The Karoubi envelope and weak idempotent completion of an extriangulated category

We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.

Independent work by [WWZZ20] has also shown that the idempotent completion of an extriangulated category is extriangulated. Although the result is the same, our work offers a different perspective. For example, the Ext 1 functor of the idempotent completion in our paper has a different description to that of [WWZZ20]. Our description of the biadditive functor has the advantage of allowing us to easily observe that the Ext 1 -groups of an idempotent completioñ A (and weak idempotent completionÂ) are subgroups of the Ext 1 -groups of A. In particular, the Ext 1 bifunctor onÃ ( andÂ) behaves like a subbifunctor of the Ext 1 bifunctor on A, in the sense of [HLN17, Definition 3.6]. Our alternative perspective also leads us to a proof of the main theorem which is quite different to the proof presented in [BS01] for the triangulated case and [WWZZ20] for extriangulated case. In our work, the role of the idempotent morphisms is clarified and the extriangles of the idempotent completion (and weak idempotent completion) have an explicit description which isn't available in the treatment of [BS01] and [WWZZ20]. As a consequence, with some additional work, we can prove that the weak idempotent completion is also extriangulated as a corollary to the fact that the idempotent completion is extriangulated. We do so by showing that the weak idempotent completion is an extension-closed subcategory of the idempotent completion; to the best of our knowledge, this additional result been shown in the triangulated or exact or extriangulated case. This paper is organised as follows: in §2, we recall the necessary theory of idempotent completions, weak idempotent completions and the theory of extriangulated categories. Finally, in §3, we show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.
In this section, we recall the basic theory of idempotent completions of additive categories, weak idempotent completions of additive categories and the theory extriangulated categories as introduced in [NP19].
Let us set the common notation for this section. Let A be an additive category. Given objects X, Y in A we will write A(X, Y ) for the group of morphisms X → Y . For an object X in A we denote the identity morphism of X by 1 X .

Idempotent completeness and weakly idempotent completeness.
Definition 2.1. [Kar68, Definition 1.21,1.2.2]. Let A be an additive category. We say that A is idempotent complete if for every idempotent morphism p : A → A ( i.e. p 2 = p) in A, there is a decomposition A ∼ = K ⊕ I of A such that p ∼ = 0 0 0 1 I with respect to this decomposition.
Proposition 2.2. [Büh10,Remark 6.2]. An additive category A is idempotent complete if and only if every idempotent morphism admits a kernel.
Every additive category A embeds fully faithfully into an idempotent complete categoryÃ. The categoryÃ is commonly referred to as the idempotent completion or as the Karoubi envelope of A.
Definition 2.3. [BS01, 1.2 Definition]. Let A be an additive category. The idempotent completion of A is denoted byÃ and is defined as follows. The objects ofÃ are the pairs (A, p) where A is an object of A and p : A → A is an idempotent morphism. A morphism inÃ from (A, p) to (B, q) is a morphism σ : A → B ∈ A such that σp = qσ = σ. For any object (A, p) inÃ, the identity morphism 1 (A,p) = p.
Proposition 2.4. [Büh10, See e.g. Remark 6.3]. Let A be an additive category. The Karoubi envelopeÃ is an idempotent complete category. The biproduct inÃ is defined as (A, p)⊕(B, q) = (A ⊕ B, p ⊕ q). There is a fully faithful additive functor i A : A →Ã defined as follows. For an object A in A, we have that i A (A) = (A, 1 A ) and for a morphism f in A, we have that i A (f ) = f . The Karoubi envelope is unique with respect to the following universal property.
Proposition 2.5. [Büh10, Proposition 6.10]. Let A be an additive category and let B be an idempotent complete category. For every additive functor F : A → B, there exists a functor F :Ã → B and a natural isomorphism α : F ⇒F i A .
We now introduce the related notion of a weakly idempotent complete category. To do, we must first recall the following definition which stems from work by Thomason on exact categories with weakly split idempotents, see [TT90, A.5.1.].
Definition 2.6. [Büh10,§7]. Let A be an arbitrary category. A morphism r : B → C is called a retraction if there exists a morphism q : C → B such that rq = 1 C . A morphism s : A → B is called a section if there exists a morphism t : B → A such that ts = 1 A .
If r : B → C is a retraction with a section s : C → B then the composition sr is an idempotent morphism. This idempotent gives a decomposition of B in the sense of Definition 2.1 if the morphism r admits a kernel. See [Büh10,Remark 7.4] for more details.
Proposition 2.7. [Büh10, Lemma 7.1]. Let A be an additive category. Then the following statements are equivalent: 1. Every section has a cokernel.

Every retraction has a kernel.
Definition 2.8. [Büh10,Definition 7.2]. Let A be an additive category. Then A is said to be weakly idempotent complete if every retraction has a kernel. Equivalently, A is weakly idempotent complete if every section has a cokernel.
Every small additive category A embeds fully faithfully into a weakly idempotent complete categoryÂ. We call the categoryÂ a weak idempotent completion of A. The construction of the weak idempotent completionÂ is similar to that of the idempotent completionÃ.
Definition 2.9. [Sel08, Definition 3.1]. Let A be any category and A an object in C. An idempotent morphism e : A → A is said to split if it admits a retraction r : A → X and a section s : X → A such that s • r = e and r • s = 1 X .
Definition 2.10. [Nee90,1.12]. Let A be a small additive category. The weak idempotent completion of A is denoted byÂ and is defined as follows. The objects ofÂ are the pairs (A, p) where A is an object of A and p : A → A is an idempotent morphism that splits. A morphism inÂ from (A, p) to (B, q) is a morphism σ : A → B ∈ A such that σp = qσ = σ. For any object (A, p) inÂ, the identity morphism 1 (A,p) = p.
Proposition 2.11. [Büh10, E.g Remark 7.8]. Let A be a small additive category. The weak idempotent completionÂ is weakly idempotent complete. The biproduct inÂ is defined as (A, p) ⊕ (B, q) = (A ⊕ B, p ⊕ q). There is a fully faithful additive functor j A : A →Â defined as follows. For an object A in A, we have that j A (A) = (A, 1 A ) and for a morphism f in A, we have that j A (f ) = f . The reader is directed to §6 and §7 of [Büh10] for a more extensive exposition of idempotent completeness and weakly idempotent completeness.

Extriangulated categories.
In this section, we will recall mostly from [NP19] the basic theory of extriangulated categories needed for this paper. Through out this subsection, C will be an additive category equipped with a biadditive functor E : C op × C → Ab, where Ab is the category of abelian groups.
Definition 2.13. [NP19, Definition 2.1]. Let A, C be objects of C. An element δ ∈ E(C, A) is called an E-extension. Formally, an E-extension is a triple (A, δ, C).
Since E is a bifunctor, for any a ∈ C(A, A ′ ) and c ∈ C(C ′ , C), we have the following Eextensions: We will abuse notation by writing E(c, −) instead of E(c op , −).
2. Any morphism c ∈ C(C ′ , C) induces a morphism of E-extensions, Definition 2.16. [NP19, Definition 2.5]. For any objects A, C in C, the zero element 0 ∈ E(C, A) is called a split E-extension.
Definition 2.17. [NP19, Definition 2.6]. Let δ ∈ E(C, A) and δ ′ ∈ E(C ′ , A ′ ) be any pair of E-extensions. Let i C : C → C ⊕ C ′ and i C ′ : C ′ → C ⊕ C ′ be the canonical inclusion maps. Let p A : A ⊕ A ′ → A, and p A ′ : A ⊕ A ′ → A ′ be the canonical projection maps. By the biadditivity of E we have the following isomorphism.
be the element corresponding to (δ, 0, 0, δ ′ ) via the above isomorphism. If A = A ′ and C = C ′ , then the sum δ + δ ′ ∈ E(C, A) is obtained by Definition 2.18. [NP19, Definition 2.7]. Let A, C be a pair of objects in C. Two sequences of

For any two equivalence classes [
Definition 2.20. [NP19, Definition 2.9]. Let s be a correspondence associating an equivalence class Then for any morphism (a, c) : δ → δ ′ of E-extensions, there exists b ∈ C(B, B ′ ) such that the following diagram commutes.
In this situation, we say that the triple of morphisms (a, b, c) realises (a, c). For δ ∈ E(C, A), we say that the sequence A 2. For any pair of extensions δ and δ ′ , we have that, We are now in a position to define an extriangulated category.
Definition 2.22. [NP19, Definition 2.12]. Let C be an additive category. An extriangulated category is a triple (C, E, s) satisfying the following axioms.
(ET1) The functor E : C op × C → Ab is a biadditive functor.
(ET2) The correspondence s is an additive realisation of E.
(ET4) Let δ ∈ E(D, A) and δ ′ ∈ E(F, B) be any pair of E-extensions, realised by the −→ E, which satisfy the following compatibilities: op The dual of (ET4).
In this case, we call s an E-triangulation of C.
There are many examples of extriangulated categories. They include, exact categories, and triangulated categories and extension-closed subcategories of triangulated subcategories. There are also extriangulated categories which are neither exact nor triangulated; for example see, Here F op is the opposite functor C op → C ′ op given by F op (A) = F (A) and F op (f op ) = (F (f )) op . Furthermore, we say that F is an extriangulated equivalence if F is an equivalence of categories.
We will conclude this section by introducing some useful terminology from [NP19] and stating results about extriangulated categories which will be helpful for the rest of the paper.

A sequence
The terminology of conflations, inflations and deflations is also used in the context of exact categories and triangulated categories analogously.

If a conflation
y −→ C, δ) an E-triangle or extriangle and denote it by the following diagram.
be any pair of Etriangles. If a triple (a, b, c) realises (a, c) : δ → δ ′ we write it as in the following commutative diagram and call (a, b, c) a morphism of E-triangles.  be any E-triangle in C. If f ∈ C(A, X) and h ∈ C(C, Z) are isomorphisms, then is again an E-triangle. be any E-triangle in C. Suppose we have the following commutative diagram, where the morphisms f, g, h are isomorphisms. Then it follows that is an E-triangle.
Proof. By Proposition 2.27, is an E-triangle. Now consider the following diagram.
Observe that it commutes, therefore it is an equivalence, which implies that The following two propositions are special cases of propositions from [HLN17] which are stated for general n-exangulated categories, but here we are restating them in the case of extriangulated categories which are in fact the same as 1-exangulated categories by [ Proposition 2.31. [NP19, Corollary 3.12]. Let (C, E, s) be an extriangulated category. For any x y δ the following sequences of natural transformations are exact.
The natural transformations δ # and δ # are defined as follows. Given any object X in C, we have that The exactness of the first sequence of natural transformations means that for any object X in C, the sequence is exact in Ab and likewise for the second sequence.

For any E-triangle A B C
x y δ , the following sequences of natural transformations are exact.
x y δ the following statements hold: be an E-triangle realising f * δ. Then there is a morphism g such that the following diagram commutes is an E-triangle. Dually, let be an E-triangle, let h : E → C be any morphism and let be an E-triangle realising h * δ. Then there is a morphism g : D → B such that the following diagram commutes is an E-triangle.
The following proposition is a special case of [HLN17, Proposition 3.5] which applies to general n-exangulated categories. But here we are restating the statement just for extriangulated categories, which are the same as 1-exangulated categories. The statement of [HLN17, Proposition 3.5] is a consequence of (ET2) and Proposition 2.34, or axioms (R0) and (EA2) respectively in the language of [HLN17]. be E-triangles. Suppose we have the following solid commutative diagram.
Then there exists a morphism w : C → Z such wb = yu, w * δ = ε and that the following is an E-triangle, 3 Idempotent completion of extriangulated categories.
For the rest of this section, let (C, E, s) be an extriangulated category and letC andĈ be the idempotent completion of C and the weak idempotent completion of C respectively. Note that in order to considerĈ, we have to further assume that C is a small category.

Idempotent completion.
Theorem 3.1. Let (C, E, s) be an extriangulated category. LetC be the idempotent completion of C. ThenC is extriangulated. Moreover, in this case the embedding i C : C →C is an extriangulated functor.
Our first step in proving the above theorem is the construction of a bifunctor F :C op ×C → Ab for the extriangulated structure. Given a pair of objects (X, p) and (Y, q) inC, we define F on objects by setting, is an abelian group. We now need to define F on morphisms. Letα : (X, p) → (Y, q) andβ : (U, e) → (V, f ) be any pair of morphisms inC. By definition these are morphisms α : For the pair (α,β) we define F(α op ,β) : F((Y, q), (U, e)) → F((X, p), (V, f )) as follows. For ε ∈ F((Y, q), (U, e)) we set F(α op ,β)(ε) := β * α * ε. It is easy to observe that F preserves identity morphisms from the above definition. Let (α 1 ,β 1 ) and (α 2 ,β 2 ) be a pair of composable morphisms inC op ×C and (α 1α2 ,β 1β2 ) be their composition. Then, so F preserves composition. This completes the definition of the bifunctor F :C op ×C → Ab. Our next step will be to verify that F :C op ×C → Ab is a biadditive functor. We will only show that F is additive in the second argument since the proof for additivity in the first argument is dual.
Now let (U, e) and (V, f ) be any pair of objects inC. Denote by F U⊕V X the abelian group Since E is a biadditive functor, there is a group isomorphism ϕ : Hence G is a group homomorphism.
Define the map H : Hence H is a group homomorphism.
Proof. The proof is dual to the proof of the previous proposition.
Having verified that the functor F is biadditive, the next thing we need to do is to define a correspondence which will be a realisation. In order to define the correspondence, we need the following lemma. This lemma is a generalisation of [BS01, Lemma 1.13] in the setting of extriangulated categories. The proof is also a straightforward adaptation. Proof.
Since (e, f ) : δ → δ is a morphism of G-extensions and t is a realisation, there exists a morphism i : B → B such that the following diagram commutes.
Let h := i 2 − i. Then we have that ha = (i 2 − i)a = 0 and bh = b(i 2 − i) = 0 from the commutativity of the above diagram and the fact that e and f are idempotent. By the exact sequences in Proposition 2.31, b is a weak cokernel of a so there existsh : C → B such that h =hb. So we observe that h 2 =hbh = 0. Let g = i + h − 2ih. Since the morphisms i and h commute and h 2 = 0 we have that g 2 = i 2 + 2ih − 4i 2 h. Since i 2 = i + h we have that g 2 = i + h + 2ih − 4ih = g. We have that ga = ia + ha − 2iha = ia = ae, since ha = 0. We likewise have that bg = bi + bh − 2bih = bi + bh − 2bhi = bi = f b. Therefore, the above diagram commutes if we replace i with g. This completes the proof. Set h := f 2 − f . Then we have that We also have that By Proposition 2.31 we have the following exact sequence in Ab.
Since h * δ = (δ # ) C (h) = 0, it follows from the exactness of the above sequence that there exists a morphismh : C → B such that h = bh. From this we can observe that h 2 = bhbh = hbh = 0. Now set g := f + h − 2f h, as f and h commute and h 2 = 0, we then have that By noting that f 2 = f + h we obtain It is then easy to check that, and We have shown that g : C → C is an idempotent morphism such that (e, g) : δ → δ is a morphism of G-extensions realised by (e, i, g). The proof of the other statement is dual.
Definition 3.7. Let r be the correspondence between F-extensions and equivalence classes of sequences of morphisms inC defined as follows. For any objects Z, X in C and idempotent morphisms p : Z → Z, q : X → X in C, let δ = p * q * ε be an extension in F((Z, p), (X, q)) such that We set where r : Y → Y is an idempotent morphism such that rx = xq and yr = py obtained by application of Lemma 3.5.
Remark 3.8. Before we can proceed any further, we need to show that the above definition of r is well-defined in the following sense. Given an F-extension δ, r(δ) is defined in terms of a choice of the representative s(δ). We will show that it is independent of this choice. Moreover, in the above definition, the idempotent morphism r : Y → Y such that rx = xq and yr = py need not be unique. We will show that all choices of such an idempotent give equivalent sequences.
Lemma 3.9. Let δ = p * q * ε be an extension in F((Z, p), (X, q)) such that For any object (A, e) ∈C, the following sequence in Ab is exact. In particular, in the sequence xq py the morphism py is a weak cokernel of xq and the morphism xq is a weak kernel of py.
Recall that r is an idempotent such that xq = rx and py = yr. Given g ∈C((Y, r), (A, e)) such that g • xq = 0, we have that gxq = grx = 0. By the exactness of the sequence in Proposition 2.31, y is a weak cokernel of x, so we have that there exists a morphism h : Z → A such that gr = hy. Since gr = hy we have that gr = gr 2 = hyr = hpy.

Proof. Since the sequences
y −→ C both realise δ, they are by definition equivalent in C. That is to say we have the following commutative diagram, where the morphism f : B → Y is an isomorphism. Now consider the following diagram.
That is to say, diagram (3) commutes. From (4) we can see that wf raq = wf aq therefore (wf r − wf )aq = 0. By Lemma 3.9, pb is a weak cokernel of aq, so there exists a morphism h : (C, p) → (Y, w) such that wf r − wf = hpb. Since r is idempotent and satisfies the relations in (1), we have that 0 = wf r − wf r = wf r 2 − wf r = hpbr = hpb, in particular wf r − wf = hpb = 0.
Hence wf r = wf . From (5) we have that py(wf r) = py(f r) therefore py(wf r − f r) = 0. By Lemma 3.9, we have that xq is a weak kernel of py and there exists a morphism g : (B, r) → (A, q) such that wf r − f r = xqg. Since w is idempotent and satisfies the relations in (1), we have that Hence wf r = f r. Now consider the morphism rf −1 w : (Y, w) → (B, r) inC. Since wf r = wf and wf r = f r, we can observe that This is to say the morphisms rf −1 w : (Y, w) → (B, r) and wf r : (B, w) → (Y, w) are mutual inverses inC. So diagram (3) does indeed give an equivalence of sequences in C. This completes the proof.
From Proposition 3.10 we conclude that r is well-defined in the sense of Remark 3.8.
where f is an isomorphism. Since the above diagram commutes f xq = u and vf = py, so set u 1 = f x and v 1 = yf −1 .
Proposition 3.12. Let r be the correspondence defined above. Then r is an additive realisation of F.
By definition we have that δ = p * q * ε and δ ′ = p ′ * q ′ * ε ′ for some ε ∈ E(Z, X) and some ε ′ ∈ E(Z ′ , X ′ ). Moreover the morphism a ∈ C(X, X ′ ) is such that aq = q ′ a = a, likewise for c ∈ C(Z, Z ′ ) we have that cp = p ′ c = c. We also have by definition that Therefore we have a morphism of E-extensions (a, c) : p * q * ε → p ′ * q ′ * ε ′ . In other words we have the following solid diagram in C.
Since s is a realisation, there exists a morphism b : Y → Y ′ making the above diagram commute. Recall that by Lemma 3.5, we have that rx = xq, yr = py, r ′ x ′ = x ′ q ′ and y ′ r ′ = p ′ y ′ . It then follows that r ′ br : (Y, r) → (Y ′ , r ′ ) makes diagram (6) commute since, So we conclude that r is a realisation of F. Now we verify additivity of r. For any pair (Z, p), (X, q), we have that 0 = p * q * 0 and By definition we have that since q = 1 (X,q) and p = 1 (Z,p) , we have that, Now take a pair of F-extensions δ = p * q * ε ∈ F((Z, p), (X, q)) and δ ′ = p ′ * q ′ * ε ′ ∈ F((Z ′ , p ′ ), (X ′ , q ′ )). Since s is an additive realisation we have that By the definition of r we have that, .
We have that It is also easy to check that r ⊕ r ′ is idempotent and satisfies the required equations arising from Lemma 3.5. So it follows that This completes the proof. Proof. Let δ = p * q * ε ∈ F((Z, p), (X, q)) and whereby rx = xq and py = yr by Lemma 3.5 and whereby r ′ x ′ = x ′ q ′ and p ′ y ′ = y ′ r ′ by Lemma 3.5. Suppose we have the following commutative diagram inC. Note that we have that q ′ a = aq = a and r ′ b = br = b by the definition of morphisms inC.
We then have the following diagram in C.
We also have that, therefore we have a morphism of F-extensions (a, p ′ cp) : δ → δ ′ , as required. This verifies (ET3). The proof for (ET3) op is dual.
Before we can prove thatC satisfies (ET4) and (ET4) op . We first need to prove the upcoming statements, which will play an important part in our proof of (ET4) and (ET4) op .  Proof. Let (A, e) be any object inC. We need to show that the following sequence in Ab is exact.
The proof of the dual statement is dual. This completes the proof.
Remark 3.15. SinceC satisfies (ET3) and (ET3) op we have thatC satisfies Proposition 2.32. By Lemma 3.14, we see thatC induces long exact sequences as in Proposition 2.31 without requiring thatC is an extriangulated category as a priori.
Then, for any object (A, e) ∈C we have the following commutative diagram, where by Lemma 3.14, the top row is exact. From the above commutative diagram, it is easy to see that the bottom row is also exact.
The following proposition is an analogue of Proposition 2.34 inC. Remarkably, we are able to prove the statement of the following proposition without requiringC to be extriangulated unlike in the statement of Proposition 2.34. We only require that (C, F, r) satisfies axioms (ET1) and (ET2).
Proposition 3.17. Let δ = p * q * ε ∈ F((C, p), (A, q)) be an F-extension where Let h : (E, w) −→ (C, p) be any morphism and suppose Then there exists a morphism g : (D, s) −→ (B, r) such that (1 (A,q) , h) : h * δ → δ is realised by (1 (A,q) , g, h). Moreover Proof. We apply Proposition 2.34 to A B C x y δ , the morphism h : Then there is a morphismḡ : D → B such that the following diagram commutes and that Since h : (E, w) −→ (C, p) is a morphism inC we have that h = hw, therefore In other words h * δ ∈ F((E, w), (A, q)), so we have that where s : D → D is an idempotent morphism such that dq = sd and we = es. Consider the following diagram. Therefore diagram (10) commutes and (1 (A,q) , h) is realised by (1 (A,q) , g, h).
Proof. Let (D, p), (A, q), (F, t) and (B, r) be objects inC and let δ ∈ F((D, p), (A, q)) and δ ′ ∈ F((F, t), (B, r)) be F-extensions with in the extriangulated category (C, E, s). Then by definition and for some idempotent s : C → C where gr = sg and tg ′ = g ′ s.
Since (C, E, s) is extriangulated we can apply (ET4) to the above E-triangles to get an object E in C, a commutative diagram in C and an E-extension δ ′′ ∈ E(E, A) where such that the following compatibilities are satisfied: (ii) d * δ ′′ = δ.
Recall that since δ ′ ∈ F((F, t), (B, r)), then by definition δ ′ = t * r * ε ′ for some ε ′ ∈ E(F, B). Also recall that pf ′ = f ′ r by (11), so we have that In other words f ′ * δ ′ ∈ F((F, t), (D, p)) and so we have by definition that where v : E → E is an idempotent such that dp = vd and te = ev.
Note that the vertical inclusion maps are due to the fact that the F-extension groups are subgroups of the respective E-extension groups and the diagram commutes. By Lemma 3.14, the top row is exact in Ab. Moreover, the sequence obtained by appending the morphism δ # :C((E, v), (D, p)) → F((E, v), (A, q)) to the top row is exact by Proposition 2.32 and Lemma 3.14.
Consider the solid part of the following diagram.
Having shown that (C, E, r) satisfies (ET4) and (ET4) op , we can now conclude that (C, E, r) is an extriangulated category. Recall that there is a fully faithful additive functor i C : C →C defined as follows. For an object A of C, we have that i C (A) = (A, 1 A ) and for a morphism f in C, we have that i C (f ) = f . We will show that this functor is an extriangulated functor in the sense of [BTS20, Definition 2.31]. In particular, the functor i C preserves the extriangulated structure of C.
Proposition 3.20. Let C be an extriangulated category andC be its idempotent completion. Then the functor i C : C →C is an extriangulated functor.
Proof. It is easy to see that the functor i C is a covariant additive functor. So all that is left is to define a natural transformation First note that by definition, F((C, 1), (A, 1)) = E(C, A). So given a pair of objects A, C in C we define Γ (C,A) : E(C, A) → F((C, 1), (A, 1)) by setting Γ (C,A) (δ) = δ for all δ ∈ E(C, A). Given a morphism (f, g) : (C, A) → (Z, X) in C op × C, consider the following diagram: E(X, Z) F((X, 1), (Z, 1)).
Theorem 3.21. Let (C, E, s) be an extriangulated category. LetC be the idempotent completion of C. ThenC is extriangulated. Moreover, the embedding i C : C →C is an extriangulated functor.

Weak idempotent completion.
Definition 3.22. [Nee90,1.12]. Let A be a small additive category. The weak idempotent completion of A is denoted byÂ and is defined as follows. The objects ofÂ are the pairs (A, p) where A is an object of A and p : A → A is an idempotent factoring as p = cr for some retraction r : A → X and some section c : X → A with rc = 1 X (i.e. p is a split idempotent). A morphism inÂ from (A, p) to (B, q) is a morphism σ : A → B ∈ A such that σp = qσ = σ.
There is fully faithful additive functor j A : A →Â from A to its weak idempotent completion defined as follows. For an object A in A, we have that j A (A) = (A, 1 A ), and for a morphism f in C, we have that j A (f ) = f .
The following lemma is key in proving the main theorem for this subsection. It is an analogue of Lemma 3.5 where we replace the idempotent morphisms with split idempotent morphisms. Proof. Since e splits, there is an object X ∈ A and morphisms e 2 : A → X and e 1 : X → A such that e = e 1 e 2 and e 2 e 1 = 1 X . Likewise, for f there is an object Z ∈ A and morphisms f 2 : C → Z and f 1 : Consider the following diagram of G-triangles.
Therefore (e 2 , f 2 ) : δ → e 2 * f * 1 δ is a morphism of G-extensions. So by axiom (ET2), there exists a morphism r 2 : B → Y such that the bottom row of the above diagram commutes. Collapsing the above diagram into the diagram below, we obtain the following morphism of G-triangles and commutative diagram. By Lemma 2.26, the morphism r 2 r 1 is an automorphism of Y . That is to say, there exists h : Y → Y such that r 2 r 1 h = 1 Y and hr 2 r 1 = 1 Y . Set g := (r 1 h)r 2 : B → B. Observe that g 2 = r 1 h(r 2 r 1 h)r 2 = r 1 h(1 Y )r 2 = r 1 hr 2 = g, so g is an idempotent morphism. Moreover, So g is in fact a split idempotent. Now consider the following diagram.
Using the fact that diagram (21) commutes, we further observe that ga = r 1 h(r 2 a) = r 1 (hx)e 2 = (r 1 x)e 2 = a(e 1 e 2 ) = ae, and bg = (br 1 )hr 2 = f 1 (yh)r 2 = (f 1 y)r 2 = (f 1 f 2 )b = f b, so diagram (23) commutes. This completes the proof. Let (C, E, s) be an extriangulated category such that C is small. We have shown that the idempotent completionC is also an extriangulated category. Now consider the weak idempotent completionĈ. From the definition of the idempotent completion and the definition of the weak idempotent completion, it is easy to see that the weak idempotent completionĈ (Definition 2.10) is a full subcategory of the idempotent completionC (Definition 2.4). As we will see in the following proposition, the weak idempotent completionĈ is extension-closed as a subcategory of the idempotent completionC.
Proposition 3.26. Let (C, E, s) be an extriangulated category such that C is small and (C, F, r) be its idempotent completion. Then the weak idempotent completionĈ of C is an extensionclosed subcategory ofC.
Proof. By Definitions 2.4, 2.10, it is easy to see thatĈ is a full additive subcategory ofC. Now consider two objects (A, q) ∈Ĉ and (B, p) ∈C. Suppose there exists an isomorphism f : (A, q) → (B, p) ∈C. We will show that (B, p) is also inĈ. Since (A, q) ∈Ĉ, we have that q is a split idempotent, which is to say there exists an object X ∈ C such that there are morphisms q 2 : A → X, q 1 : X → A satisfying q = q 1 q 2 and q 2 q 1 = 1 X .
Since f : (A, q) → (B, p) ∈C is an isomorphism, there exists a morphism f −1 : (B, p) → (A, q) such that f f −1 = p and f −1 f = q. Moreover f satisfies the relations f q = pf = f since it is a morphism inC. Set p 1 = f q 1 : X → B and p 2 = q 2 f −1 : B → X, then we observe that p = p 1 p 2 and p 2 p 1 = 1 X . Hence, the idempotent p splits, so (B, p) ∈Ĉ. We conclude thatĈ is closed under isomorphisms.
Let (A, q) and (C, p) be objects inĈ and consider a conflation (A, q) of some F-extension δ = p * q * ε in F((C, p), (A, q)). We will show that (M, m) is also inĈ. By  ) is an isomorphism. Moreover (B, w) lies inĈ since w is a split idempotent, so it follows that (M, m) ∈Ĉ sinceĈ is closed under isomorphims. This completes the proof.
Theorem 3.27. Let (C, E, s) be an extriangulated category such that C is small. LetĈ be the weak idempotent completion of C. ThenĈ is extriangulated. Moreover, the embedding j C : C →Ĉ is an extriangulated functor.
Proof. Since the idempotent completionC is extriangulated and the weak idempotent completion C is an extension-closed subcategory ofC by Proposition 3.26,Ĉ is also extriangulated by Lemma 3.25. The other statement follows as an easy consequence of this and Theorem 3.21.