enriched category theory

category theory

monoidal category, closed monoidal category

cosmos, multicategory, bicategory, double category, virtual double category

enriched category

enriched functor, profunctor

enriched natural transformation

enriched adjoint functor

enriched product category

enriched functor category

weighted limit

end, coend

copowering (tensoring)

powering (cotensoring)

monoidal enriched category

cartesian closed enriched category

locally cartesian closed enriched category

enriched homotopical category

enriched model category

model structure on homotopical presheaves

additive and abelian categories

enriched category theory

homological algebra

stable homotopy theory

Ab-enriched category

pre-additive category

additive category,

(AB1) pre-abelian category

(AB2) abelian category

(AB5) Grothendieck category

additive functor

left/right exact functor

derived category

derived functor

homotopy category

triangulated category

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homological algebra

(also nonabelian homological algebra)

Introduction

Context

additive and abelian categories

Ab-enriched category

pre-additive category

additive category

pre-abelian category

abelian category

Grothendieck category

abelian sheaves

semi-abelian category

Basic definitions

kernel, cokernel

complex

differential

homology

category of chain complexes

chain complex

chain map

chain homotopy

chain homology and cohomology

quasi-isomorphism

homological resolution

simplicial homology

generalized homology

exact sequence,

injective object, projective object

injective resolution, projective resolution

flat resolution

Stable homotopy theory notions

derived category

triangulated category, enhanced triangulated category

stable (∞,1)-category

stable model category

pretriangulated dg-category

A-∞-category

(∞,1)-category of chain complexes

derived functor, derived functor in homological algebra

Tor, Ext

homotopy limit, homotopy colimit

abelian sheaf cohomology

Constructions

double complex

Koszul-Tate resolution, BRST-BV complex

spectral sequence

spectral sequence of a filtered complex

spectral sequence of a double complex

Grothendieck spectral sequence

Leray spectral sequence

Serre spectral sequence

Hochschild-Serre spectral sequence

Lemmas

diagram chasing

3x3 lemma

four lemma, five lemma

snake lemma, connecting homomorphism

horseshoe lemma

Baer's criterion

Schanuel's lemma

Homology theories

singular homology

cyclic homology

Theorems

Dold-Kan correspondence / monoidal, operadic

Eilenberg-Zilber theorem

universal coefficient theorem

Künneth theorem

An additive category is a category which is

an Ab-enriched category;

(sometimes called a pre-additive category–this means that each hom-set carries the structure of an abelian group and composition is bilinear)

which admits finite coproducts

(and hence, by prop. below, finite products which coincide with the coproducts, hence finite biproducts).

The natural morphisms between additive categories are additive functors.

A pre-abelian category is an additive category which also has kernels and cokernels. Equivalently, it is an Ab-enriched category with all finite limits and finite colimits. An especially important sort of additive category is an abelian category, which is a pre-abelian one satisfying the extra exactness property that all monomorphisms are kernels and all epimorphisms are cokernels. See at additive and abelian categories for more.

The Ab-enrichment of an additive category does not have to be given a priori. Every semiadditive category (a category with finite biproducts) is automatically enriched over commutative monoids (as described at biproduct), so an additive category may be defined as a category with finite biproducts whose hom-monoids happen to be groups. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire AbAb-enriched structure follows automatically for abelian categories.

Some authors use additive category to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a CMonCMon-enriched (commutative monoid enriched) category, with or without assumptions of products.

In an Ab-enriched category (or even just a CMonCMon-enriched category), a finite product is also a coproduct, and dually (hence a biproduct).

This statement includes the zero-ary case: any terminal object is also an initial object, hence a zero object (and dually), hence every additive category has a zero object.

More precisely, for {X i} i∈I\{X_i\}_{i \in I} a finite set of objects in an Ab-enriched category, the unique morphism

whose components are identities for i=ji = j and are zero otherwise is an isomorphism.

Consider first the nullary (i.e., zero-ary) case. Given a terminal object *\ast, the unique morphism id *:*→*id_\ast: \ast \to \ast is the zero morphism 00 in its hom-object. For any object AA, the zero morphism 0 A:*→A0_A: \ast \to A must equal any morphism f:*→Af: \ast \to A on account of f=fid *=f0=0 Af = f id_\ast = f 0 = 0_A where the last equation is by CMonCMon-enrichment. Hence *\ast is initial. (N.B.: this argument applies more generally to categories enriched in pointed sets, and is self-dual.)

Consider now the case of binary (co-)products. Using zero morphisms, in addition to its canonical projection maps p i:X 1×X 2→X ip_i \colon X_1 \times X_2 \to X_i, any binary product also admits “injection” maps X i→X 1×X 2X_i \to X_1 \times X_2, and dually for the coproduct:

Observe some basic compatibility of the AbAb-enrichment with the product:

First, for (α 1,β 1),(α 2,β 2):R→X 1×X 2(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2 then

(using that the projections p 1p_1 and p 2p_2 are linear and by the universal property of the product).

Second, (id,0)∘p 1(id,0) \circ p_1 and (0,id)∘p 2(0,id) \circ p_2 are two projections on X 1×X 2X_1\times X_2 whose sum is the identity:

(We may check this, via the Yoneda lemma on generalized elements: for (α,β):R→X 1×X 2(\alpha, \beta) \colon R \to X_1\times X_2 any morphism, then (id,0)∘p 1∘(α,β)=(α,0)(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0) and (0,id)∘p 2∘(α,β)=(0,β)(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta), so the statement follows with equation (⋆)(\star).)

Now observe that for f i:X i→Qf_i \;\colon\; X_i \to Q any two morphisms, the sum

gives a morphism of cocones

Moreover, this is unique: suppose ϕ′\phi' is another morphism filling this diagram, then, by using equation (⋆⋆)(\star \star), we get

This means that X 1×X 2X_1\times X_2 satisfies the universal property of a coproduct.

By a dual argument, the binary coproduct X 1⊔X 2X_1 \sqcup X_2 is seen to also satisfy the universal property of the binary product. By induction, this implies the statement for all finite (co-)products. (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)

Such products which are also coproducts as in prop. are sometimes called biproducts or direct sums; they are absolute limits for Ab-enrichment.

The coincidence of products with biproducts in prop. does not extend to infinite products and coproducts.) In fact, an Ab-enriched category is Cauchy complete just when it is additive and moreover its idempotents split.

Conversely:

A semiadditive category is a category that has all finite products which, moreover, are biproducts in that they coincide with finite coproducts as in def. .

In a semiadditive category, def. , the hom-sets acquire the structure of commutative monoids by defining the sum of two morphisms f,g:X⟶Yf,g \;\colon\; X \longrightarrow Y to be

With respect to this operation, composition is bilinear.

The associativity and commutativity of ++ follows directly from the corresponding properties of ⊕\oplus. Bilinearity of composition follows from naturality of the diagonal Δ X\Delta_X and codiagonal ∇ X\nabla_X:

Given an additive category according to def. , then the enrichment in commutative monoids which is induced on it via prop. and prop. from its underlying semiadditive category structure coincides with the original enrichment.

By the proof of prop. , the codiagonal on any object in an additive category is the sum of the two projections:

Therefore (checking on generalized elements, as in the proof of prop. ) for all morphisms f,g:X→Yf,g \colon X \to Y we have commuting squares of the form

Prop. says that being an additive category is an extra property on a category, not extra structure. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.

semiadditive category

abelian category

additive (∞,1)-category

triangulated category

Textbook accounts:

Discussion in homological algebra:

See also:

Discussion of model category structures on additive categories is around def. 4.3 of

Formalization of additive categories as univalent categories in univalent foundations of mathematics (homotopy type theory):

A characterisation of preadditive categories in terms of commutative monoids in cartesian multicategories is given in:

Last revised on January 25, 2024 at 09:12:00. See the history of this page for a list of all contributions to it.