Correction to: The Karoubi envelope and weak idempotent completion of an extriangulated category

A Correction to this paper has been published: 10.1007/s10485-021-09664-8


Introduction
In the original article [2], we showed that given an extriangulated category (C, E, s), its idempotent completionC is also an extriangulated category (C, F, r). An important technical result is [2, Proposition 3.10], which states that the correspondence r is well-defined. The proof of the proposition given in the original article was incorrect. We will give a correct proof of this statement in this corrigendum. The statement of the proposition is as follows.
Proposition 3.10 Let δ be an extension in F((C, p), (A, q)) realised under s by the following sequences, Then given idempotents r : B → B and w : Y → Y such that aq = ra, pb = br and xq = wx, py = yw (3) the sequences are equivalent. That is to say, r is well-defined.
To prove the equivalence of the sequences (4) and (5) inC, the strategy used in [2] was to prove that the morphism w f r : (B, r ) → (Y , w) is an isomorphism, where f : B → Y is an isomorphism in C which gives the equivalence of the sequences (1) and (2) in C. We claimed The original article can be found online at https://doi.org/10.1007/s10485-021-09664-8.
B Dixy Msapato mmdmm@leeds.ac.uk this could be done by first showing that w f r = w f , using the fact that w f raq = w f aq and hence (w f r − w f )aq = 0, then employing the fact that py is a weak cokernel of aq inC to further deduce that w f r = w f . However, it is not clear if w f is a morphism inC, so we cannot take advantage of the fact that py is a weak cokernel of aq inC in this way.

Corrigendum
Recall for an extriangulated category (C, E, s), we defined the biadditive functor F :C op × C → Ab on the idempotent completion as follows. Given a pair of objects (X , p) and (Y , q) inC, we define F on objects by setting, )). Before we can give the proof of Proposition 3.10, we first need to collect some lemmas which will be needed.
be a pair of complexes in C. Suppose that the following sequences of functors are exact, and likewise for Y • . Then for any commutative diagram f the following statements are equivalent.

Lemma 2.2 [1, Proposition 2.21] Let δ ∈ E(C, A) be an extension, and let X
−→ C be a pair of complexes in C. Suppose that the following sequences of functors are exact, Then f • is a homotopic equivalence.
We also need to strengthen [2, Lemma 3.9] as follows.

Lemma 2.3
Let δ = p * q * ε be an extension in F((Z , p), (X , q)) such that Then the following sequences of functors are exact;

is an idempotent morphism such that r x = xq and yr = py, obtained by an application of [2, Lemma 3.5].
Proof We will only show the exactness of the first sequence. The proof of the exactness of the second sequence is dual. Exactness atC((Y , r ), −) is as in [2,Lemma 3.9]. So what is left is to prove exactness atC((X , q), −). Since (C, E, r) is an extriangulated category, the following sequence is exact. Let (A, e) be an arbitrary object inC. Take any morphism f : (Y , r ) → (A, e) ∈ C((Y , r ), (A, e)). Then by the exactness of (6). We conclude that im(C(xq, (A, e)) ⊆ ker(δ # (A,e) ). Now take any morphism g : (X , q) → (A, e) ∈C ((X , q), (A, e)). Recall that this means g is a morphism g : X → A in C such that gq = eg = g. Suppose δ # (A,e) (g) = g * δ = 0. Since g is also a morphism in C and δ is an E-extension, we have by the exactness of (6) that there exists h : Y → A such that g = hx. Now consider the morphism h = ehr : (Y , r ) → (A, e). We have that h xq = (ehr)xq = eh(r x)q = eh(xq)q = e(hx)q = e(g)q = g.
We are now able to give a proof of Proposition 3.10.

Proof of Proposition 3.10
Proof Since the sequences A a −→ B b −→ C, and A x −→ Y y −→ C both realise δ, they are by definition equivalent in C. That is to say we have the following commutative diagram,