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Additive category

In category theory, an additive category is a preadditive category that possesses a zero object and admits all finite biproducts, enabling the formation of direct sums for finite collections of objects.^[1]^[2] A preadditive category, in turn, is one where the hom-sets between any pair of objects form abelian groups under addition, and the composition of morphisms is bilinear over the integers, meaning it distributes over addition in both arguments and is compatible with scalar multiplication by integers.^[1]^[2] This structure generalizes the notion of categories of abelian groups or modules, where morphisms can be added and scaled, and provides a foundational framework for homological algebra.^[2] Key properties of additive categories include the existence of kernels and cokernels for every morphism, facilitated by the biproduct structure, although these do not necessarily satisfy the full exactness conditions required for abelian categories.^[2] In an additive category, the zero object serves both as an initial and terminal object, and direct sums coincide with both products and coproducts, ensuring a symmetric treatment of "sums" in the categorical sense.^[1] Additive functors between such categories preserve the abelian group structure on hom-sets, the zero object, and biproducts, making them the natural morphisms in this context.^[1]^[2] Prominent examples of additive categories include the category of abelian groups Ab, the category of left modules over a ring R-Mod, and categories of vector spaces over a field, all of which support the required additive operations and biproducts via direct sums.^[2] These categories often serve as models for studying chain complexes and exact sequences, bridging abstract category theory with concrete algebraic structures.^[2] While additive categories capture essential linearity, they fall short of abelian categories by not guaranteeing that every monomorphism is a kernel or that images equal coimages, highlighting their role as an intermediate structure in homological contexts.^[2]^[3]

Preliminaries

Basic category theory prerequisites

A category \mathcal{C} consists of a collection \mathrm{Ob}(\mathcal{C}) of objects and, for every pair of objects A, B \in \mathrm{Ob}(\mathcal{C}), a set \Hom_{\mathcal{C}}(A, B) of morphisms from A to B, together with a composition operation: for any three objects A, B, C \in \mathrm{Ob}(\mathcal{C}), a function \Hom_{\mathcal{C}}(B, C) \times \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{C}}(A, C) denoted (f, g) \mapsto g \circ f, such that composition is associative, i.e., (h \circ g) \circ f = h \circ (g \circ f) whenever defined, and such that for every object A, there exists an identity morphism \mathrm{id}_A \in \Hom_{\mathcal{C}}(A, A) satisfying \mathrm{id}_A \circ f = f = f \circ \mathrm{id}_A for all f \in \Hom_{\mathcal{C}}(A, B) or \Hom_{\mathcal{C}}(C, A).^[4]^[5] The sets \Hom_{\mathcal{C}}(A, B) are called the hom-sets (or morphism sets) of the category, and they collect all arrows between the specified objects; the collection of all such hom-sets, along with the objects and composition, fully specifies the structure of \mathcal{C}.^[4] A functor F: \mathcal{C} \to \mathcal{D} between categories \mathcal{C} and \mathcal{D} is a map that sends objects of \mathcal{C} to objects of \mathcal{D} and morphisms of \mathcal{C} to morphisms of \mathcal{D} in a way that preserves identities, i.e., F(\mathrm{id}_A) = \mathrm{id}_{F(A)} for all A \in \mathrm{Ob}(\mathcal{C}), and preserves composition, i.e., F(g \circ f) = F(g) \circ F(f) whenever g \circ f is defined in \mathcal{C}.^[4]^[6] A natural transformation \eta: F \Rightarrow G between two functors F, G: \mathcal{C} \to \mathcal{D} assigns to each object A \in \mathrm{Ob}(\mathcal{C}) a morphism \eta_A: F(A) \to G(A) in \mathcal{D} such that for every morphism f: A \to B in \mathcal{C}, the diagram

\begin{CD} F(A) @>{\eta_A}>> G(A) \\ @V{F(f)}VV @VV{G(f)}V \\ F(B) @>>{\eta_B}> G(B) \end{CD}

commutes, meaning G(f) \circ \eta_A = \eta_B \circ F(f).^[4]^[6] Categories are classified as small or large based on the size of their collections: a category is small if both its collection of objects and its collection of all morphisms form sets, whereas a large category has at least one of these collections as a proper class; most categories encountered in practice, such as those of sets or groups, are large due to the vastness of their objects. A skeletal category is one in which isomorphic objects are equal, providing a minimal representative structure equivalent to a broader category by selecting one object from each isomorphism class while preserving all morphisms between them.^[7]

Abelian groups and bilinear maps

An abelian group is a group (G, +) equipped with a binary operation + that is associative, has an identity element $0 \in Gsuch thatg + 0 = gfor allg \in G, and every element g \in Ghas an [additive inverse](/page/Additive_inverse)-g \in Gsuch thatg + (-g) = 0, with the additional property that the operation is commutative: g + h = h + gfor allg, h \in G$.^[8] This commutativity distinguishes abelian groups from non-abelian groups, where the order of elements matters under the operation.^[8] Prominent examples of abelian groups include the integers \mathbb{Z} under addition, where the identity is $0and the inverse ofnis-n, satisfying all group axioms with commutativity m + n = n + m. Another example is any [vector space](/page/Vector_space) Vover a [field](/page/Field)k, considered as an abelian group under vector addition, with 0_Vas the identity and-vas the inverse forv \in V; here, commutativity follows from u + v = v + ufor allu, v \in V$.^[9] A bilinear map between abelian groups is a function f: A \times B \to C, where A, B, and C are abelian groups, that preserves the group operations in each argument separately. Specifically, for all a, a' \in A and b \in B,

f(a + a', b) = f(a, b) + f(a', b),

and for all b, b' \in B and a \in A,

f(a, b + b') = f(a, b) + f(a, b').

^[9] These additivity conditions ensure that f is a group homomorphism when fixed in one variable: the map a \mapsto f(a, b) is a homomorphism A \to C for fixed b \in B, and similarly for fixed a \in A.^[10] In the more general setting of modules over a commutative ring R, where abelian groups are viewed as \mathbb{Z}-modules, a bilinear map f: M \times N \to P between R-modules additionally preserves scalar multiplication: for r \in R, m \in M, and n \in N, f(rm, n) = r f(m, n) = f(m, rn).^[9] This homogeneity property extends the additivity over integers inherent to abelian groups, allowing bilinear maps to interact compatibly with ring actions in algebraic structures like tensor products.^[11]

Definition

Definition via preadditive structure

An additive category can be defined as a preadditive category equipped with a zero object and such that every finite collection of objects admits a biproduct. A preadditive category is a category \mathcal{C} in which, for every pair of objects A, B \in \mathcal{C}, the Hom-set \Hom_{\mathcal{C}}(A, B) is an abelian group, and the composition of morphisms is bilinear with respect to the group structures on the Hom-sets.^[5]^[12] The abelian group structure on \Hom_{\mathcal{C}}(A, B) equips each set of morphisms between A and B with an addition operation, denoted f + g for f, g \in \Hom_{\mathcal{C}}(A, B). In concrete realizations of preadditive categories, such as the category of abelian groups or modules over a ring, this addition is defined pointwise: (f + g)(x) = f(x) + g(x) for all x in the domain where f and g are defined. Each Hom-group also contains a zero element, called the zero morphism $0_{A,B} \in \Hom_{\mathcal{C}}(A, B), which acts as the identity for addition and satisfies f \circ 0_{B,C} = 0_{A,C}and0_{B,C} \circ f = 0_{A,B}for composable morphismsf$.^[5] Bilinearity of composition means that, for morphisms f, g: A \to B and h: B \to C, the following distributive laws hold:

h \circ (f + g) = h \circ f + h \circ g, \quad (f + g) \circ h = f \circ h + g \circ h.

More generally, for f, f': A \to B and g, g': B \to C, bilinearity extends to

(g + g') \circ (f + f') = g \circ f + g \circ f' + g' \circ f + g' \circ f'.

This implies that addition of composite morphisms satisfies (f \circ g) + (f' \circ g') = (f + f') \circ (g + g') only in special cases, such as when f = f' or g = g'; otherwise, the right-hand side expands to the full sum of cross-compositions as above.^[5]^[12] To obtain an additive category from a preadditive one, there must exist a zero object $0 \in \mathcal{C}that is both initial and terminal: for every objectA \in \mathcal{C}, the Hom-sets \Hom_{\mathcal{C}}(0, A)and\Hom_{\mathcal{C}}(A, 0)each consist of exactly one morphism, which is the zero morphism0_{0,A}or0_{A,0}$. This zero object, together with the existence of finite biproducts for every finite collection of objects, ensures a consistent zero morphism across the category and distinguishes additive categories from mere preadditive ones.^[5]^[12]

Equivalent formulations

An equivalent formulation defines an additive category as a semiadditive category, which is a category possessing all finite biproducts, meaning that finite products and coproducts coincide for every finite family of objects.^[5] In such a category, the biproduct of objects A and B, denoted A \oplus B, comes equipped with injection morphisms i_A: A \to A \oplus B and i_B: B \to A \oplus B, and projection morphisms p_A: A \oplus B \to A and p_B: A \oplus B \to B, satisfying the relations p_A i_A = \mathrm{id}_A, p_B i_B = \mathrm{id}_B, and i_A p_A + i_B p_B = \mathrm{id}_{A \oplus B}.^[13] This structure induces an abelian monoid operation on each hom-set \mathrm{Hom}(X, Y), turning the category into a preadditive one where composition is bilinear with respect to the induced addition.^[5] A more precise equivalent definition requires that every finite family of objects admits a biproduct, and that the induced maps \mathrm{Hom}(A \oplus B, C) \to \mathrm{Hom}(A, C) \times \mathrm{Hom}(B, C), given by precomposition with the injections i_A and i_B, are isomorphisms of abelian groups.^[13] The inverse sends a pair (f, g) with f: A \to C and g: B \to C to the unique morphism [f, g]: A \oplus B \to C such that [f, g] \circ i_A = f and [f, g] \circ i_B = g, ensuring the bilinearity of composition.^[5] To see the equivalence to the preadditive formulation, note that the biproducts induce an abelian group structure on each hom-set via a subtraction operation. Specifically, for parallel morphisms f, g: A \to B, the difference f - g is defined using the cokernel of the diagonal morphism or equivalently as the composite involving projections and injections, with addition then given by f + g = f - (0 - g), where 0 denotes the zero morphism induced by the structure.^[13] This operation is associative and commutative, and composition distributes over it, yielding bilinearity; conversely, in a preadditive category with finite biproducts, the existing group structure aligns uniquely with the one induced by the biproducts.^[5] Moreover, the existence of finite biproducts implies a zero object, obtained as the empty biproduct, which serves as both initial and terminal object.^[13] As a concrete illustration of the induced addition, consider two morphisms f, g: A \to C; their sum is f + g = [f, g] \circ \Delta_A, where \Delta_A: A \to A \oplus A is the diagonal morphism \Delta_A = \langle \mathrm{id}_A, \mathrm{id}_A \rangle and [f, g]: A \oplus A \to C is the pairing morphism.^[5] This construction ensures that the addition is independent of choices and compatible with the biproduct axioms.

Generalizations to enriched categories

Enriched category theory provides a framework for generalizing categorical structures by replacing ordinary hom-sets with objects in a monoidal category V, allowing for more abstract notions of composition and morphisms. An enriched category \mathcal{C} over a monoidal category V consists of a class of objects, together with hom-objects \mathcal{C}(A, B) \in V for each pair of objects A, B, a composition morphism \circ_{A,B,C}: \mathcal{C}(B, C) \otimes_V \mathcal{C}(A, B) \to \mathcal{C}(A, C) in V for all A, B, C, and identity morphisms I_V \to \mathcal{C}(A, A) from the unit object I_V of V, satisfying associativity and unit axioms via diagrams in V.^[14] This setup captures additive categories when V = \mathrm{Ab}, the category of abelian groups under the cartesian monoidal structure (direct products).^[15] In the case V = \mathrm{Ab}, an \mathrm{Ab}-enriched category has hom-objects that are abelian groups, with composition given by bilinear maps in \mathrm{Ab}, preserving addition and scalar multiplication by integers. Such categories are precisely the preadditive categories, where the hom-sets form abelian groups and composition is bilinear, satisfying the usual associativity and unit conditions. Additive categories then arise as those \mathrm{Ab}-enriched categories equipped with a zero object and finite biproducts, which coincide with finite products and coproducts.^[16]^[14] This notion generalizes further to enrichment over the category of commutative monoids, denoted \mathrm{CommMon}, where hom-objects are commutative monoids (under addition) and composition is bilinear with respect to the monoidal structure on \mathrm{CommMon}. In such settings, the additive structure on morphisms is captured by the monoid operation, often replacing abelian group addition with more general semigroup or monoid addition, while preserving bilinearity for composition. For instance, powerset enrichment over the monoid of subsets under union allows modeling relational structures where "addition" corresponds to union of relations, generalizing additivity to non-invertible settings.^[17] A key difference from ordinary additive categories lies in the role of the monoidal tensor \otimes_V: in \mathrm{Ab}-enrichment, direct products in the underlying category are replaced by tensors with the biproduct structure of \mathrm{Ab}, enabling the abstraction of addition to arbitrary V. For the theory to support internal homs, V must be closed monoidal, providing an internal hom-object [X, Y]_V such that morphisms from Z \otimes_V X to Y correspond to morphisms from Z to [X, Y]_V, which facilitates the definition of enriched functors and natural transformations in the additive context.^[14]^[15]

Examples

Standard examples from algebra

The category of abelian groups, denoted \mathbf{Ab}, is a fundamental example of an additive category. In \mathbf{Ab}, the objects are abelian groups, and the morphisms are group homomorphisms. The Hom-sets \Hom(A, B) form abelian groups under pointwise addition: for f, g \in \Hom(A, B), the sum (f + g)(a) = f(a) + g(a) for all a \in A. Composition of morphisms is bilinear, meaning (f + g) \circ h = f \circ h + g \circ h and f \circ (g + h) = f \circ g + f \circ h for compatible morphisms f, g, h. Biproducts in \mathbf{Ab} are given by direct sums of abelian groups.^[18] More generally, for any ring R (not necessarily commutative), the category \mathbf{R}-\mathbf{Mod} of left R-modules is additive. Here, objects are R-modules, and morphisms are R-linear maps. The Hom-sets \Hom(M, N) are abelian groups under pointwise addition: (f + g)(m) = f(m) + g(m) for m \in M. Composition remains bilinear with respect to this addition, and scalar multiplication by elements of R is incorporated into the module structure. Biproducts correspond to direct sums of modules. The subcategory of finitely presented R-modules, often denoted \mathbf{mod}-R, inherits these properties as a full additive subcategory.^[19]^[20] When R = k is a field, \mathbf{R}-\mathbf{Mod} specializes to the category \mathbf{Vect}_k of vector spaces over k. Morphisms are linear maps, with Hom-sets forming vector spaces (hence abelian groups) under pointwise addition. In the full subcategory of finite-dimensional vector spaces, biproducts are finite direct sums, emphasizing the finite nature of these structures.^[21] The category of chain complexes of abelian groups, \mathbf{Ch}(\mathbf{Ab}), provides another algebraic example. Objects are sequences of abelian groups with differentials satisfying d^2 = 0, and morphisms are chain maps. This category is additive, with Hom-sets being abelian groups under pointwise addition of maps, and composition bilinear. Biproducts are given by direct sums of complexes componentwise.^[22]

Examples from topology and geometry

In topology, the category of topological abelian groups provides a natural example of an additive category. Here, the objects are abelian groups equipped with a topology making the group operations continuous, and the morphisms are continuous group homomorphisms. The Hom-sets form abelian groups under pointwise addition, as the sum of two continuous homomorphisms is again continuous and the zero morphism is the constant map to the identity element. This category admits a zero object, the trivial group with the indiscrete topology, and finite biproducts given by the product topology on direct sums.^[23] A prominent example from algebraic topology is the category Ch(R) of chain complexes of R-modules, where R is a commutative ring. The objects are chain complexes, sequences of R-modules connected by differentials satisfying d² = 0, and the morphisms are chain maps of degree zero, which preserve the differential. This category is additive because the Hom-sets between chain complexes form abelian groups under pointwise addition of chain maps, inheriting the additive structure from the underlying category of R-modules, which is itself additive. Finite biproducts exist as the direct sum of chain complexes, componentwise, and the zero object is the zero complex with trivial modules in each degree.^[24] In sheaf theory, the category Sh(X, Ab) of sheaves of abelian groups on a topological space X exemplifies an additive structure arising from local data. Objects are sheaves where sections over open sets form abelian groups, with the sheaf condition ensuring gluing, and morphisms are sheaf homomorphisms preserving the abelian group structure pointwise. The Hom-sets are abelian groups under pointwise addition of sections, as addition is compatible with the sheaf topology and restrictions. Biproducts are given by the sheafification of the direct sum of presheaves, yielding the sheaf direct sum, and the zero sheaf serves as the zero object. This setup underscores the additive nature in contexts involving continuous variation over spaces.^[25] Geometric contexts yield the category VB(X) of vector bundles over a topological space X, which is additive via the Whitney sum operation. Objects are vector bundles, locally trivial bundles of vector spaces, and morphisms are bundle maps, linear on fibers. The Hom-sets form abelian groups under pointwise addition of bundle maps, reflecting the vector space structure on fibers, and composition distributes over this addition. The Whitney sum E ⊕ F of two bundles E and F provides the biproduct, combining fibers directly and inheriting the topology from the base, while the trivial line bundle acts as the zero object when rank zero is allowed. This structure facilitates K-theory computations in geometry.^[26] Derived categories in homological algebra and geometry, such as the derived category D(A) of an abelian category A, underlie additive categories with additional triangulated structure. Objects are complexes in A, modulo quasi-isomorphisms, and morphisms are roofs of chain maps, but the underlying Hom-sets inherit the abelian group structure from the homotopy category of complexes, which is additive. Biproducts are preserved from the category of complexes, given by direct sums, and the zero object is the zero complex. While triangulated for capturing exactness, the core additive framework supports cohomology computations in topological settings, like singular cohomology.^[27]

Properties

Internal characterization of addition

In an additive category, the addition of parallel morphisms can be defined intrinsically using the biproducts and zero object, without presupposing an external abelian group structure on the hom-sets.^[5] For morphisms f, g: A \to B, their sum f + g: A \to B is given by

f + g = \nabla_B \circ (f \oplus g) \circ \Delta_A,

where \Delta_A: A \to A \oplus A is the diagonal morphism and \nabla_B: B \oplus B \to B is the codiagonal morphism.^[5] This construction leverages the universal properties of the biproducts A \oplus A and B \oplus B, ensuring that the operation is canonical within the category.^[5] The diagonal \Delta_A is the unique morphism induced by the copairing of the identity maps, satisfying p_1 \circ \Delta_A = \mathrm{id}_A and p_2 \circ \Delta_A = \mathrm{id}_A, where p_1, p_2: A \oplus A \to A are the projections.^[5] Similarly, the codiagonal \nabla_B is the unique pairing of the identity maps, satisfying \nabla_B \circ i_1 = \mathrm{id}_B and \nabla_B \circ i_2 = \mathrm{id}_B, with i_1, i_2: B \to B \oplus B the injections.^[5] The zero morphism $0_{A,B}: A \to B is then defined as the unique map factoring through the zero object, $0_{A,B} = !_{B} \circ \eta_A, where \eta_A: A \to 0 and !_B: 0 \to B are the unique morphisms to and from the zero object.^[5] This internal definition satisfies the key universal property of addition: for any morphism h: C \to A, (f + g) \circ h = f \circ h + g \circ h, where the addition on the right is again defined via the biproduct structure on C \oplus C.^[5] Analogously, addition is right-distributive over composition: for k: B \to D, k \circ (f + g) = k \circ f + k \circ g.^[5] These properties ensure bilinearity of the addition with respect to composition, as expressed by the equation

k \circ (f + g) + k' \circ (f + g) = (k + k') \circ (f + g) = k \circ f + k \circ g + k' \circ f + k' \circ g

for k, k': B \to D, following from the universal properties of the biproducts and the associativity of the operation.^[5] The addition is well-defined and unique because the biproducts provide a universal characterization: any two morphisms agreeing on the components after diagonalization must coincide, as \Delta_A is an epimorphism and \nabla_B a monomorphism in this context.^[5] Moreover, this structure endows each hom-set \hom(A, B) with the axioms of an abelian group, where inverses exist uniquely such that f + (-f) = 0 for each f, and the operation is commutative due to the symmetry isomorphisms in the biproducts.^[5] When an external abelian group structure is already present on the hom-sets (as in a preadditive category with biproducts), this internal definition coincides with the given addition, as both satisfy the same universal properties with respect to composition and the zero morphism.^[5] The proof proceeds by verifying that the internal sum distributes over the external one via the projections and inclusions of the biproducts, confirming equivalence.^[5]

Matrix representations of morphisms

In additive categories equipped with finite biproducts, morphisms between direct sums of objects can be represented in a matrix form analogous to linear transformations between vector spaces.^[5]^[18] Specifically, consider a morphism f: \bigoplus_{i=1}^n A_i \to \bigoplus_{j=1}^m B_j, where the biproducts are formed using the canonical inclusions \iota_i: A_i \to \bigoplus_{k=1}^n A_k and projections \pi_j: \bigoplus_{l=1}^m B_l \to B_j. This morphism f is uniquely determined by its component morphisms f_{ji} = \pi_j \circ f \circ \iota_i: A_i \to B_j for $1 \leq j \leq m and $1 \leq i \leq n, which form the entries of an m \times n matrix (f_{ji}).^[5]^[18] The universal property of biproducts ensures that every such matrix corresponds to a unique morphism f, establishing an isomorphism of abelian groups between \hom(\bigoplus A_i, \bigoplus B_j) and the direct sum of the \hom(A_i, B_j).^[5] The addition of morphisms in this representation corresponds to entrywise addition of matrices. If f and g are morphisms represented by matrices (f_{ji}) and (g_{ji}), respectively, then f + g is represented by (f_{ji} + g_{ji}), reflecting the abelian group structure on hom-sets.^[5]^[18] Composition of morphisms aligns with matrix multiplication, facilitated by the bilinearity of composition over the addition of morphisms, which ensures distributivity: for composable morphisms f: \bigoplus A_i \to \bigoplus B_j and g: \bigoplus C_k \to \bigoplus A_i, the composite f \circ g has matrix entries given by

(f \circ g)_{ik} = \sum_j f_{ji} \circ g_{kj}.

This formula mirrors standard matrix multiplication, with the sum taken in the abelian group of morphisms.^[5]^[18] A concrete example arises in the category \mathbf{Vect}_k of vector spaces over a field k, where biproducts are direct sums and morphisms are linear maps. Choosing standard bases for finite-dimensional spaces V = \bigoplus_{i=1}^n V_i and W = \bigoplus_{j=1}^m W_j, a linear map f: V \to W is represented by the matrix whose (j,i)-entry is the scalar matrix of the component map f_{ji}: V_i \to W_j with respect to the chosen bases. Addition and composition then coincide with the usual matrix operations.^[5] However, this matrix representation is not intrinsic to the category but depends on the choice of direct sum decompositions (or bases in the vector space case), which may not be unique up to canonical isomorphism. Different choices can yield equivalent but non-identical matrices, limiting its use to computational or illustrative purposes rather than invariant structural analysis.^[5]^[18]

Additive functors and natural transformations

Definition and properties of additive functors

In category theory, an additive functor between additive categories \mathcal{C} and \mathcal{D} is a functor F: \mathcal{C} \to \mathcal{D} that preserves the abelian group structure on the Hom-sets, meaning that for any objects A, B \in \mathcal{C} and morphisms f, g: A \to B, the induced map F: \Hom_{\mathcal{C}}(A, B) \to \Hom_{\mathcal{D}}(F(A), F(B)) is a group homomorphism, so F(f + g) = F(f) + F(g).^[5] This additivity implies that F sends zero morphisms to zero morphisms, i.e., F(0_{A,B}) = 0_{F(A),F(B)} for all A, B \in \mathcal{C}.^[28] Since additive categories are equipped with finite biproducts (direct sums), an additive functor F preserves these biproducts up to natural isomorphism: for any objects A, B \in \mathcal{C}, there exists a natural isomorphism F(A \oplus B) \cong F(A) \oplus F(B) that respects the inclusion and projection morphisms.^[29] Equivalently, the canonical map F(A) \oplus F(B) \to F(A \oplus B) (or its converse) is an isomorphism in \mathcal{D}.^[28] This preservation ensures that F maps the zero object of \mathcal{C} to the zero object of \mathcal{D}, as the zero object is the biproduct $0 \cong 0 \oplus 0.^[29] Additive functors automatically preserve the bilinear nature of composition in additive categories. Specifically, if f: A \to B and g: B \to C are morphisms in \mathcal{C}, then the additivity of F on Hom-sets, combined with the bilinearity of composition (f' + f'') \circ g = f' \circ g + f'' \circ g and f \circ (g' + g'') = f \circ g' + f \circ g'' in \mathcal{C}, yields F(f \circ g) = F(f) \circ F(g), though the functoriality of F already ensures this equality; the key additive property reinforces the group homomorphism aspect across compositions.^[5] Moreover, the composite of two additive functors is again additive, making the collection of additive functors between additive categories form a subcategory of the functor category.^[28] A standard example is the forgetful functor U: R\text{-Mod} \to \text{[Ab](/page/AB)} from the category of modules over a ring R to the category of abelian groups, which forgets the R-action while preserving the underlying abelian group structure on objects and the addition of Hom-sets.^[5] This functor is additive because it induces group homomorphisms on Hom-sets and preserves direct sums, as the direct sum of R-modules underlies the direct sum of abelian groups.^[29]

Additive natural transformations

An additive natural transformation \eta: F \Rightarrow G between additive functors F, G: \mathcal{C} \to \mathcal{D}, where \mathcal{C} and \mathcal{D} are additive categories, is a natural transformation whose components \eta_A: F(A) \to G(A) are morphisms in \mathcal{D} and satisfy the additivity condition \eta_{A \oplus B} = \eta_A \oplus \eta_B for all objects A, B \in \mathrm{Ob}(\mathcal{C}), where \oplus on the right denotes the unique morphism induced by the biproduct in \mathcal{D}.^[5] This condition ensures that \eta respects the biproduct structure preserved by F and G.^[5] The additivity condition arises naturally from the naturality squares applied to the inclusion and projection morphisms of biproducts in \mathcal{C}, combined with the preservation of biproducts by the additive functors F and G.^[5] Specifically, for the inclusion i_A: A \to A \oplus B, naturality yields \eta_{A \oplus B} \circ F(i_A) = G(i_A) \circ \eta_A, and since F(i_A) and G(i_A) are the corresponding inclusions in the biproducts F(A) \oplus F(B) and G(A) \oplus G(B), along with the analogous condition for projections, uniqueness of biproduct morphisms implies the desired equality.^[5] The collection of all such additive natural transformations, denoted \mathrm{Nat}^+(F, G), forms an abelian group under pointwise addition defined by (\eta + \zeta)_A = \eta_A + \zeta_A for \eta, \zeta \in \mathrm{Nat}^+(F, G), where + on the right is the group operation in the Hom-group \mathrm{Hom}_\mathcal{D}(F(A), G(A)).^[5] The zero element is the zero transformation with all components the zero morphism, and the inverse of \eta is -\eta with components -\eta_A. This group structure is itself natural and additive, as the pointwise operations commute with the functors' preservation of abelian group structures on hom-sets.^[5] This construction ensures compatibility with the Hom-group structures in additive categories, as the components of \eta lie in these groups and the naturality condition preserves the bilinear composition and addition of morphisms.^[5] For instance, the identity transformation \mathrm{Id}_F: F \Rightarrow F on an additive functor F is additive, since \mathrm{Id}_{A \oplus B} = \mathrm{Id}_F(A \oplus B) = F(A \oplus B) \to F(A \oplus B) coincides with \mathrm{Id}_A \oplus \mathrm{Id}_B by the preservation of biproducts.^[5]

Special cases and relations

Preadditive and semiadditive categories

A preadditive category is a category in which the collection of morphisms between any two objects forms an abelian group, and the composition of morphisms is bilinear with respect to the group structures on the Hom-sets.^[1] Unlike a full additive category, a preadditive category does not require the existence of a zero object.^[1] A semiadditive category, in contrast, is one equipped with all finite biproducts, which provide a zero object and induce a commutative monoid structure on each Hom-set via the universal properties of the biproducts; composition is bilinear with respect to this monoidal addition.^[30] The biproducts serve both as finite products and finite coproducts, enabling the addition of objects and morphisms in a compatible way.^[31] The key distinction from a full additive category lies in the structure on the Hom-sets: while an additive category combines the abelian group enrichment of a preadditive category with a zero object (equivalently, all finite biproducts), a semiadditive category only guarantees monoidal Hom-sets without additive inverses for morphisms.^[1] Thus, semiadditive categories represent a weaker form where addition is present but not necessarily invertible.^[30] An example of a semiadditive category that is not preadditive is the category of abelian monoids, where finite biproducts exist as direct sums, inducing commutative monoid structures on Hom-sets, but these lack additive inverses in general.^[31] The category of sets equipped only with its finite coproducts fails to be semiadditive, as its products and coproducts do not coincide to form biproducts.^[30] The concept of semiadditive categories emerged as a foundational stepping stone in the development of additive category theory during the 1960s, notably in Barry Mitchell's systematic treatment that bridged earlier work on abelian categories to broader homological structures.^[30]

Abelian categories as a special case

An abelian category is an additive category in which every morphism admits a kernel and a cokernel, and the canonical morphism from the coimage to the image of every morphism is an isomorphism.^[32] This condition ensures that every monomorphism is the kernel of its cokernel and every epimorphism is the cokernel of its kernel, making monomorphisms and epimorphisms normal.^[33] The concept was introduced to generalize homological methods from module categories, providing a framework where exact sequences and derived functors behave coherently. In an abelian category, images exist for all morphisms and coincide with their coimages, allowing for a well-defined notion of exactness at objects in sequences.^[32] Short exact sequences—those where the morphism to the second object is a kernel of the subsequent morphism and a cokernel of the prior one—capture the essential homological structure, enabling the development of tools like the snake lemma and long exact sequences in cohomology.^[33] These properties make abelian categories foundational in homological algebra, as they support the construction of projective and injective resolutions without requiring additional set-theoretic assumptions beyond smallness. Abelian categories are additive categories and thus possess abelian group structures on hom-sets, a zero object, and finite biproducts.^[32] Prominent examples include the category Ab of abelian groups, where morphisms are group homomorphisms, and the category R-Mod of modules over a commutative ring R, both of which satisfy the abelian axioms through their module-theoretic kernels and cokernels.^[33] A key result embedding abelian categories into familiar settings is the Freyd–Mitchell embedding theorem, which states that every small abelian category admits an exact, fully faithful functor into the category of modules over some ring. This theorem, proved independently by Freyd and Mitchell, underscores the concrete representability of abstract abelian structures and facilitates the application of module-theoretic techniques in broader homological contexts.

Karoubian categories

A Karoubian category, also known as an idempotent-complete category, is an additive category in which every idempotent endomorphism e: A \to A (satisfying e \circ e = e) admits a kernel, which serves as a direct summand of A. This kernel, often denoted as the image of e, splits A as a biproduct A \cong \operatorname{im}(e) \oplus \operatorname{coker}(e), ensuring that formal direct summands are realized internally within the category.^[34]^[35] The Karoubi envelope provides a universal construction to idempotent-complete any additive category C, yielding a Karoubian category \overline{C} that embeds C fully and faithfully while preserving the additive structure and biproducts. Objects in \overline{C} are pairs (X, e) where X \in \mathrm{Ob}(C) and e: X \to X is an idempotent morphism, with morphisms \phi: (X, e) \to (Y, f) required to satisfy f \circ \phi = \phi = \phi \circ e. This envelope is functorial and ensures that every idempotent in the image of the embedding splits in \overline{C}.^[36] In the context of quiver representations, the category of representations over a field is additive but not necessarily Karoubian; its Karoubi envelope completes it by splitting idempotents, yielding indecomposable representations as direct summands. Similarly, the category of projective modules over a ring arises as the Karoubi envelope of the full subcategory of free modules, preserving the additive and biproduct structures inherent to the original category.^[36]^[37] A key property of Karoubian categories is that every object decomposes uniquely (up to isomorphism and permutation) into a direct sum of indecomposable objects, provided the category satisfies additional conditions like the Krull-Schmidt property; this decomposition arises from iteratively splitting all nontrivial idempotents in the endomorphism ring.^[38]^[35] The notion originates from Max Karoubi's work on K-theory, where such categories (termed pseudo-abelian) were introduced to handle idempotent splittings in vector bundle contexts, and they play a foundational role in completing triangulated categories to ensure idempotent completeness.^[39]

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