nLab additive and abelian categories

Definitions

There is a sequence of extra structure and property on a category that makes this category behave like a general context for homological algebra. In order of increasing structure and property this is:

1. Ab-enriched category: a category that is Ab-enriched;

2. pre-additive category: an $Ab$-enriched category that has a terminal object or initial object and therefore a zero object; Notice however that many authors (e.g. Weibel, Popescu) by preadditive (or pre-additive) simply mean $Ab$-enriched.

   • pseudo-abelian category: a pre-additive category such that every idempotent has a kernel and cokernel

3. additive category: a pre-additive category that admits binary products or binary coproducts and therefore binary biproducts, hence is a semiadditive category (equivalently, an $Ab$-enriched category with all finite products or coproducts);

4. pre-abelian category: an additive category that admits kernels and cokernels (equivalently, an $Ab$-enriched category with all finite limits and colimits);

5. abelian category: a pre-abelian category such that every monomorphism is a kernel and every epimorphism is a cokernel (and many other equivalent definitions).

Pre-abelian and abelian categories are sometimes called (AB1) and (AB2) categories, after the sequence of additional axioms on top of additive categories introduced by Grothendieck in Tohoku. AB1 and AB2 are self-dual axioms (AB1 is existence of kernels and cokernels, and AB2 requires that, for any $f$, the canonical morphism $\mathrm{Coim}\,f\to \mathrm{Im}\,f$ is an isomorphism). These continue in non-selfdual manner:

• AB3: an abelian category with all coproducts (hence with all colimits);

• AB4: an (AB3) category in which coproducts of monomorphisms are monic;

• AB5: an (AB3) category in which filtered colimits of exact sequences exist and are exact;

• AB6: an (AB3) category such that

  • For any object $A$ in the category and any family $(B^j)_{j\in J}$ of increasing directed families of $B^i=(B_i^j)_{i\in I_j}$ of subobjects $B^j$ of $A$, we have $\bigcap_{j\in J}\left(\sum_{i\in I_j}B_j^i\right)=\sum_{(i_j)\in \prod I_j}\left(\bigcap_{j\in J}B_{i_j}^i\right)$.

  • Notice that this implies that inf for any family of subobjects exists.

We have a chain of implications AB6$\Rightarrow$AB5$\Rightarrow$AB4, see (Nicolae Popescu 1973, Sect. 2.8, Corollaries 8.9, 8.13).

The concepts (AB3–AB6) also have dual forms (AB3*–AB6*).

There are further refinements along these lines. In particular

• Grothendieck category: an (AB5) category with a generator.

Theorem

(Nicolae Popescu 1973, Sect. 2.8, Theorem 8.6)
Let $\mathcal{C}$ be an AB3-category. The following assertions are equivalent:

1. $\mathcal{C}$ is AB5.

2. For any direct set $\left\{X_i\right\}_i$ of subobjects of any object $X$, the unique morphism $j: \underset{\rightarrow}{\lim} X_i \rightarrow X$ defined such that $j u_i=j_i$, (where $u_i: X_i \rightarrow \underset{\rightarrow}{\lim} X_i$ are structural morphisms and $j_i: X_i \rightarrow X$ are canonical inclusions) is an isomorphism of $\underset{\rightarrow}{\lim} X_i$ onto $\sum_i X_i$.

3. Let $X$ be any object, $\left\{X_i\right\}_i$ a direct set of subobjects, and $X^{\prime}$ a subobject of $X$. Then:

   $$\sum_i\left(X_i \cap X^{\prime}\right)=\left(\sum_i X_i\right) \cap X^{\prime}$$

4. Let $f: Y \rightarrow X$ be a morphism and $\left\{X_i\right\}_i$ a direct set of subobjects of $X$. Then:

   $$f^{-1}\left(\sum_i X_i\right)=\sum_i f^{-1}\left(X_i\right)$$

5. Let $\left\{X_i\right\}_{i \in I}$ be a set of objects. For any finite subset $F$ of $I$, we denote by $X_F$ image of canonical morphism $u_F: \coprod_{i^{\prime} \in F} X_{i^{\prime}} \rightarrow \coprod_i X_i$, defined such that $u_F u_{i^{\prime}}{ }^{\prime}=u_{i^{\prime}}$, where $u_{i^{\prime}}{ }^{\prime}$ and $u_{i^{\prime}}$ are respectively structural morphisms. Then for any subobject $X$ of $\coprod_i X_i$ we have:

   $$X=\sum_{F \in T}\left(X \cap X_F\right)$$

   $T$ being the set of all finite subsets of $I$.

Further refinements

Various further axiom structures are considered for additive (sometimes abelian) categories.

• suspended category

• triangulated category

• Quillen exact category

• Grothendieck category: an AB5-category with a generator

• Gabriel’s property sup

Generalizations

• Abelian groups are to general groups as abelian categories are to semi-abelian categories.

Examples

Various generic classes of examples of additive and abelian categories are of relevance:

Related concepts

• delta-functor
• category of chain complexes
• derived category
• stable (infinity,1)-category

References

• Nicolae Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press (1973)