Grothendieck category

0.1 Generator, Generator Family and Cogenerator

Let 𝒞 be a category. Moreover, let = _i∈I be a family of objects of 𝒞. The family is said to be a family of generators of the category 𝒞 if for any object A of 𝒞 and any subobject B of A, distinct from A, there is at least an index i ∈ I, and a morphism, u : U_i → A, that cannot be factorized through the canonical injection i : B → A. Then, an object U of 𝒞 is said to be a generator of the category 𝒞 provided that U belongs to the family of generators _i∈I of 𝒞 ([4]).

By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a family of cogenerators of the same category 𝒞, and also the notion of cogenerator U^∗ of 𝒞, if all of the required, reverse arrows exist. Notably, in a groupoid– regarded as a small category with all its morphisms invertible– this is always possible, and thus a groupoid can always be cogenerated via duality. Moreover, any generator in the dual category 𝒞^op is a cogenerator of 𝒞.

0.2 Ab-conditions: Ab3 and Ab5 conditions

1.

(Ab3). Let us recall that an Abelian category 𝒜b is cocomplete (or an 𝒜b3-category) if it has arbitrary direct sums.

2.

(Ab5). A cocomplete Abelian category 𝒜b is said to be an 𝒜b5-category if for any directed family _i∈I of subobjects of 𝒜, and for any subobject B of 𝒜, the following equation holds

(∑ _i∈IA_i) ⋂ B = ∑ _i∈I(A_i ⋂ B).

0.2.1 Remarks

• One notes that the condition Ab3 is equivalent to the existence of arbitrary direct limits.
• Furthermore, Ab5 is equivalent to the following proposition: there exist inductive limits and the inductive limits over directed families of indices are exact, that is, if I is a directed set and 0 → A_i → B_i → C_i → 0 is an exact sequence for any i ∈ I, then
  

  is also an exact sequence.

• By duality, one readily obtains conditions Ab3* and Ab5* simply by reversing the arrows in the above conditions defining Ab3 and Ab5.

0.3 Grothendieck and co-Grothendieck Categories

As an example consider the category 𝒜b of Abelian groups such that if _i∈I is a family of abelian groups, then a direct product Π is defined by the Cartesian product Π_i(X_i) with addition defined by the rule: (x_i) + (y_i) = (x_i + y_i). One then defines a projection ρ : Π_i(X_i) → X_i given by p_i((x_i)) = x_i. A direct sum is obtained by taking the appropriate subgroup consisting of all elements (x_i) such that x_i = 0 for all but a finite number of indices i. Then one also defines a structural injection , and it is straightforward to prove that 𝒜b is an 𝒜b6 and 𝒜b4^∗ category. (viz. p 61 in ref. [4]).

0.3.1 Remarks

1.

Let 𝒜 be an Abelian category and 𝒞 a small category. One defines then a functor k_c : 𝒜→ [𝒞,𝒜] as follows: for any X ∈ Ob𝒜, k_𝒞(X) : 𝒞 →𝒜 is the constant functor which is associated to X. Then 𝒜 is an 𝒜b5 category (respectively, 𝒜b5^∗), if and only if for any directed set I, as above, the functor k_I has an exact left (or respectively, right) adjoint.

2.

With 𝒜b4, 𝒜b5, 𝒜b4^∗, and 𝒜b6 one can construct categories of (pre) additive functors.

3.

A preabelian category is an additive category with the additional (𝒜b1) condition that for any morphism f in the category there exist also both kerf and cokerf;

4.

An Abelian category can be then also defined as a preabelian category in which for any morphism f : X → Y , the morphism f : coimf → imf is an isomorphism (the 𝒜b2 condition).

References