducts for all small families of objects.^[11] A category has all binary coproducts if every pair of objects admits a coproduct, and it has all finite coproducts if every finite family does; the latter follows from the former via iterated applications of binary coproducts, with associativity holding up to unique isomorphism by the uniqueness theorem. Categories with all small coproducts extend this to families indexed by arbitrary small sets, but finite coproducts suffice for many constructions in algebra and topology.^[11]

Infinite Coproducts

In category theory, an infinite coproduct of a family of objects \{A_i\}_{i \in I} in a category \mathcal{C}, where I is an indexing set, is an object \coprod_{i \in I} A_i together with morphisms \iota_j \colon A_j \to \coprod_{i \in I} A_i for each j \in I, such that for any object X in \mathcal{C} and any family of morphisms \{f_j \colon A_j \to X\}_{j \in I}, there exists a unique morphism f \colon \coprod_{i \in I} A_i \to X satisfying f \circ \iota_j = f_j for all j \in I.^[1]^[3] This universal property generalizes the finite case to arbitrary indexing sets, ensuring the coproduct is unique up to canonical isomorphism when it exists.^[1] In the category \mathbf{Set} of sets, the infinite coproduct \coprod_{i \in I} A_i is constructed as the disjoint union \bigcup_{i \in I} (A_i \times \{i\}), where each A_i is embedded via the injection \iota_i(a) = (a, i).^[1]^[3] The cardinality of this coproduct is the cardinal sum \sum_{i \in I} |A_i|, which for infinite I accounts for the possible uncountability depending on the sizes of the A_i.^[1] This construction satisfies the universal property by mapping families of functions to the unique function on the disjoint union that respects the tags.^[3] In the category \mathbf{Ab} of abelian groups, the infinite coproduct \bigoplus_{i \in I} A_i is the direct sum, realized as the subgroup of the Cartesian product \prod_{i \in I} A_i consisting of elements with finite support (i.e., tuples where all but finitely many components are zero).^[1]^[3] The inclusions \iota_j send a \in A_j to the tuple with a in the j-th position and zeros elsewhere, and the universal property holds for homomorphisms into any abelian group, as infinite sums are not generally defined without finite support.^[1] This direct sum forms a bifunctor in additive categories like \mathbf{Ab}.^[1] In the category \mathbf{Grp} of groups, the infinite coproduct \ast_{i \in I} G_i is the free product, constructed as the quotient of the free group on the disjoint union of the underlying sets of the G_i by relations preserving the group operations within each G_i, resulting in reduced words alternating elements from distinct G_i.^[1]^[3] The inclusions \iota_j embed each G_j monomorphically, with images intersecting trivially at the identity, and the universal property ensures unique homomorphisms factoring through families from the G_i.^[1] For infinite I, the free product can be expressed as the direct limit of finite free products over finite subsets of I.^[3] Infinite coproducts exist in a category \mathcal{C} whenever \mathcal{C} admits colimits over the discrete diagram indexed by I, a condition often met in categories that are cocomplete or have small coproducts for arbitrary index sets.^[1]^[3] Functors preserving coproducts, such as left adjoints, map infinite coproducts to coproducts in the target category, facilitating constructions in algebraic and topological contexts.^[1]

Duality with Products

In category theory, the coproduct construction exhibits a profound duality with the product, mediated by the opposite category functor. Specifically, for a category \mathcal{C}, the coproduct of objects A and B in \mathcal{C}, denoted A \sqcup B, is isomorphic to the product A \times B in the opposite category \mathcal{C}^{\mathrm{op}}, where all morphisms are reversed in direction.^[1] This correspondence arises because the opposite functor \mathrm{Op}: \mathcal{C} \to \mathcal{C}^{\mathrm{op}} is contravariant and preserves universal properties by inverting the hom-sets: the bijection \mathcal{C}(A \sqcup B, C) \cong \mathcal{C}(A, C) \times \mathcal{C}(B, C) in \mathcal{C} becomes \mathcal{C}^{\mathrm{op}}(C, A \times B) \cong \mathcal{C}^{\mathrm{op}}(C, A) \times \mathcal{C}^{\mathrm{op}}(C, B) in \mathcal{C}^{\mathrm{op}}.^[12] This duality manifests explicitly in the reversing of morphism directions: the inclusion maps i_A: A \to A \sqcup B and i_B: B \to A \sqcup B of the coproduct in \mathcal{C} correspond to the projection maps p_A: A \times B \to A and p_B: A \times B \to B of the product in \mathcal{C}^{\mathrm{op}}.^[1] Consequently, any mediating morphism [f, g]: A \sqcup B \to C in \mathcal{C}, defined by the coproduct's universal property such that f = [f,g] \circ i_A and g = [f,g] \circ i_B, dualizes to a pairing morphism \langle f, g \rangle: C \to A \times B in \mathcal{C}^{\mathrm{op}}, satisfying the product's universal property.^[12] Illustrative examples highlight this duality in familiar categories. In the category of sets \mathbf{Set}, the coproduct is the disjoint union A \sqcup B, which dualizes to the Cartesian product A \times B in \mathbf{Set}^{\mathrm{op}}, with inclusions tagging elements to distinguish origins and projections selecting components.^[1] Similarly, in the category of abelian groups \mathbf{Ab}, the coproduct (direct sum) A \oplus B coincides with the product (direct product), forming a biproduct where inclusions embed into the first and second components and projections extract them, reflecting the self-dual nature under the duality.^[12] In other categories, coproducts and their dual products may diverge, particularly for infinite families. For instance, in the category of groups \mathbf{Grp}, finite coproducts are free products A * B, but infinite coproducts exist via colimits of finite sub-coproducts and require careful construction to ensure the universal property holds, unlike the straightforward disjoint unions in \mathbf{Set}.^[1] In the category of posets \mathbf{Poset}, where morphisms are order-preserving functions, the coproduct is the disjoint union of posets, with inclusions as order-preserving embeddings and elements from different posets incomparable; the product is the Cartesian product equipped with the componentwise order. These interchange under the opposite category functor.^[1]

Relations to Other Concepts

As a Colimit

In category theory, the binary coproduct of two objects A and B in a category \mathcal{C} is defined as the colimit of the discrete diagram consisting of A and B (two objects with no morphisms between them).^[3] This colimit comes equipped with inclusion maps i_A: A \to A + B and i_B: B \to A + B such that for any object X with maps f: A \to X and g: B \to X, there exists a unique map h: A + B \to X making the diagram commute, satisfying the universal property of the colimit cocone.^[3] In categories with an initial object $0, the binary coproduct A + B can be realized as the pushout of the span A \leftarrow 0 \rightarrow B along the unique morphisms from $0 to A and from $0 to B. This construction equates the coproduct to a specific instance of a pushout where the "gluing" object is initial, ensuring no overlap between the components of A and B.^[13] More generally, an indexed coproduct \coprod_{i \in I} A_i over an index set I is the colimit of a diagram F: I \to \mathcal{C} where I is viewed as a discrete category (with only identity morphisms) and F(i) = A_i.^[14] This extends the binary case to arbitrary families, with inclusion maps i_j: A_j \to \coprod_{i \in I} A_i for each j \in I, universal among all families of maps from the A_i to some object.^[14] In a cocomplete category, which has all small colimits, coproducts exist as a particular type of colimit over discrete diagrams, but not every colimit is a coproduct; for instance, coequalizers arise from diagrams with parallel arrows, distinct from the morphism-free structure of coproducts. For example, in the category of simplicial sets, the coproduct of two simplicial sets X and Y is their disjoint union, given levelwise by the disjoint union of the sets of n-simplices X_n \sqcup Y_n for each n, preserving the face and degeneracy maps independently.^[15]

Coproducts in Enriched Categories

In a V-enriched category \mathbf{B}, where V is a symmetric monoidal closed category, an enriched coproduct of objects A_i (for i \in I) is an object C together with enriched natural transformations \iota_i: A_i \to C (the inclusions) such that for any object B in \mathbf{B}, the induced map

\mathbf{B}(C, B) \to \prod_{i \in I} \mathbf{V}(I_i, \mathbf{B}(A_i, B))

is an isomorphism in V, where I_i denotes the representable V(I, -) and the product is taken in V.^[16] This satisfies the weighted colimit property, where the coproduct is the colimit weighted by the coproduct of representables \coprod_i V(I, I_i)^{op}, ensuring a universal enriched cone.^[16] More generally, for a diagram G: \mathbf{K} \to \mathbf{B} weighted by F: \mathbf{K}^{op} \to \mathbf{V}, the enriched coproduct \{F, G\} exists if \mathbf{B}(\{F, G\}, B) \cong [\mathbf{K}^{op}, \mathbf{V}](F, \mathbf{B}(G(-), B)) naturally in V.^[16] In Ab-enriched categories, where V = Ab (the category of abelian groups), enriched coproducts coincide with the ordinary coproducts in the underlying category provided that Ab has finite or small coproducts, which it does via direct sums.^[16] For instance, the enriched coproduct of abelian groups A and B is their direct sum A \oplus B, with the hom-Ab-objects \mathbf{B}(A \oplus B, C) isomorphic to \mathbf{Ab}( \mathbb{Z}, \mathbf{B}(A, C) \times \mathbf{B}(B, C) ), reflecting the additive structure.^[16] This alignment holds because finite coproducts in Ab-enriched categories are absolute, preserving the universal property under the forgetful functor to Set.^[17] In monoidal categories such as Vect_k (vector spaces over a field k), enriched over itself, coproducts are realized via the tensor coproduct structure when the category is tensored and cotensored.^[16] Specifically, the enriched coproduct of vector spaces corresponds to their direct sum, which coincides with the tensor product with the coproduct in V under the closed structure, as Vect_k is symmetric monoidal closed and cocomplete.^[16] This requires the weight to be representable and the diagram to be small, ensuring the colimit exists pointwise in the underlying category.^[17] Not all V-enriched categories admit coproducts, even if V is cocomplete; existence depends on the enriched structure and the diagram's size, often requiring V to be locally small and complete.^[16] The enriched coproduct, when it exists, induces an ordinary coproduct in the underlying category \mathbf{B}_0, but the converse does not hold, as the enriched universal property is stricter and involves V-objects rather than sets.^[16] In 2-categories, coproducts generalize to pseudo-coproducts, where the inclusions are 1-cells and the universal property involves invertible 2-cells ensuring coherence up to isomorphism, rather than strict equality.^[17] This accounts for the weak equivalences inherent in higher-dimensional structures, as seen in bicategory-enriched settings.^[17]