1. Additive categories

1.1. Definition By a pre-additive category one understands a category
A \mathsf{A}
enriched over the category of abelian groups. This means that for each pair of objects
A,B A,B
in
A \mathsf{A}
the morphism set
Mor(A,B) \Mor (A,B)
carries an abelian group structure
+ (A,B):Mor(A,B)×Mor(A,B)→Mor(A,B),(f,g)↦f+g +_{(A,B)} : \: \Mor (A,B) \times \Mor (A,B) \rightarrow \Mor (A,B), \quad (f,g) \mapsto f + g
such that composition of morphisms in
categoryA \category{A}
is bilinear in the following sense:
  1. If
    A,B,C A,B,C
    are objects of
    categoryA \category{A}
    ,
    f,f 1,f 2∈Mor(A,B) f,f_1,f_2 \in \Mor (A,B)
    and
    g,g 1,g 2∈Mor(B,C) g,g_1,g_2 \in \Mor (B,C)
    , then
    g∘(f 1+f 2)=(g∘f 1)+(g∘f 2)and(g 1+g 2)∘f=(g 1∘f)+(g 2∘f). g \circ (f_1 + f_2) = (g\circ f_1) + (g\circ f_2) \quad \text{and} \quad (g_1 + g_2) \circ f = (g_1 \circ f) + (g_2\circ f) .
Usually one denotes the set of morphism between objects
A A
and
B B
of a pre-additive category
categoryA \category{A}
by
Hom(A,B) \Hom (A,B)
instead of
Mor(A,B) \Mor (A,B)
. We will follow this conention from now on. The zero element of
Hom(A,B) \Hom (A,B)
will be denoted by
0 (A,B) 0_{(A,B)}
or briefly by
0 0
, if no confusion can arise. In general, and as done already in the definition, we will abbreviate the group operation
+ (A,B) +_{(A,B)}
on
Hom(A,B) \Hom (A,B)
by
+ +
for clearity of exposition. A pre-additive structure on a category imposes quite a useful relation between finite products and coproducts of its objects, namely that they have to coincide when they exist.
1.1. Proposition Let
A \mathsf{A}
be a pre-additive category, and
A 1,…,A n A_1, \ldots , A_n
a finite family of objects in
categoryA \category{A}
.
  1. If
    ∏ l=1 nA l \prod_{l=1}^n A_l
    is a product with canonical projections
    p k:∏ l=1 nA l→A k p_k : \prod_{l=1}^n A_l \to A_k
    ,
    k=1,…,n k=1,\ldots , n
    , then it is also a coproduct where the canonical injections are given by the uniquely determined morphisms
    i k:A k↦∏ l=1 nA l i_k : A_k \mapsto \prod_{l=1}^n A_l
    such that
    p l∘i k={id A k, ifk=l, 0, else. p_l \circ i_k = \begin{cases} \id_{A_k}, & \text{if}\ k=l, \\ 0, & \text{else}. \end{cases}
    In addition, the equality
    ∑ l=1 ni l∘p l=id ∏ l=1 nA l \sum_{l=1}^n i_l \circ p_l = \id_{\prod_{l=1}^n A_l} (1.1)
    holds true.
  2. If
    ∐ l=1 nA l \coprod_{l=1}^n A_l
    is a coproduct with canonical injections
    i k:∐ l=1 nA l→A k i_k : \coprod_{l=1}^n A_l \to A_k
    ,
    k=1,…,n k=1,\ldots , n
    , then it is also a product with canonical projections given by the uniquely determined morphisms
    p k:∐ l=1 nA l↦A k p_k : \coprod_{l=1}^n A_l \mapsto A_k
    such that
    p k∘i l={id A k ifk=l, 0 else. p_k \circ i_l = \begin{cases} \id_{A_k} & \text{if}\ k=l, \\ 0 & \text{else}. \end{cases}
    In addition, the equality
    ∑ l=1 ni l∘p l=id ∐ l=1 nA l \sum_{l=1}^n i_l \circ p_l = \id_{\coprod_{l=1}^n A_l} (1.2)
    holds true.
    1. test 1
    2. test 2
Proof: Let us first show . So assume that
∏ l=1 nA l \prod_{l=1}^n A_l
is a product with canonical projections
p k p_k
, and define the
i k i_k
as in . Then we have, for
k=1,…,n k=1,\ldots, n
,
p k∘(∑ l=1 ni l∘p l)=∑ l=1 np k∘i l∘p l=p k. p_k \circ \Big( \sum_{l=1}^n i_l \circ p_l \Big) = \sum_{l=1}^n p_k \circ i_l \circ p_l = p_k .
By the universal property of the product, follows. Now let
f k:A k→X f_k : A_k \to X
,
k=1,…,n k=1,\ldots ,n
, be a family of morphisms in
categoryA \category{A}
. Define
f:∏ l=1 nAl→X f: \prod_{l=1}^nA_l \to X
by
f=∑ l=1 nf l∘p l f = \sum_{l=1}^n f_l \circ p_l
and compute
f∘i k=(∑ l=1 nf l∘p l)∘i k=∑ l=1 nf l∘p l∘i k=f k. f \circ i_k = \Big( \sum_{l=1}^n f_l \circ p_l \Big) \circ i_k = \sum_{l=1}^n f_l \circ p_l \circ i_k = f_k .
If
f˜:∏ l=1 nA l→X \tilde{f}: {\prod_{l=1}^n} A_l \to X
is another morphism satisfying
f˜∘i k=f k \tilde{f} \circ i_k = f_k
for all
i i
, then
f−f˜=(f−f˜)∘(∑ l=1 ni l∘p l)=∑ l=1 n(f−f˜)∘i l∘p l=∑ l=1 n(f l−f l)∘p l=0. f - \tilde{f} \, = \big( f - \tilde{f} \big) \circ \big( \sum_{l=1}^n i_l \circ p_l \big) = \sum_{l=1}^n \big( f - \tilde{f} \big) \circ i_l \circ p_l = \sum_{l=1}^n \big( f_l - f_l \big) \circ p_l = 0 .
But this entails that
∏ l=1 nA l {\prod_{l=1}^n} A_l
together with the morphisms
i k i_k
fulfills the universal property of a coproduct of the family
(A l) l=1 n (A_l)_{l=1}^n
. One shows by an analogous but dual argument.
Since by the proposition the product and the coproduct of finitely many objects
A k A_k
,
k=1,…,n k=1,\ldots,n
in a pre-additive category
categoryA \category{A}
coincide (up to canonical isomorphism), one denotes them by the same symbol, namely by
⨁ k=1 nA k, \bigoplus_{k=1}^n A_k ,
and calls the resulting object the direct sum of the
A k A_k
. The proposition tells also that an initial or terminal object in
categoryA \category{A}
has to be a zero object which we then denote by
0 categoryA 0_\category{A}
or
0 0
if no confusion can arise.
1.2. Definition A pre-additive category
categoryA \category{A}
is called additive, if it has the following properties:
  1. categoryA \category{A}
    has a zero object.
  2. Every finite family of objects has a product.
  1. Every finite family of objects has a coproduct.
1.1. Example The category
categoryAb \category{Ab}
of abelian groups carries in a natural way the structure of an additive category. Likewise, if
R R
is a (unital) ring, the category
R-categoryMod R\text{-}\category{Mod}
of
R R
-left modules is additive.