Idempotent Completions of n-Exangulated Categories

Abstract

Suppose $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an n-exangulated category. We show that the idempotent completion and the weak idempotent completion of $\mathcal {C}$ are again n-exangulated categories. Furthermore, we also show that the canonical inclusion functor of $\mathcal {C}$ into its (resp. weak) idempotent completion is n-exangulated and 2-universal among n-exangulated functors from $(\mathcal {C},\mathbb {E},\mathfrak {s})$ to (resp. weakly) idempotent complete n-exangulated categories. Furthermore, we prove that if $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is n-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and $(n+2)$-angulated cases. However, our constructions recover the known structures in the established cases up to n-exangulated isomorphism of n-exangulated categories.

1 Introduction

Idempotent completion began with Karoubi’s work [20] on additive categories. It was shown that an additive category embeds into an associated one which is idempotent complete, that is, in which all idempotent morphisms admit a kernel. Particularly nice examples of idempotent complete categories include Krull-Schmidt categories, which can be characterised as idempotent complete additive categories in which each object has a semi-perfect endomorphism ring (see Chen–Ye–Zhang [12, Thm. A.1], Krause [24, Cor. 4.4]). Other examples include the vast class of pre-abelian categories (see e.g. [33, Rem. 2.2]); e.g. a module category, or the category of Banach spaces (over the reals, say).

Suppose $\mathcal {C}$ is an additive category. The objects of the idempotent completion $\widetilde{\mathcal {C}}$ of $\mathcal {C}$ are pairs (X, e), where X is an object of $\mathcal {C}$ and $e:X \rightarrow X$ is an idempotent morphism in $\mathcal {C}$, i.e. $e^2 = e$. What is particularly nice is that if $\mathcal {C}$ has a certain kind of structure, then in several cases this induces the same structure on $\widetilde{\mathcal {C}}$. For example, Karoubi had already shown that the idempotent completion of an additive category is again additive (see [20, (1.2.2)]). Furthermore, it has been shown for the following, amongst other, extrinsic structures that if $\mathcal {C}$ has such a structure, then so too does $\widetilde{\mathcal {C}}$:

• (i) triangulated (see Balmer–Schlichting [7, Thm. 1.5]);

• (ii) exact (see Bühler [11, Prop. 6.13]);

• (iii) extriangulated (see [27, Thm. 3.1]); and

• (iv) $(n+2)$-angulated, where $n\geqslant 1$ is an integer (see Lin [25, Thm. 3.1]).

See also Liu–Sun [26] and Zhou [35].

Idempotent complete exact and triangulated categories are verifiably important in algebra and algebraic geometry. As a classical example, in Neeman [28] an idempotent complete exact category $\mathcal {E}$ is needed to give a clean description of the kernel of the localisation functor from the homotopy category of $\mathcal {E}$ to its derived category. And, more generally, many equivalences only hold up to direct summands, i.e. up to idempotents (see, for example, Orlov [31, Thm. 2.11], or Kalck–Iyama–Wemyss–Yang [21, Thm. 1.1]). Therefore, it is usually helpful to view an algebraic structure as sitting inside its idempotent completion.

The idempotent completion $\widetilde{\mathcal {C}}$ comes equipped with an inclusion functor given by on objects. Moreover, in several of the cases above it has been shown that this functor is 2-universal in an appropriate sense; see e.g. Proposition 2.8 for a precise formulation. For example, without any assumptions other than additivity, the functor is additive and 2-universal amongst additive functors from $\mathcal {C}$ to idempotent complete additive categories. On the other hand, if e.g. $\mathcal {C}$ has an exact structure, then is exact and 2-universal amongst exact functors from $\mathcal {C}$ to idempotent complete exact categories.

In homological algebra two parallel generalisations have been made from the classical settings of exact and triangulated categories. One of these has been the introduction of extriangulated categories as defined by Nakaoka–Palu [30]. An extriangulated category is a triplet $(\mathcal {C},\mathbb {E},\mathfrak {s})$, where $\mathcal {C}$ is an additive category, is a biadditive functor to the category of abelian groups, and $\mathfrak {s}$ is a so-called additive realisation of $\mathbb {E}$. The realisation $\mathfrak {s}$ associates to each $\delta \in \mathbb {E}(Z,X)$ a certain equivalence class of a 3-term complex. As an example, each triangulated category $(\mathcal {C},\Sigma ,{{\triangle }})$, where $\Sigma$ is a suspension functor and ${{\triangle }}$ is a triangulation, is an extriangulated category. Indeed, one defines the corresponding bifunctor by . See [30, Prop. 3.22] for more details. In addition, each suitable exact category is extriangulated; see [30, Exam. 2.13]. A particular advantage of this theory is that the collection of extriangulated categories is closed under taking extension-closed subcategories. Although an extension-closed subcategory of an exact category is again exact, the same does not hold in general for triangulated categories.

We note here that, importantly, it was shown in [27, Sec. 3.1] that the extriangulated structure on $\widetilde{\mathcal {C}}$ produced from case (iii) above is compatible with the more classical constructions of (i) and (ii). For instance, given a triangulated category $\mathcal {C}$, one can equip its idempotent completion $\widetilde{\mathcal {C}}$ with a triangulation by (i) or with an extriangulation by (iii), but these structures are the same in the sense of [30, Prop. 3.22]. Analogously, (iii) also recovers (ii) if one starts with an extriangulated category that is exact.

Let $n\geqslant 1$ be an integer. The other aforementioned generalisation in homological algebra has been the development of higher homological algebra. This includes the introduction of n-exact and n-abelian categories by Jasso [19], and $(n+2)$-angulated categories by Geiss–Keller–Oppermann [14]. Respectively, these generalise exact, abelian and triangulated categories, in that one recovers the classical notions by setting $n=1$. For instance, an $(n+2)$-angulated category is a triplet satisfying some axioms, where $\Sigma$ is still an automorphism of $\mathcal {C}$, but now consists of a collection of $(n+2)$-angles each of which has $n+3$ terms.

The focal point of this paper is on the idempotent completion of an n-exangulated category. These categories were axiomatised by Herschend–Liu–Nakaoka [16], and simultaneously generalise extriangulated, $(n+2)$-angulated, and suitable n-exact categories (see [16, Sec. 4]). Like an extriangulated category, an n-exangulated category $(\mathcal {C},\mathbb {E},\mathfrak {s})$ consists of an additive category $\mathcal {C}$, a biadditive functor , and a so-called exact realisation $\mathfrak {s}$ of $\mathbb {E}$, which satisfy some axioms (see Sect. 3.1). The realisation $\mathfrak {s}$ now associates to each a certain equivalence class (see Sect. 3.1)

of an $(n+2)$-term complex. In this case, the pair is called an $\mathfrak {s}$-distinguished n-exangle. We recall that structure-preserving functors between n-exangulated categories were defined in [10, Def. 2.32]. They are known as n-exangulated functors and they send distinguished n-exangles to distinguished n-exangles.

Suppose that $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an n-exangulated category. Let $\widetilde{\mathcal {C}}$ denote the idempotent completion of $\mathcal {C}$ as an additive category. We define a biadditive functor as follows. For any pair of objects $(X,e),(Z,e')\in \widetilde{\mathcal {C}}$, we let $\mathbb {F}((Z,e'),(X,e))$ consist of triplets $(e,\delta ,e')$ where $\delta \in \mathbb {E}(Z,X)$ such that $\mathbb {E}(Z,e)(\delta ) = \delta = \mathbb {E}(e',X)(\delta )$. On morphisms $\mathbb {F}$ is essentially a restriction of $\mathbb {E}$; see Definition 4.4 for details. Now we define a realisation $\mathfrak {t}$ of $\mathbb {F}$. For $(e,\delta ,e') \in \mathbb {F}((Z,e'),(X,e))$, we have that for some $(n+2)$-term complex with and since $\mathfrak {s}$ is a realisation of $\mathbb {E}$. We choose an idempotent morphism of complexes, such that and ; see Corollary 4.13. Lastly, we set $\mathfrak {t}((e,\delta ,e'))$ to be the equivalence class of the complex

in $\widetilde{\mathcal {C}}$. We say that an n-exangulated category is idempotent complete if its underlying additive category is (see Definition 4.31).

Theorem A

(Theorem 4.32, Theorem 4.39) The triplet $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is an idempotent complete n-exangulated category. Furthermore, the inclusion functor extends to an n-exangulated functor , which is 2-universal among n-exangulated functors from $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ to idempotent complete n-exangulated categories.

An n-exact category $(\mathcal {C}, \mathcal {X})$ (see [19, Def. 4.2]) induces an n-exangulated category $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ if, for each pair of objects $A,C\in \mathcal {C}$, the collection of n-extensions of C by A forms a set; see [16, Prop. 4.34]. As in [23, Def. 4.6], we say that an n-exangulated category $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ is n-exact if its n-exangulated structure arises in this way. Combining Theorem A with [23, Cor. 4.12], we deduce the following.

Corollary B

(Corollary 4.34) If $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an n-exangulated category that is n-exact, then the idempotent completion $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is n-exact.

We explain in Remark 4.40 how Theorem A unifies the constructions in cases (i)–(iv) above. Furthermore, we comment on some obstacles faced in proving the n-exangulated case in Remark 4.41.

From Theorem A we deduce the following corollary, giving a way to produce Krull-Schmidt n-exangulated categories.

Corollary C

(Corollary 4.33) If each object in $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ has a semi-perfect endomorphism ring, then the idempotent completion $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is a Krull-Schmidt n-exangulated category.

Finally, we note that analogues of Theorem A and Corollary B are shown for the weak idempotent completion in Sect. 5. The importance of being weakly idempotent complete for extriangulated categories was very recently demonstrated in [23, Prop. 2.7]. It turns out that for an extriangulated category, the underlying category being weakly idempotent complete is equivalent to the condition (WIC) defined in [30, Cond. 5.8]. Moreover, (WIC) is a key assumption in many results on extriangulated categories, e.g. [30, §§5–7], [17, §3], Zhao–Zhu–Zhuang [36]. We remark that the analogue of (WIC) for n-exangulated categories is automatic if $n\geqslant 2$, but it is not equivalent to the weak idempotent completeness of the underlying category; see [23, Thm. B] for more details.

2 On the Splitting of Idempotents

In this section we recall some key definitions regarding idempotents and idempotent completions of categories. We focus on the idempotent completion of an additive category in Sect. 2.1 and on the weak idempotent completion in Sect. 2.2. Throughout this section, we let $\mathcal {A}$ denote an additive category. For a more in-depth treatment, we refer the reader to [11, Secs. 6–7].

2.1 Idempotent Completion

Recall that by an idempotent (in $\mathcal {A}$) we mean a morphism $e:X\rightarrow X$ satisfying $e^2 = e$ for some object $X\in \mathcal {A}$.

The following definition is from Borceux [6].

Definition 2.1

[6, Defs. 6.5.1, 6.5.3] An idempotent $e :X \rightarrow X$ in $\mathcal {A}$ is said to split if there exist morphisms $r :X \rightarrow Y$ and $s :Y \rightarrow X$, such that $sr=e$ and . The category $\mathcal {A}$ is idempotent complete, or has split idempotents, if every idempotent in $\mathcal {A}$ splits.

If $\mathcal {A}$ has split idempotents and $e:X \rightarrow X$ is an idempotent in $\mathcal {A}$, then the object X admits a direct sum decomposition (see e.g. Auslander [3, p. 188]). In particular, the idempotent e and its counterpart each admit a kernel. Idempotent complete additive categories can be characterised by such a criterion and its dual.

Proposition 2.2

[6, Prop. 6.5.4] An additive category is idempotent complete if and only if every idempotent admits a kernel, if and only if every idempotent admits a cokernel.

From this point of view, idempotent complete categories sit between additive categories and pre-abelian categories, the latter being additive categories in which every morphism admits a kernel and a cokernel; see for example Bucur–Deleanu [4, §5.4].

Every additive category can be viewed as a full subcategory of an idempotent complete one. This goes back to Karoubi [20, Sec. 1.2], so the idempotent completion of $\mathcal {A}$ is also often referred to as the Karoubi envelope of $\mathcal {A}$.

Definition 2.3

The idempotent completion $\widetilde{\mathcal {A}}$ of $\mathcal {A}$ is the category defined as follows. Objects of $\widetilde{\mathcal {A}}$ are pairs (X, e), where X is an object of $\mathcal {A}$ and is idempotent. For objects $(X,e), (Y,e') \in \hbox {obj}\widetilde{\mathcal {A}}$, a morphism from (X, e) to $(Y,e')$ is a triplet $(e',r,e)$, where $r \in \mathcal {A}(X,Y)$ satisfies

$$\begin{aligned} re = r = e'r \end{aligned}$$

in $\mathcal {A}$. Composition of morphisms is defined by

$$\begin{aligned}(e'',s,e')\circ (e',r,e) :=(e'', sr, e),\end{aligned}$$

whenever $(e',r,e)\in \widetilde{\mathcal {A}}((X,e), (Y, e'))$ and $(e'',s,e')\in \widetilde{\mathcal {A}}((Y,e'), (Z, e''))$. The identity of an object $(X,e)\in \hbox {obj}\widetilde{\mathcal {A}}$ will be denoted and is the morphism (e, e, e).

A morphism $(e',r,e) :(X,e) \rightarrow (Y,e')$ in the idempotent completion $\widetilde{\mathcal {A}}$ of $\mathcal {A}$ is usually denoted more simply as r; see e.g. [7, Def. 1.2] and [11, Rem. 6.3]. However, for precision in Sects. 4–5, we use triplets for morphisms in $\widetilde{\mathcal {A}}$ so that we can easily distinguish morphisms in $\mathcal {A}$ from morphisms in its idempotent completion. Our choice of notation also has the added benefit of keeping track of the (co)domain of a morphism in $\widetilde{\mathcal {A}}$. This becomes important later when different morphisms in $\widetilde{\mathcal {A}}$ have the same underlying morphism; see Notation 4.37.

By a functor we always mean a covariant functor. The inclusion functor is defined as follows. An object $X \in \hbox {obj}\mathcal {A}$ is sent to and a morphism $r \in \mathcal {A}(X,Y)$ is mapped to .

Lemma 2.4

If is a split idempotent, with a splitting $e=sr$ where $r :X \rightarrow Y$ and $s :Y \rightarrow X$, then .

Proof

We have and . Hence, there are morphisms and in $\widetilde{\mathcal {A}}$ with and . Hence, $\tilde{r}$ and $\tilde{s}$ are mutually inverse isomorphisms in $\widetilde{\mathcal {A}}$. $\square$

If $\mathcal {A}$ is an idempotent complete category, then the functor is an equivalence of categories; see e.g. [11, Rem. 6.5]. But more generally we have the following.

Proposition 2.5

[11, Rem. 6.3] The idempotent completion $\widetilde{\mathcal {A}}$ is an idempotent complete additive category with biproduct given by $(X, e) \oplus (Y, e') = (X \oplus Y, e \oplus e')$. The inclusion functor is fully faithful and additive.

Remark 2.6

Let (X, e) be an arbitrary object of $\widetilde{\mathcal {A}}$. Then (X, e) is a direct summand of . Indeed, there is an isomorphism . The canonical inclusion of (X, e) into is given by the morphism , and the projection of onto (X, e) by . Similarly for .

The functor is 2-universal in some sense; see Proposition 2.8. For this we recall the notion of whiskering a natural transformation by a functor. We will use Hebrew letters (e.g. (beth), (tsadi), (daleth), (mem)) for natural transformations. Suppose $\mathcal {B},\mathcal {C},\mathcal {D}$ are categories and that we have a diagram

where $\mathscr {F},\mathscr {G},\mathscr {H}$ are functors and is a natural transformation.

Definition 2.7

The whiskering of $\mathscr {F}$ and is the natural transformation defined by for each $X\in \mathcal {B}$.

The next proposition explains the 2-universal property satisfied by .

Proposition 2.8

[11, Prop. 6.10] For any additive functor $\mathscr {F}:\mathcal {A}\rightarrow \mathcal {B}$ with $\mathcal {B}$ idempotent complete:

• (i) there is an additive functor $\mathscr {E}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}$ and a natural isomorphism and, in addition,

• (ii) for any functor $\mathscr {G}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}$ and any natural transformation , there exists a unique natural transformation with .

2.2 Weak Idempotent Completion

A weaker notion than being idempotent complete is that of being weakly idempotent complete. This was introduced in the context of exact categories by Thomason–Trobaugh [34, Axiom A.5.1]. It is, however, a property of the underlying additive category and gives rise to the following definition.

Definition 2.9

[11, Def. 7.2] An additive category is weakly idempotent complete if every retraction has a kernel.

Definition 2.9 is actually self-dual. Indeed, in an additive category, every retraction has a kernel if and only if every section has a cokernel; see e.g. [11, Lem. 7.1].

If $r :X \rightarrow Y$ is a retraction in $\mathcal {A}$, with corresponding section $s :Y \rightarrow X$, and r admits a kernel k, then the split idempotent also has kernel k. Conversely, if $e:X\rightarrow X$ is a split idempotent, with splitting given by $e=sr$ where $r :X \rightarrow Y$, then a kernel of e is also a kernel of r. Therefore, weakly idempotent complete categories are those additive categories in which split idempotents admit kernels, in contrast to idempotent complete categories in which all idempotents admit kernels (see Proposition 2.2).

Definition 2.10

The weak idempotent completion $\widehat{\mathcal {A}}$ of $\mathcal {A}$ is the full subcategory of $\widetilde{\mathcal {A}}$ consisting of all objects $(X, e)\in \widetilde{\mathcal {A}}$ such that is a split idempotent in $\mathcal {A}$.

Remark 2.11

We note that Definition 2.10 above differs slightly from the definition of the weak idempotent completion of $\mathcal {A}$ suggested in [11, Rem. 7.8]. If, as in [11], we ask that objects of $\widehat{\mathcal {A}}$ are pairs (X, e) where $e:X\rightarrow X$ splits, then $\widehat{\mathcal {A}}$ is equivalent to $\mathcal {A}$. Indeed, if $sr=e$ and , where $r:X\rightarrow Y$ and $s:Y\rightarrow X$, then in $\widetilde{\mathcal {A}}$ by Lemma 2.4. That is, we have not added any objects that are not already isomorphic to some object of . On the other hand, if we take objects in $\widehat{\mathcal {A}}$ to be pairs (X, e) where splits (as in Definition 2.10), then we have in $\widehat{\mathcal {A}}$, where and , where $r':X\rightarrow Y'$ and $s':Y'\rightarrow X$. In this case, since in $\widetilde{\mathcal {A}}$, we see that a “complementary” summand of in has been added. This discrepancy has been noticed previously; see e.g. Henrard–van Roosmalen [18, Prop. A.11].

It follows that $\widehat{\mathcal {A}}$ is an additive subcategory of $\widetilde{\mathcal {A}}$ and that it is weakly idempotent complete; see e.g. [11, Rem. 7.8] or [18, Sec. A.2]. From this observation, we immediately have the next lemma.

Lemma 2.12

Suppose $\widetilde{X}, \widetilde{Y}, \widetilde{Z} \in \widetilde{\mathcal {A}}$ with $\widetilde{X} \oplus \widetilde{Y} \cong \widetilde{Z}$. Then any two of $\widetilde{X}, \widetilde{Y}, \widetilde{Z}$ being isomorphic to objects in $\widehat{\mathcal {A}}$ implies that the third object is also isomorphic to an object in $\widehat{\mathcal {A}}$.

Analogously to the construction in Sect. 2.1, there is an inclusion functor , given by on objects, which is 2-universal among additive functors from $\mathcal {A}$ to weakly idempotent complete categories; see e.g. [28, Rem. 1.12] or [11, Rem. 7.8].

Proposition 2.13

For any additive functor $\mathscr {F}:\mathcal {A}\rightarrow \mathcal {B}$ with $\mathcal {B}$ weakly idempotent complete:

• (i) there is an additive functor $\mathscr {E}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}$ and a natural isomorphism ; and, in addition,

• (ii) for any additive functor $\mathscr {G}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}$ and any natural transformation , there exists a unique natural transformation with .

Let denote the inclusion functor of the subcategory $\widehat{\mathcal {A}}$ into $\widetilde{\mathcal {A}}$. The functor factors through as . An additive functor $\mathscr {F}:\widehat{\mathcal {A}} \rightarrow \mathcal {B}$ to a weakly idempotent complete category $\mathcal {B}$ is determined up to unique natural isomorphism by its behaviour on the image of $\mathcal {A}$ in $\widehat{\mathcal {A}}$; similarly, a natural transformation of additive functors $\widehat{\mathcal {A}}\rightarrow \mathcal {B}$ is also completely determined by its action on ; see [11, Rems. 6.7, 6.9].

Remark 2.14

In [11, Rem. 7.9], it is remarked that there is a subtle set-theoretic issue regarding the existence of the weak idempotent completion of an additive category. Let NBG denote von Neumann-Bernays-Gödel class theory (see Fraenkel–Bar-Hillel–Levy [13, p. 128]), and let (AGC) denote the Axiom of Global Choice [13, p. 133]. The combination NBG + (AGC) is a conservative extension of ZFC [13, p. 131–132, 134]. If one chooses an appropriate class theory to work with, such as NBG + (AGC), then the weak idempotent completion always exists as a category. This would follow from the Axiom of Predicative Comprehension for Classes (see [13, p. 123]); this is also known as the Axiom of Separation (e.g. Smullyan–Fitting [32, p. 15]). Furthermore, a priori it is not clear to the authors if Proposition 2.8 and 2.13 follow in an arbitrary setting without (AGC). This is because in showing that, for example, an additive functor $\mathscr {F}:\widetilde{\mathcal {A}} \rightarrow \mathcal {B}$, where $\mathcal {B}$ is idempotent complete, is determined by its values on , one must choose a kernel and an image of the idempotent $\mathscr {F}(e)$ for each idempotent e in $\mathcal {A}$.

3 n-Exangulated Categories, Functors and Natural Transformations

Let $n\geqslant 1$ be an integer. In this section we recall the theory of n-exangulated categories established in [16], n-exangulated functors as defined in [10], and n-exangulated natural transformations as recently introduced in [9]. We also use this opportunity to set up some notation.

3.1 n-Exangulated Categories

The definitions in this subsection and more details can be found in [16, Sec. 2]. For this subsection, suppose that $\mathcal {C}$ is an additive category and that is a biadditive functor.

Let A, C be objects in $\mathcal {C}$. We denote by the identity element of the abelian group $\mathbb {E}(C,A)$. Suppose $\delta \in \mathbb {E}(C,A)$ and that $a:A\rightarrow B$ and $d:D\rightarrow C$ are morphisms in $\mathcal {C}$. We put and . Since $\mathbb {E}$ is a bifunctor, we have that .

An $\mathbb {E}$-extension is an element $\delta \in \mathbb {E}(C,A)$ for some $A,C\in \mathcal {C}$. A morphism of $\mathbb {E}$-extensions from $\delta \in \mathbb {E}(C,A)$ to $\rho \in \mathbb {E}(D,B)$ is given by a pair (a, c) of morphisms $a:A\rightarrow B$ and $c:C\rightarrow D$ in $\mathcal {C}$ such that .

Let be a product and be a coproduct in $\mathcal {C}$, and let ${\delta \in \mathbb {E}(C,A)}$ and $\rho \in \mathbb {E}(D,B)$ be $\mathbb {E}$-extensions. The direct sum of $\delta$ and $\rho$ is the unique $\mathbb {E}$-extension $\delta \oplus \rho \in \mathbb {E}(C\oplus D,A\oplus B)$ such that the following equations hold.

From the Yoneda Lemma, each $\mathbb {E}$-extension $\delta \in \mathbb {E}(C,A)$ induces two natural transformations. The first is given by for all objects $B\in \mathcal {C}$ and all morphisms $a:A\rightarrow B$. The second is and defined by for all objects $D\in \mathcal {C}$ and all morphisms $d:D\rightarrow C$.

Let $\textsf {\textrm{Ch}}({\mathcal {C}})$ be the category of complexes in $\mathcal {C}$. Its full subcategory consisting of complexes concentrated in degrees $0,1,\ldots , n,n+1$ is denoted . If , we depict as

omitting the trails of zeroes at each end.

Definition 3.1

Let be complexes, and suppose that and are $\mathbb {E}$-extensions.

• (i) The pair is known as an $\mathbb {E}$-attached complex if and . An $\mathbb {E}$-attached complex is called an n-exangle (for $(\mathcal {C},\mathbb {E})$) if, further, the sequences

  

  of functors are exact.

• (ii) A morphism of $\mathbb {E}$-attached complexes is given by a morphism such that . Such an is called a morphism of n-exangles if and are both n-exangles.

• (iii) The direct sum of the $\mathbb {E}$-attached complexes (or the n-exangles) and is the pair .

From the definition above, one can form the additive category of $\mathbb {E}$-attached complexes, and its additive full subcategory of n-exangles.

Given a pair of objects $A,C\in \mathcal {C}$, we define a subcategory of in the following way. An object is an object of that satisfies and . For , a morphism is a morphism with and . Note that this implies is not necessarily a full subcategory of , nor necessarily additive.

Let be complexes. Two morphisms in are said to be homotopic if they are homotopic in the standard sense viewed as morphisms in . This induces an equivalence relation $\sim$ on . We define as the category with the same objects as and with .

A morphism is called a homotopy equivalence if its image in the category is an isomorphism. In this case, and are said to be homotopy equivalent. The isomorphism class of in (equivalently, its homotopy class in ) is denoted . Since the (usual) homotopy class of in $\textsf {\textrm{Ch}}({\mathcal {C}})$ may differ from its homotopy class in , we reserve the notation specifically for its isomorphism class in .

Notation 3.2

For $X \in \mathcal {C}$ and $i \in \{ 0, \dots , n\}$, we denote by the object in given by for $j = i, i+1$ and for $0 \leqslant j \leqslant i-1$ and $i+2 \leqslant j \leqslant n+1$, as well as .

Definition 3.3

Let $\mathfrak {s}$ be an assignment that, for each pair of objects $A,C\in \mathcal {C}$ and each $\mathbb {E}$-extension ${\delta \in \mathbb {E}(C,A)}$, associates to $\delta$ an isomorphism class in . The correspondence $\mathfrak {s}$ is called an exact realisation of $\mathbb {E}$ if it satisfies the following conditions.

• (R0) For any morphism $(a,c):\delta \rightarrow \rho$ of $\mathbb {E}$-extensions with $\delta \in \mathbb {E}(C,A)$, $\rho \in \mathbb {E}(D,B)$, and , there exists such that $f_{0}=a$ and . In this setting, we say that realises $\delta$ and is a lift of (a, c).

• (R1) If , then is an n-exangle.

• (R2) For each object $A\in \mathcal {C}$, we have and .

In case $\mathfrak {s}$ is an exact realisation of $\mathbb {E}$ and the following terminology is used. The morphism is said to be an $\mathfrak {s}$-inflation and the morphism an $\mathfrak {s}$-deflation. The pair is known as an $\mathfrak {s}$-distinguished n-exangle.

Suppose $\mathfrak {s}$ is an exact realisation of $\mathbb {E}$ and . We will often use the diagram

to express that is an $\mathfrak {s}$-distinguished n-exangle. If we also have that and is a morphism of n-exangles, then we call a morphism of $\mathfrak {s}$-distinguished n-exangles and we depict this by the following commutative diagram.

We need one last definition before being able to define an n-exangulated category.

Definition 3.4

Suppose is a morphism in , such that for some . The mapping cone of is the complex

with , , and for $i\in \{1,\ldots ,n-1\}$.

We are in position to state the main definition of this subsection.

Definition 3.5

An n-exangulated category is a triplet $(\mathcal {C},\mathbb {E},\mathfrak {s})$, consisting of an additive category $\mathcal {C}$, a biadditive functor and an exact realisation $\mathfrak {s}$ of $\mathbb {E}$, such that the following conditions are met.

($\hbox {EA1}$):

The collection of $\mathfrak {s}$-inflations is closed under composition. Dually, the collection of $\mathfrak {s}$-deflations is closed under composition.

($\hbox {EA2}$):

Suppose $\delta \in \mathbb {E}(D,A)$ and $c\in \mathcal {C}(C,D)$. If and , then there exists a morphism lifting , such that . In this case, the morphism is called a good lift of .

:

The dual of (EA2).

Notice that the definition of an n-exangulated category is self-dual. In particular, the dual statements of several results in Sects. 4–5 are used without proof.

3.2 n-Exangulated Functors and Natural Transformations

In order to show that the canonical functor from an n-exangulated category $(\mathcal {C},\mathbb {E},\mathfrak {s})$ to its idempotent completion is 2-universal among structure-preserving functors from $(\mathcal {C},\mathbb {E},\mathfrak {s})$ to idempotent complete n-exangulated categories, we will need the notion of a morphism of n-exangulated categories and that of a morphism between such morphisms.

For this subsection, suppose $(\mathcal {C},\mathbb {E},\mathfrak {s})$, $(\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ and $(\mathcal {C}'',\mathbb {E}'',\mathfrak {s}'')$ are n-exangulated categories. If $\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'$ is an additive functor, then it induces several other additive functors, e.g. and obvious restrictions thereof. These are all defined in the usual way. However, by abuse of notation, we simply write $\mathscr {F}$ for each of these.

Definition 3.6

[10, Def. 2.32] Suppose that $\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'$ is an additive functor and that $\Gamma :\mathbb {E}(-,-) \Rightarrow \mathbb {E}'(\mathscr {F}-, \mathscr {F}-)$ is a natural transformation of functors . The pair $(\mathscr {F},\Gamma ) :(\mathcal {C},\mathbb {E},\mathfrak {s}) \rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ is called an n-exangulated functor if, for all $A,C\in \mathcal {C}$ and each $\delta \in \mathbb {E}(A,C)$, we have that whenever .

If we have a sequence of n-exangulated functors, then the composite of $(\mathscr {F},\Gamma )$ and $(\mathscr {L},\Phi )$ is defined to be

This is an n-exangulated functor $(\mathcal {C},\mathbb {E},\mathfrak {s})\rightarrow (\mathcal {C}'',\mathbb {E}'',\mathfrak {s}'')$; see [9, Lem. 3.19(ii)].

The next result implies that n-exangulated functors preserve finite direct sum decompositions of distinguished n-exangles. It will be used in the main result of Sect. 4.5.

Proposition 3.7

Let $\mathscr {F}:\mathcal {C}\rightarrow \mathcal {C}'$ be an additive functor and ${\Gamma :\mathbb {E}(-,-) \Rightarrow \mathbb {E}'(\mathscr {F}-,\mathscr {F}-)}$ a natural transformation. Suppose $\delta \in \mathbb {E}(C,A)$ and $\rho \in \mathbb {E}(D,B)$ are $\mathbb {E}$-extensions, and and are $\mathfrak {s}$-distinguished.

• (i) If is a morphism of $\mathbb {E}$-attached complexes, then the induced morphism is a morphism of $\mathbb {E}'$-attached complexes.

• (ii) We have as $\mathbb {E}'$-attached complexes.

Proof

(i)  Note that since $\Gamma$ is natural and is an $\mathbb {E}$-attached complex. Similar computations show that both and are $\mathbb {E}'$-attached complexes. As is a morphism of complexes, it suffices to prove

This follows immediately from and the naturality of $\Gamma$.

(ii)  This follows from applying (i) to the morphisms in the appropriate biproduct diagram of $\mathbb {E}$-attached complexes. $\square$

Lastly, we recall the notion of a morphism of n-exangulated functors. The extriangulated version was defined in Nakaoka–Ogawa–Sakai [29, Def. 2.11(3)].

Definition 3.8

[9, Def. 4.1] Suppose $(\mathscr {F},\Gamma ), (\mathscr {G},\Lambda ):(\mathcal {C},\mathbb {E},\mathfrak {s})\rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ are n-exangulated functors. A natural transformation of functors is said to be n-exangulated if, for all $A,C\in \mathcal {C}$ and each $\delta \in \mathbb {E}(C,A)$, we have

(3.1)

We denote this by . In addition, if has an n-exangulated inverse, then it is called an n-exangulated natural isomorphism. It is straightforward to check that has an n-exangulated inverse if and only if is an isomorphism for each $X\in \mathcal {C}$.

4 The Idempotent Completion of an n-Exangulated Category

Throughout this section we work with the following setup.

Setup 4.1

Let $n\geqslant 1$ be an integer. Let $(\mathcal {C},\mathbb {E},\mathfrak {s})$ be an n-exangulated category. We denote by the inclusion of the category $\mathcal {C}$ into its idempotent completion $\widetilde{\mathcal {C}}$; see Sect. 2.

In this section, we will construct a biadditive functor (see Sect. 4.1) and an exact realisation $\mathfrak {t}$ of $\mathbb {F}$ (see Sect. 4.2), and then show that $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ is an n-exangulated category (see Sects. 4.3–4.5). For $n=1$, we recover the main results of [27]. First, we establish some notation to help our exposition.

Notation 4.2

We reserve notation with a tilde for objects and morphisms in $\widetilde{\mathcal {C}}$.

• (i) If $\widetilde{X}\in \widetilde{\mathcal {A}}$ is some object, then we will denote the identity morphism of $\widetilde{X}$ by . Recall from Definition 2.3 that the identity of an object $(X,e)\in \widetilde{\mathcal {A}}$ is .

• (ii) Given a morphism $(e',r,e)\in \widetilde{\mathcal {C}}((X,e),(Y,e'))$, we call $r:X\rightarrow Y$ the underlying morphism of $(e',r,e)$.

• (iii) Suppose $(X,e), (Y,e') \in \widetilde{\mathcal {C}}$ and $r \in \mathcal {C}(X, Y)$ with $e' r = r = r e$. Then there is a unique morphism $\tilde{r} \in \widetilde{\mathcal {C}}((X,e), (Y,e'))$ with underlying morphism r. This morphism $\tilde{r}$ is the triplet $(e', r, e)$. Moreover, we will use this notation specifically for this correspondence. That is, we write $\tilde{s}:(X,e) \rightarrow (Y,e')$ is a morphism in $\widetilde{\mathcal {C}}$ if and only if we implicitly mean that the underlying morphism of $\tilde{s}$ is denoted s, i.e. we have $\tilde{s} = (e',s,e)$.

Remark 4.3

By Notation 4.2(iii), two morphisms $\tilde{r}, \tilde{s} \in \widetilde{\mathcal {C}} ((X,e), (Y,e'))$ are equal if and only if their underlying morphisms r and s, respectively, are equal in $\mathcal {C}$. Thus, for all objects $\widetilde{X},\widetilde{Y}\in \widetilde{\mathcal {C}}$, removing the tilde from morphisms in $\widetilde{\mathcal {C}}(\widetilde{X},\widetilde{Y})$ defines an injective abelian group homomorphism $\widetilde{\mathcal {C}}(\widetilde{X},\widetilde{Y}) \rightarrow \mathcal {C}(X,Y)$. In particular, a diagram in $\widetilde{\mathcal {C}}$ commutes if and only if its diagram of underlying morphisms commutes.

4.1 Defining the Biadditive Functor $\mathbb {F}$

The following construction is the higher version of the one given in [27, Sec. 3.1] for extriangulated categories.

Definition 4.4

We define a functor as follows. For objects and in $\widetilde{\mathcal {C}}$, we put

For morphisms in $\widetilde{\mathcal {C}}$, we define

Remark 4.5

We make some comments on Definition 4.4.

• (i) The assignment $\mathbb {F}$ on morphisms takes values where claimed due to the following. For morphisms and , and an $\mathbb {F}$-extension , we have

  $$\begin{aligned} \mathbb {E}(e''_{n+1}, e'_{0})\mathbb {E}(c,a)(\delta )&= \mathbb {E}(ce''_{n+1},e'_{0}a)(\delta ) \\&= \mathbb {E}(c,a)(\delta ). \end{aligned}$$

  Therefore, lies in . It is then straightforward to verify that $\mathbb {F}$ is indeed a functor.

• (ii) The set is an abelian group by defining

  

  for . The additive identity element of is . The inverse of is . Notice that we get an abelian group monomorphism:

  

  This homomorphism plays a role later in the proof of Theorem 4.39.

• (iii) It follows from the definition of $\mathbb {F}$ that it is biadditive since $\mathbb {E}$ is.

• (iv) Given , the pair is a morphism of $\mathbb {E}$-extensions $\delta \rightarrow \delta$. Indeed, we have that from Definition 4.4.

Notation 4.6

As for objects and morphisms in $\widetilde{\mathcal {C}}$, we use tilde notation for $\mathbb {F}$-extensions, which gives us a way to pass back to $\mathbb {E}$-extensions.

• (i) We will denote an $\mathbb {F}$-extension of the form by ${\tilde{\delta }}$. We call the underlying $\mathbb {E}$-extension of ${\tilde{\delta }}$.

• (ii) For and with , there is a unique $\mathbb {F}$-extension with underlying $\mathbb {E}$-extension $\delta$. This $\mathbb {F}$-extension is . Again, we use this instance of the tilde notation for this correspondence: we write if and only if the underlying $\mathbb {E}$-extension of ${\tilde{\rho }}$ is $\rho$, i.e. .

Remark 4.7

Analogously to our observations in Remark 4.3, we note that by Notation 4.6(ii) any two $\mathbb {F}$-extensions are equal if and only if their underlying $\mathbb {E}$-extensions are equal. Hence, removing the tilde from $\mathbb {F}$-extensions defines an injective abelian group homomorphism $\mathbb {F}((Y,e'), (X,e)) \rightarrow \mathbb {E}(Y,X)$ for $(X,e), (Y,e') \in \widetilde{\mathcal {C}}$.

4.2 Defining the Realisation $\mathfrak {t}$

To define an exact realisation $\mathfrak {t}$ of the functor $\mathbb {F}$ defined in Sect. 4.1, given a morphism of extensions consisting of two idempotents, we will need to lift this morphism to an $(n+2)$-tuple of idempotents. That is, we require a higher version of the idempotent lifting trick (see [27, Lem. 3.5] and [7, Lem. 1.13]). This turns out to be quite non-trivial and requires an abstraction of the case when $n=1$ in order to understand the mechanics of why this trick is successful.

We start with two lemmas related to the polynomial ring $\mathbb {Z}[x]$. Recall that $\mathbb {Z}[x]$ has the universal property that for any (unital, associative) ring R and any element $r \in R$ there is a unique (identity preserving) ring homomorphism $\varphi _r :\mathbb {Z}[x] \rightarrow R$ with $\varphi _r(x) = r$. For $p=p(x) \in \mathbb {Z}[x]$, we denote $\varphi _r(p)$ by p(r) as is usual.

Lemma 4.8

For each $m \in \mathbb {N}$, the ideals $(x^m) = {(x)}^m$ and $((x-1)^m) = {(x-1)}^m$ of $\mathbb {Z}[x]$ are coprime.

Proof

The ideals $\sqrt{{(x)}^m} = (x)$ and $\sqrt{{(x-1)}^m} = (x-1)$ are coprime in $\mathbb {Z}[x]$. Hence, $(x^m)$ and $( (x-1)^m )$ are also coprime by Atiyah–MacDonald [2, Prop. 1.16]. $\square$

Lemma 4.9

For each $m \in \mathbb {N}_{\geqslant 1}$, there is a polynomial $p_m \in (x^m) \unlhd \mathbb {Z}[x]$, such that for every (unital, associative) ring R we have:

• (i) $p_m(e) = e$ for each idempotent $e \in R$; and

• (ii) the element $p_m(r) \in R$ is an idempotent for each $r \in R$ satisfying $(r^2-r)^m = 0$.

Proof

Fix an integer $m \geqslant 1$. By Lemma 4.8, we can write $1 = x^m p_m' + (x-1)^m q_m'$ for some polynomials $p_m'$ and $q_m'$ in $\mathbb {Z}[x]$. We set $p_m :=x^m p_m'$.

Let R be a ring. For any idempotent $e \in R$, evaluating $x = x^{m+1} p_m' + x(x-1)^m q_m'$ at e and using $e(e-1) = 0$ yields $e = e^{m+1} p_m'(e) = e^m p_m'(e) = p_m(e)$, proving (i).

Now suppose $r \in R$ is an element with $(r^2-r)^m = 0$. Evaluation of

$$\begin{aligned} p_m = (x^m p'_m) \cdot 1 = (x^m p'_m) \cdot (x^m p'_m + (x-1)^m q_m') = p_m^2 + (x^2-x)^m p_m' q_m' \end{aligned}$$

at r shows $p_m(r)^2 = p_m(r)$ since $(r^2-r)^m = 0$, which finishes the proof. $\square$

The following is an abstract formulation of [27, Lem. 3.5] and [7, Lem. 1.13].

Lemma 4.10

Let be a complex in an additive category $\mathcal {A}$ and suppose is a weak cokernel of . Suppose is a morphism of complexes with and both idempotent. Then there exists a morphism , such that the following hold.

• (i) The triplet is a morphism of complexes.

• (ii) The element is idempotent and satisfies .

• (iii) The triplet is an idempotent morphism of complexes.

• (iv) If is a homotopy of morphisms , then the pair yields a homotopy .

Proof

Choose a polynomial as obtained in Lemma 4.9. Define $q :=xp'_{2}$ and set . We show this morphism satisfies the claims in the statement. For this, we will make use of the following. Let $p=p(x)\in \mathbb {Z}[x]$ be any polynomial. Since is a morphism of complexes, we have that is also a morphism of complexes, i.e. the diagram

commutes.

(i)  Note that , where the last equality follows from Lemma 4.9(i). Similarly, . Thus, using $p=q$ in the commutative diagram (4.1) shows that is a morphism of complexes.

(ii)  Since $f_1' = q(f_1)$ is a polynomial in $f_1$, we immediately have that . Furthermore, we see that . Thus, to show that is idempotent, it is enough to show that $(f_1^2 - f_1)^2 = 0$ by Lemma 4.9(ii). Let $r(x) = x^2 -x$. We see that and vanish as and are idempotents. Therefore, by choosing $p = r$ in (4.1) we have and so there is with , because is a weak cokernel of . This implies as , and hence is idempotent.

(iii)  Note that is a morphism of complexes using in (4.1).

(iv)  Suppose is a homotopy. Then we see that

Hence, is a null homotopy as desired. $\square$

Remark 4.11

Let $p'_2 = -2x + 3$ and $q'_2 = 2x + 1$. Then indeed $1 = x^2p'_2 + (x-1)^2q'_2$. Hence, is a possible choice for $m=2$ in Lemma 4.9. Letting $h = x^2 - x$ and $i = x$, we see that . Then the idempotent obtained in Lemma 4.10 is the idempotent obtained through the idempotent lifting trick in [27, Lem. 3.5].

Lemma 4.12

Suppose is an $\mathfrak {s}$-distinguished n-exangle and is an idempotent with . Then can be extended to a null homotopic, idempotent morphism with for $2 \leqslant i \leqslant n+1$. Further, the null homotopy of $e_{\bullet }$ can be chosen to be of the shape .

Proof

We have so is a morphism of $\mathbb {E}$-extensions. The solid morphisms of the diagram

clearly commute, so we need to find a morphism making the two leftmost squares commute. Since is an $\mathfrak {s}$-distinguished n-exangle, there is an exact sequence

The morphism is in the kernel of as . Therefore, there exists with . If we put , then is morphism of $\mathfrak {s}$-distinguished n-exangles and is a homotopy. By Lemma 4.10, using that and 0 are idempotents, there is an idempotent , such that is an idempotent morphism of complexes and that is a homotopy. Finally, is a morphism of $\mathfrak {s}$-distinguished n-exangles since . $\square$

Corollary 4.13

Suppose and that is an $\mathfrak {s}$-distinguished n-exangle. The morphism of $\mathbb {E}$-extensions has a lift that is idempotent and satisfies for all $2 \leqslant i \leqslant n -1$, such that there is a homotopy .

Proof

Define and , we have and so . Therefore, by Lemma 4.12 we can extend of $\mathfrak {s}$-distinguished n-exangles with $e'_{i} = 0$ for $i\in \{ 2,\ldots ,n+1 \}$, having a homotopy . Similarly, by the dual of Lemma 4.12, we can extend $e''_{n+1}$ to an idempotent morphism with $e''_{i} = 0$ for $i\in \{ 0,\ldots ,n-1 \}$, such that there is a homotopy . Consider the morphism and hence is a homotopy.

If $n = 1$, then and is a homotopy. Lemma 4.10 yields an idempotent morphism and a homotopy . Then and are the desired idempotent morphism and homotopy, respectively.

If $n \geqslant 2$, then the compositions $e'_{\bullet }e''_{\bullet }$ and $e''_{\bullet } e'_{\bullet }$ are zero. This implies that is idempotent. Hence, and are the desired idempotent morphism and homotopy, respectively. $\square$

The following simple lemma will be used several times.

Lemma 4.14

Suppose that $(X,e), (Y,e')$ are objects in $\widetilde{\mathcal {C}}$ and $r:X\rightarrow Y$ is a morphism in $\mathcal {C}$. Setting $s :=e're$ yields a morphism $\tilde{s} = (e',s,e) :(X,e)\rightarrow (Y,e')$ in $\widetilde{\mathcal {C}}$.

The previous result allows us to view a complex in $\mathcal {C}$ that is equipped with an idempotent endomorphism as a complex in the idempotent completion $\widetilde{\mathcal {C}}$, as follows.

Definition 4.15

Suppose is a complex in $\mathcal {C}$ and is an idempotent morphism of complexes. We denote by the complex in $\widetilde{\mathcal {C}}$ with object in degree i and differential .

In the notation of Definition 4.15, the underlying morphism of the differential satisfies

(4.2)

since is a morphism of complexes and consists of idempotents. Furthermore, whenever we write to denote a complex in $\widetilde{\mathcal {C}}$, we always mean that is an idempotent morphism in $\textsf {\textrm{Ch}}({\mathcal {C}})$ and that is the induced object in $\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})$ as described in Definition 4.15.

We make a further remark on the notation . Because of the need to tweak the differentials in according to (4.2), one cannot recover the original complex with differentials from the pair defined in Definition 4.15. This is in contrast to the description of an object in $\widetilde{\mathcal {C}}$ as a pair (X, e) where one can recover $X\in \mathcal {C}$ uniquely. Thus, is an abuse of notation but should hopefully cause no confusion.

Lemma 4.14 allows us to induce morphisms of complexes in $\widetilde{\mathcal {C}}$ given a morphism between complexes in $\mathcal {C}$ if the complexes involved come with idempotent endomorphisms. The proof is also straightforward.

Lemma 4.16

Suppose that are objects in $\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})$ and that is a morphism in . Then defining for each $i\in \mathbb {Z}$ gives rise to a morphism in $\textsf {\textrm{Ch}}({\widetilde{\mathcal {C}}})$ with .

Notation 4.17

In the setup of Lemma 4.16, the composite is a morphism of complexes . In this case, we call the underlying morphism of .

We need two more lemmas before we can define a realisation of the functor $\mathbb {F}$.

Lemma 4.18

Assume . Further, suppose that is an $\mathfrak {s}$-distinguished n-exangle and is an idempotent lift of . Then is an n-exangle for $(\widetilde{\mathcal {C}}, \mathbb {F})$.

Proof

Let $(Y,e')\in \widetilde{\mathcal {C}}$ be arbitrary. We will show that the induced sequence

where $(\tilde{d}_i^{(X,e)})_{*} = \widetilde{\mathcal {C}}((Y,e'), \tilde{d}_i^{(X,e)})$, is exact. The exactness of the dual sequence can be verified similarly. Checking the above sequence is a complex is straightforward using that and that .

To check exactness at for some $1\leqslant i \leqslant n$, suppose we have a morphism with $\tilde{d}_i^{(X,e)} \tilde{r} = 0$, that is, . As , we see that $d_i^Xr=0$, whence there exists such that $d_{i-1}^Xs = r$ because is an $\mathfrak {s}$-distinguished n-exangle. By Lemma 4.14, there is a morphism with $t = e_{i-1} s e'$. Then we observe that , whence $\tilde{d}_{i-1}^{(X,e)} \tilde{t} = \tilde{r}$.

Lastly, suppose is a morphism with . Then we have . Hence, there is a morphism such that as is an $\mathfrak {s}$-distinguished n-exangle. Then the morphism with satisfies $\tilde{d}^{(X,e)}_n \tilde{w} = \tilde{u}$, as required. $\square$

Lemma 4.19

Suppose and that in $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$. If and are idempotent lifts of , then are isomorphic in , i.e. .

Proof

We will use [16, Prop. 2.21]. To this end, note that and are both n-exangles in $(\widetilde{\mathcal {C}}, \mathbb {F})$ by Lemma 4.18. Hence, we only have to show that

are both non-empty. Since we have , there are morphisms and in (with and ). We then obtain morphisms and in with and by Lemma 4.16. Note that and . So, since , we have that and $\tilde{k}_{\bullet }$ are morphisms in and we are done. $\square$

Hence, the following is well-defined.

Definition 4.20

For , pick so that and, by Corollary 4.13, an idempotent morphism lifting . We put .

Remark 4.21

For , the definition of depends on neither the choice of with , nor on the choice of lifting by Lemma 4.19. By Corollary 4.13, for each , we can find an $\mathfrak {s}$-distinguished n-exangle and an idempotent morphism , such that and is null homotopic in .

Proposition 4.22

The assignment $\mathfrak {t}$ is an exact realisation of $\mathbb {F}$.

Proof

(R0)  Suppose and , and let $(\tilde{a},\tilde{c}):{\tilde{\delta }} \rightarrow {\tilde{\rho }}$ be a morphism of $\mathbb {F}$-extensions. Suppose and . Since (a, c) is a morphism of $\mathbb {E}$-extensions, there is a lift of it using that $\mathfrak {s}$ is an exact realisation of $\mathbb {E}$. As and are morphisms in $\widetilde{\mathcal {C}}$, we have that and . Hence, by Lemma 4.16, it follows that with lifts $(\tilde{a},\tilde{c})$.

(R1)  This is Lemma 4.18.

(R2)  Let $(X,e)\in \widetilde{\mathcal {C}}$ be arbitrary. By Remark 4.5(ii), we have that the zero element of $\mathbb {F}((0,0),(X,e))$ has the zero element $_{X}0_{0}$ of $\mathbb {E}(0,X)$ as its underlying $\mathbb {E}$-extension. Since $\mathfrak {s}$ is an exact realisation of $\mathbb {E}$, we know

The tuple is an idempotent morphism lifting . Thus, by Definition 4.20 and using , we see that

Dually, . $\square$

4.3 The Axiom (EA1) for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$

Now that we have a biadditive functor and an exact realisation $\mathfrak {t}$ of $\mathbb {F}$, we can begin to verify axioms (EA1), (EA2) and (EA2). In this subsection, we will check that the collection of $\mathfrak {t}$-inflations is closed under composition. One can dualise the results here to see that $\mathfrak {t}$-deflations compose to $\mathfrak {t}$-deflations.

The following result only needs that $\mathfrak {s}$ is an exact realisation of . It is an analogue of [22, Lem. 2.1] for n-exangulated categories, allowing us to complete a “partial” lift of a morphism of extensions.

Lemma 4.23

(Completion Lemma) Let and be $\mathfrak {s}$-distinguished n-exangles. Let l, r be integers with $0 \leqslant l \leqslant r-2 \leqslant n-1$. Suppose there are morphisms and , where , such that is a morphism of $\mathbb {E}$-extensions and the solid part of the diagram

commutes. Then there exist morphisms such that (4.3) commutes.

Proof

We proceed by induction on $l \geqslant 0$. Suppose $l=0$. We induct downwards on $r \leqslant n+1$. If $r = n+1$, then the result follows from axiom (R0) for $\mathfrak {s}$ since is a morphism of $\mathbb {E}$-extensions. Now assume that the result holds for $l=0$ and some $3 \leqslant r\leqslant n+1$. Suppose we are given morphisms and such that for $i\in \{ r-1,\ldots , n \}$. By the induction hypothesis, we obtain a morphism

of $\mathfrak {s}$-distinguished n-exangles. We will denote this morphism by . Next, note that we have . Since is a weak kernel of , there exists so that . Set for . Notice that we have for $i \notin \{r-1, r-2\}$. We claim that (4.3) commutes. By construction, we only need to check commutativity of the two squares involving . These indeed commute since

and

using the commutativity of (4.4). This concludes the base case $l=0$.

The inductive step for $l\geqslant 0$ is carried out in a similar way to the inductive step above on r, using that is a weak cokernel of . $\square$

From the Completion Lemma 4.23 and some earlier results from this section we derive the following, which is used in the main result of this subsection.

Lemma 4.24

Suppose is an $\mathfrak {s}$-distinguished n-exangle. Assume and are idempotents, such that and

commutes. Then and can be extended to an idempotent morphism with for .

Proof

First, suppose $n = 1$. Then the solid morphisms of the diagram

form a commutative diagram, and by [16, Prop. 3.6(1)] there is a morphism such that is a morphism of $\mathfrak {s}$-distinguished 1-exangles. Recall the polynomial from Lemma 4.9. We will show that , where , is the desired idempotent morphism of $\mathfrak {s}$-distinguished n-exangles.

Since is an $\mathfrak {s}$-distinguished 1-exangle, there is an exact sequence

As , there exists a morphism with . This shows that because is idempotent. Hence, is an idempotent morphism of complexes by Lemma 4.9(ii). Furthermore, using Lemma 4.9(i), so that is a morphism of $\mathbb {E}$-extensions. This computation also shows the existence of with underlying $\mathbb {E}$-extension $\delta$.

Now suppose $n \geqslant 2$. We have . Therefore, is an element of with underlying $\mathbb {E}$-extension $\delta$. The solid morphisms of the diagram

form a commutative diagram, and is a morphism of $\mathbb {E}$-extensions as . Since the rows are the $\mathfrak {s}$-distinguished n-exangle , by Lemma 4.23 we can find a morphism , so that the diagram above is a morphism . Furthermore, as and are idempotent, we may assume that is an idempotent by Lemma 4.10. $\square$

Given a $\mathfrak {t}$-inflation $\tilde{f}$ that fits into a $\mathfrak {t}$-distinguished n-exangle , we cannot a priori say too much about how might look. This is one of the main issues in trying to prove (EA1) for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$. The next lemma gives us a way to deal with this and is the last preparatory result we need before the main result of this subsection.

Lemma 4.25

Let be a $\mathfrak {t}$-inflation. Then there is an $\mathfrak {s}$-distinguished n-exangle $\langle X'_{\bullet }, \delta \rangle$ with and for some $C \in \mathcal {C}$, such that and for some .

Proof

Since is an $\mathfrak {t}$-inflation, there is a $\mathfrak {t}$-distinguished n-exangle with , and $\tilde{d}_0^{\widetilde{Y}} = \tilde{f}$. By definition of $\mathfrak {t}$, this means there is an $\mathfrak {s}$-distinguished n-exangle $\langle Y'_{\bullet }, \delta ' \rangle$ with an idempotent morphism $e'_{\bullet } :\langle Y'_{\bullet }, \delta ' \rangle \rightarrow \langle Y'_{\bullet }, \delta ' \rangle$, such that and , and there are mutually inverse homotopy equivalences which satisfy and . Note that we, thus, have and . In particular, we have a commutative diagram

in $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$, where .

Consider the complex and the $\mathbb {E}$-extension . Note that if $n=1$, then ; otherwise we have ${{Y''_{n+1}}} =0$. In either case, we have an $\mathfrak {s}$-distinguished n-exangle $\langle Y''_{\bullet }, \delta '' \rangle$ using the axiom (R2) for $\mathfrak {s}$, and hence also an $\mathfrak {s}$-distinguished n-exangle $\langle Y''_{\bullet } \oplus Y'_{\bullet }, \delta '' \oplus \delta ' \rangle$ by [16, Prop. 3.3]. Using the canonical isomorphism we see that the complex

realises in $(\mathcal {C},\mathbb {E},\mathfrak {s})$ by [16, Cor. 2.26(2)]. Consider the diagram

in $\mathcal {C}$, where and . This diagram commutes since

and

Notice that the composition ba is an automorphism of , and so the complex

forms part of an $\mathfrak {s}$-distinguished n-exangle $\langle X'_{\bullet }, \delta \rangle$ by [16, Cor. 2.26(2)]. We have

as , and . Setting and $C :=Y'_{1}$ finishes the proof. $\square$

We close this subsection with the following result, which together with its dual demonstrates that axiom (EA1) holds for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$.

Proposition 4.26

Suppose and are $\mathfrak {t}$-inflations with . Then is a $\mathfrak {t}$-inflation.

Proof

By Lemma 4.25, there exists an $\mathfrak {s}$-distinguished n-exangle $\langle X'_{\bullet }, \delta \rangle$ with and for some $C \in \mathcal {C}$, so that for some and . Similarly, there is also an $\mathfrak {s}$-distinguished n-exangle $\langle Y'_{\bullet }, \delta ' \rangle$ with and for some $C' \in \mathcal {C}$, so that for some and . Setting , we also have the n-exangle by axiom (R2) for $\mathfrak {s}$. Then is $\mathfrak {s}$-distinguished by [16, Prop. 3.3]. We have and , and

is the $\mathfrak {s}$-inflation of with respect to the given decompositions. Since and $d^{Y'\oplus Y''}_0$ are $\mathfrak {s}$-inflations, by (EA1) for $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$, we have that the morphism

is an $\mathfrak {s}$-inflation, where we used that . Therefore, there is an $\mathfrak {s}$-distinguished n-exangle $\langle Z''_{\bullet }, \delta '' \rangle$ with , and .

Our next aim is to apply Lemma 4.24 to $\langle Z''_{\bullet }, \delta '' \rangle$. Thus, we claim that . Since , we can apply [16, Prop. 3.6(1)] to obtain a morphism

of $\mathfrak {s}$-distinguished n-exangles. In particular, we have that

(4.5)

As and is idempotent, we see that

(4.6)

This implies that

Since $\langle X'_{\bullet }, \delta \rangle$ is an $\mathfrak {s}$-distinguished n-exangle, by [16, Lem. 3.5] there is an exact sequence

As seen above, vanishes under , so there is a morphism with . Since , this implies

showing that .

Now consider the idempotent . A quick computation yields the equality . Therefore, by Lemma 4.24, there is an idempotent morphism $e''_{\bullet } :\langle Z''_{\bullet }, \delta '' \rangle \rightarrow \langle Z''_{\bullet }, \delta '' \rangle$ with , $e''_{1} = e'_{1} \oplus 0 \oplus 0$ as well as an $\mathbb {F}$-extension with underlying $\mathbb {E}$-extension $\rho = \delta ''$. We obtain a $\mathfrak {t}$-distinguished n-exangle $\langle (Z''_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle$. Then the $\mathfrak {t}$-inflation of this n-exangle is given by the morphism satisfying

As is an isomorphism in $\widetilde{\mathcal {C}}$, the complex

with $\mathfrak {t}$-inflation and forms part of the $\mathfrak {t}$-distinguished n-exangle $\langle \widetilde{X}''_{\bullet }, {\tilde{\rho }} \rangle$ by [16, Cor. 2.26(2)]. $\square$

4.4 The Axiom (EA2) for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$

The goal of this subsection is to show that axiom (EA2) holds for the triplet $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$. Again, by dualising one can deduce that axiom (EA2) also holds. We need two key technical lemmas first.

Lemma 4.27

Suppose that:

• (i) is an $\mathbb {F}$-extension;

• (ii) is a morphism in $\widetilde{\mathcal {C}}$ for some ;

• (iii) and are $\mathfrak {s}$-distinguished n-exangles with ;

• (iv) is an idempotent morphism lifting , such that is null homotopic; and

• (v) is an idempotent morphism lifting , such that is null homotopic.

Then a good lift of the morphism of $\mathbb {E}$-extensions exists, so that

is commutative in $\mathcal {C}$. In particular, we have as morphisms .

Remark 4.28

Notice that and $c e'_{n+1}= c$ imply

(4.8)

Therefore, is indeed a morphism of $\mathbb {E}$-extensions and condition (v) makes sense. Condition (iv) makes sense due to Remark 4.5(iv).

Proof of Lemma 4.27

Since is a morphism of $\mathbb {E}$-extensions, it admits a good lift using axiom (EA2) for the n-exangulated category $(\mathcal {C},\mathbb {E},\mathfrak {s})$. Define for $0\leqslant i \leqslant n+1$. Note that and by assumption. For $i=0$, we have . On the other hand, for $i=n+1$ we have . Therefore, the morphism is of the form .

The squares on the top and bottom faces in (4.7) commute as and , respectively, are morphisms of complexes. The squares on the front and back faces in (4.7) commute because is the sum of morphisms of complexes from to . This also implies is a lift of . Of the remaining squares, the leftmost clearly commutes and the rightmost commutes as is a morphism in $\widetilde{\mathcal {C}}$. For $1\leqslant i \leqslant n$, we have

Therefore, diagram (4.7) commutes and, further, the last assertion follows.

It remains to show that is a good lift of . Recall that and are both null homotopic by assumption, and so is also null homotopic. Then it follows from [16, Rem. 2.33(1)] that is a good lift of since $g'_{\bullet }$ is. $\square$

The next result allows us to define a good lift in $\widetilde{\mathcal {C}}$ from the one we created in Lemma 4.27.

Lemma 4.29

In the setup of Lemma 4.27, the morphism with under-lying morphism is a good lift of the morphism of $\mathbb {F}$-extensions.

Proof

From (4.8), we see that is indeed an $\mathbb {F}$-extension and a morphism of $\mathbb {F}$-extensions. Using and , as well as the commutativity of (4.7), we see that is a morphism of $\mathfrak {t}$-distinguished n-exangles, lifting .

Recall from Definition 3.4 that denotes the mapping cone of in $\mathcal {C}$, and that is $\mathfrak {s}$-distinguished as is a good lift of . Using the commutativity of (4.7), that and are morphisms of complexes, and that , one can verify that the diagram

commutes. Thus, the vertical morphisms form an idempotent morphism ${e''_{\bullet } :{M_{g}^\mathcal {C}}_{\bullet } \rightarrow {M_{g}^\mathcal {C}}_{\bullet }}$ of complexes. Furthermore, (4.9) is a morphism of $\mathfrak {s}$-distinguished n-exangles as

This calculation also shows that . Thus, by definition of $\mathfrak {t}$, we have that $\mathfrak {t}({\tilde{\rho }}) = [ ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }) ]$, i.e. $\langle ({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet }), {\tilde{\rho }} \rangle$ is $\mathfrak {t}$-distinguished.

It is straightforward to verify that the object $({M^{\mathcal {C}}_g}_{\bullet }, e''_{\bullet })$ is equal to the mapping cone ${M^{\widetilde{\mathcal {C}}}_{\tilde{h}}}_{\bullet }$ of in , so $\langle {M^{\widetilde{\mathcal {C}}}_{\tilde{h}}}_{\bullet }, {\tilde{\rho }} \rangle$ is $\mathfrak {t}$-distinguished. Lastly, we note that because

Hence, is a $\mathfrak {t}$-distinguished n-exangle. $\square$

We are in position to prove axiom (EA2) for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$. Axiom (EA2) can be shown dually.

Proposition 4.30

(Axiom (EA2) for $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$) Let -extension and suppose is a morphism in $\widetilde{\mathcal {C}}$. Suppose are $\mathfrak {t}$-distinguished n-exangles. Then has a good lift .

Proof

Notice that the underlying $\mathbb {E}$-extension of is . By definition of $\mathfrak {t}$ and Remark 4.21, there are $\mathfrak {s}$-distinguished n-exangles $\langle X'_{\bullet }, \delta \rangle$ and and idempotent morphisms and , such that and $\mathfrak {t}(\tilde{c}^\mathbb {F}{\tilde{\delta }}) = [(Y'_{\bullet }, e'_{\bullet })]$, and so that and are null homotopic in . We note that since and , we have that and, in particular, that . Moreover, it follows that all the hypotheses of Lemma 4.27 are satisfied.

Therefore, by Lemma 4.29, the morphism of $\mathbb {F}$-extensions has a good lift . Since and , there is a homotopy equivalence in and a homotopy equivalence in . By [16, Cor. 2.31], the composite is then also a good lift of . $\square$

4.5 Main Results

In this subsection we present our main results regarding the idempotent completion and an n-exangulated structure we can impose on it.

Definition 4.31

We call an n-exangulated category $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ (resp. weakly) idempotent complete if the underlying additive category $\mathcal {C}$ is (resp. weakly) idempotent complete.

In [5, Prop. 2.5] a characterisation of weakly idempotent complete extriangulated categories is given. Next we note that the first part of Theorem A from Sect. 1 summarises our work from Sects. 4.1–4.4.

Theorem 4.32

Let $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ be an n-exangulated category. Then the triplet $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is an idempotent complete n-exangulated category.

Proof

This follows from Propositions 2.5, 4.22, 4.26 and 4.30, and the duals of the latter two. $\square$

And Corollary C from Sect. 1 is a nice consequence of this.

Corollary 4.33

Let $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ be an n-exangulated category, such that each object in $\mathcal {C}$ has a semi-perfect endomorphism ring. Then the idempotent completion $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is a Krull-Schmidt n-exangulated category.

Proof

By Theorem 4.32, the idempotent completion $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is an idempotent complete n-exangulated category. By [12, Thm. A.1] (or [24, Cor. 4.4]), it is enough to show that endomorphism rings of objects in $\widetilde{\mathcal {C}}$ are semi-perfect rings. Let (X, e) be an object in $\widetilde{\mathcal {C}}$. We have that is semi-perfect since is fully faithful (see Proposition 2.5). By Remark 2.6, we have that . In particular, we see that is an idempotent subring of the semi-perfect ring . Hence, by Anderson–Fuller [1, Cor. 27.7], we have that the endomorphism ring of each object in $\widetilde{\mathcal {C}}$ is semi-perfect. $\square$

We recall that, by [23, Cor. 4.12], an n-exangulated category is n-exact if and only if its inflations are monomorphisms and its deflations are epimorphisms.

Corollary 4.34

Suppose $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ is n-exact. Then $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is n-exact.

Proof

We use [23, Cor. 4.12] and only show that $\mathfrak {t}$-inflations are monomorphisms; showing $\mathfrak {t}$-deflations are epimorphisms is dual. Let be a $\mathfrak {t}$-inflation and suppose there is a morphism in $\widetilde{\mathcal {C}}$ with $\tilde{f} \tilde{g} = \widetilde{0}$. By Lemma 4.25, there is an $\mathfrak {s}$-inflation , which is monic as $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ is n-exact. We have $fg = 0$ as $\tilde{f} \tilde{g} = \widetilde{0}$, and we also have because the underlying morphism g of $\tilde{g}$ satisfies . Thus, we see that and this implies $g = 0$ as is monic. Hence, $\tilde{g} = 0$ and we are done. $\square$

The main aim of this subsection is to establish the relevant 2-universal property of the inclusion functor . We will show that forms part of an n-exangulated functor , and that this is 2-universal in an appropriate sense. The next lemma is straightforward to check.

Lemma 4.35

The family of abelian group homomorphisms

for , defines a natural isomorphism .

Proposition 4.36

The pair is an n-exangulated functor from $(\mathcal {C},\mathbb {E},\mathfrak {s})$ to $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$.

Proof

We verify that if is an $\mathfrak {s}$-distinguished n-exangle, then is $\mathfrak {t}$-distinguished, where . We have the idempotent morphism , so from Definition 4.20 we see that . $\square$

We lay out some notation that will be used in the remainder of this section and also in Sect. 5.

Notation 4.37

Let (X, e) be an object in the idempotent completion $\widetilde{\mathcal {C}}$ of $\mathcal {C}$. Then (X, e) is a direct summand of by Remark 2.6. By and , we denote the canonical inclusion and projection morphisms, respectively.

Recall that, for an additive category $\mathcal {C}'$ and a biadditive functor , the $\mathbb {E}'$-attached complexes and morphisms between them were defined in Definition 3.1, and together they form an additive category.

Lemma 4.38

Let be an $\mathbb {F}$-extension. Suppose and is an idempotent morphism. With , we have that

(4.10)

as $\mathfrak {t}$-distinguished n-exangles.

Proof

Let . First, note that and are $\mathbb {F}$-attached complexes, and is a $\mathfrak {t}$-distinguished n-exangle since $(\mathscr {F},\Gamma )$ is an n-exangulated functor.

Consider the morphisms and of complexes induced by , as well as the corresponding ones $\tilde{i}_{e'_{\bullet }}$ and $\tilde{p}_{e'_{\bullet }}$ for $e'_{\bullet }$. We claim that there is a biproduct diagram

in the category of $\mathbb {F}$-attached complexes. To see that is a morphism of $\mathbb {F}$-attached complexes, we just need to check that . By Remark 4.7, it is enough to see that holds, and this is indeed true because is a morphism of $\mathbb {F}$-attached complexes. To see that $\tilde{i}_{e'_{\bullet }}$ and $\tilde{p}_{e'_{\bullet }}$ are morphisms of $\mathbb {F}$-attached complexes, one uses that . Furthermore, we have the identities , and , so (4.11) is a biproduct diagram in the additive category of $\mathbb {F}$-attached complexes. Therefore, we have that (4.10) is an isomorphism as $\mathbb {F}$-attached complexes.

Lastly, since is $\mathfrak {t}$-distinguished, it follows from [16, Prop. 3.3] that (4.10) is an isomorphism of $\mathfrak {t}$-distinguished n-exangles. $\square$

Thus, we can now present and prove the main result of this section, which shows that the n-exangulated inclusion functor is 2-universal amongst n-exangulated functors from $(\mathcal {C},\mathbb {E},\mathfrak {s})$ to idempotent complete n-exangulated categories.

Theorem 4.39

Suppose $(\mathscr {F},\Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ is an n-exangulated functor to an idempotent complete n-exangulated category $(\mathcal {C}', \mathbb {E}', \mathfrak {s}')$. Then the following statements hold.

• (i) There is an n-exangulated functor $(\mathscr {E}, \Psi ) :(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ and an n-exangulated natural isomorphism .

• (ii) In addition, for any n-exangulated functor $(\mathscr {G}, \Theta ) :(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ and any n-exangulated natural transformation , there is a unique n-exangulated natural transformation with .

Proof

(i)  By Proposition 2.8(i), there exists an additive functor $\mathscr {E}:\widetilde{\mathcal {C}} \rightarrow \mathcal {C}'$ and a natural isomorphism . It remains to show that $\mathscr {E}$ forms part of an n-exangulated functor $(\mathscr {E},\Psi )$ and that is n-exangulated.

First, we define a natural transformation $\Psi :\mathbb {F}(-,-) \Rightarrow \mathbb {E}'(\mathscr {E}-, \mathscr {E}-)$ as the composition of several abelian group homomorphisms. For , we set

For and in $\widetilde{\mathcal {C}}$, we define an abelian group homomorphism

and put

For morphisms and in $\mathcal {C}$, we have

(4.12)

as is natural. For morphisms , we have

(4.13)

using how $\mathbb {F}$ is defined on morphisms (see Definition 4.4). We claim that the family of abelian group homomorphisms

for defines a natural transformation $\Psi :\mathbb {F}(-,-) \Rightarrow \mathbb {E}'(\mathscr {E}-, \mathscr {E}-)$. To this end, fix objects and , and morphisms and in $\widetilde{\mathcal {C}}$. First, note that we have

(4.14)

and, similarly,

(4.15)

Therefore, we see that

Next, we must show that $(\mathscr {E},\Psi )$ sends $\mathfrak {t}$-distinguished n-exangles to $\mathfrak {s}'$-distinguished n-exangles. Thus, let and , and suppose . We need that , which will follow from seeing that is a direct summand of an $\mathfrak {s}'$-distinguished n-exangle.

By Remark 4.21, we may take a complex in with and an idempotent morphism lifting , such that . Note for later that we thus have , and hence . Let . Since $(\mathscr {F}, \Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ is an n-exangulated functor, the n-exangle is $\mathfrak {s}'$-distinguished. As we have an isomorphism of complexes , the $\mathbb {E}'$-attached complex is $\mathfrak {s}'$-distinguished by [16, Cor. 2.26(2)]. Since is just the identity homomorphism, a quick computation yields

(4.16)

In particular, this implies that is $\mathfrak {s}'$-distinguished.

Note that as $\mathbb {F}$-attached complexes by Lemma 4.38, where . We see that is a direct summand of the $\mathfrak {s}'$-distinguished n-exangle by Proposition 3.7(ii). Hence, by [16, Prop. 3.3], and so $(\mathscr {E},\Psi )$ is an n-exangulated functor.

Lastly, it follows immediately from (4.16) that is an n-exangulated natural transformation .

(ii)  By Proposition 2.8(ii), there exists a unique natural transformation with , so it remains to show that induces an n-exangulated natural transformation $(\mathscr {E}, \Psi ) \Rightarrow (\mathscr {G}, \Theta )$. For this, let and be arbitrary. Note that we have

(4.17)

Hence, we obtain

and the proof is complete. $\square$

We close this section with some remarks on our main results and constructions.

Remark 4.40

Before commenting on how our results unify the constructions in cases in the literature and on how our proof methods compare, we set up and recall a little terminology. Suppose $(\mathcal {C},\mathbb {E},\mathfrak {s})$ and $(\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ are n-exangulated categories. We call an n-exangulated functor $(\mathscr {F},\Gamma ) :(\mathcal {C},\mathbb {E},\mathfrak {s}) \rightarrow (\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ an n-exangulated isomorphism if $\mathscr {F}$ is an isomorphism of categories and $\Gamma$ is a natural isomorphism. This terminology is justified by [9, Prop. 4.11]. Lastly, we recall that n-exangulated functors between $(n+2)$-angulated categories are $(n+2)$-angulated in the sense of [10, Def. 2.7] (or exact as in Bergh–Thaule [8, Sec. 4]), and that n-exangulated functors between n-exact categories are n-exact in the sense of [10, Def. 2.18]; see [10, Thms. 2.33, 2.34].

It has been shown that a triplet $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is a 1-exangulated category if and only if it is extriangulated (see [16, Prop. 4.3]). Suppose that $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an extriangulated category and consider the idempotent completion $\widetilde{\mathcal {C}}$ of $\mathcal {C}$. By [27, Thm. 3.1], there is an extriangulated structure $(\mathbb {F}',\mathfrak {t}')$ on $\widetilde{\mathcal {C}}$. By our Theorem 4.32, there is a 1-exangulated (or extriangulated) category $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$. By direct comparison of the constructions, one can check that $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ and $(\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')$ are n-exangulated isomorphic. Indeed, the bifunctors $\mathbb {F}$ and $\mathbb {F}'$ differ only by a labelling of the elements due to our convention in Definition 4.4; and, ignoring this re-labelling, the realisations $\mathfrak {s}$ and $\mathfrak {s}'$ are the same by Lemma 4.19. Furthermore, since $(\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')$ recovers the triangulated and exact category cases, we see that our construction agrees with the classical (i.e. $n=1$) cases up to n-exangulated isomorphism.

For larger n, we just need to compare $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ with the construction in [25]. Thus, suppose $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is the n-exangulated category coming from an $(n+2)$-angulated category . Recall that in this case $\mathbb {E}(Z,X) = \mathcal {C}(Z,\Sigma X)$ for $X,Z\in \mathcal {C}$. Using [25, Thm. 3.1], one obtains an $(n+2)$-angulated category , where $\widetilde{\Sigma }$ is induced by $\Sigma$. From this $(n+2)$-angulated category, just like above, we obtain an induced n-exangulated category $(\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')$. Notice that $\mathbb {F}'(-,-) = \widetilde{\mathcal {C}}(-,\widetilde{\Sigma }-)$. Comparing $(\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')$ to the n-exangulated category $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ found from Theorem 4.32, again we see that $\mathbb {F}$ and $\mathbb {F}'$ differ by the labelling convention we chose in Definition 4.4. By [16, Prop. 4.8] we have that $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ induces an $(n+2)$-angulated category , and therefore the n-exangulated inclusion functor is, moreover, $(n+2)$-angulated. It follows from [25, Thm. 3.1(2)] that and must be equal, and hence $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ and $(\widetilde{\mathcal {C}},\mathbb {F}',\mathfrak {t}')$ are n-exangulated isomorphic.

Remark 4.41

Our proofs in this article differ from the proofs in both the extriangulated and the $(n+2)$-angulated cases. First, the axioms for an n-exangulated category look very different from the axioms for an extriangulated category. Therefore, the proofs from [27] cannot be directly generalised to the $n>1$ case. Even of the results that seem like they might generalise nicely, one comes across immediate obstacles. Indeed, Lin [25, p. 1064] already points out that lifting idempotent morphisms of extensions to idempotent morphisms of n-exangles is non-trivial. Despite this, we are able to overcome this here. This, amongst other problems, forces Lin to use another approach, and hence demonstrates why our methods are distinct.

Remark 4.42

He–He–Zhou [15] have considered idempotent completions of n-exangulated categories in a specific setup. In their setup, there is an ambient Krull-Schmidt $(n+2)$-angulated category $\mathcal {C}$ and an additive subcategory $\mathcal {A}$ that is n-extension-closed (see Definition 5.2) and closed under direct summands in $\mathcal {C}$. The main aim of [15] is to show that the idempotent completion $\widetilde{\mathcal {A}}$ of $\mathcal {A}$ is an n-exangulated subcategory of $\widetilde{\mathcal {C}}$.

Since $\mathcal {A}$ is an additive subcategory of and closed under direct summands in a Krull-Schmidt category, it is Krull-Schmidt itself. In particular, $\mathcal {A}\simeq \widetilde{\mathcal {A}}$ is already idempotent complete by [24, Cor. 4.4]. Moreover, in the setup of [15], it already follows that $\mathcal {A}$ inherits an n-exangulated structure from $\mathcal {C}\simeq \widetilde{\mathcal {C}}$. Indeed, (EA1) is proven in [22, Lem. 3.8], and (EA2) and (EA2) are straightforward to check directly. It is then clear that $\mathcal {A}$ inherits an n-exangulated structure from $\mathcal {C}$.

5 The Weak Idempotent Completion of an n-Exangulated Category

Just as in Sect. 4, we assume $n\geqslant 1$ is an integer and that $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an n-exangulated category. By Theorems 4.32 and 4.39, the idempotent completion of $(\mathcal {C},\mathbb {E},\mathfrak {s})$ is an n-exangulated category $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ and the inclusion functor of $\mathcal {C}$ into $\widetilde{\mathcal {C}}$ is part of an n-exangulated functor , which satisfies the 2-universal property from Theorem 4.39. In this section, we turn our attention to the weak idempotent completion $\widehat{\mathcal {C}}$ of $\mathcal {C}$ and we show that it forms part of a triplet $(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})$ that is n-extension-closed (see Definition 5.2) in $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$. It will then follow that $(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})$ is itself n-exangulated, and, moreover, there is an analogue of Theorem 4.39 for $(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})$; see Theorem 5.5.

We begin with the following proposition, which is an analogue of Lemma 4.38 for the weak idempotent completion.

Proposition 5.1

Suppose are objects, is an $\mathbb {F}$-extension and . Then there is a $\mathfrak {t}$-distinguished n-exangle with and an $\mathfrak {s}$-distinguished n-exangle , such that

as $\mathfrak {t}$-distinguished n-exangles.

Proof

By Corollary 4.13, there exists an idempotent morphism with for $2 \leqslant i \leqslant n-1$, as well as a homotopy , where . Notice for $i = 0, n+1$ by assumption. Furthermore, for . By Lemma 4.38 we have -distinguished n-exangles. We will show that there is an isomorphism in for some , as well as an isomorphism in for some object .

If $i = 0,n+1$, then is split by assumption, so by Lemma 2.4 there are objects $Y'_i \in \mathcal {C}$ and isomorphisms . For $2\leqslant i \leqslant n-1$, we see that , so by Lemma 2.4 again we have isomorphisms , but now where $Y'_{i} = 0 \in \mathcal {C}$. Since by assumption and because for $2 \leqslant i \leqslant n-1$, we put and for $i \in \{ 0,n+1\} \cup \{2,\ldots , n-1\}$. It remains to find appropriate isomorphisms and $\tilde{s}'_i$ for $i=1,n$.

We have a morphism with underlying morphism and another with underlying morphism by Lemma 4.14. Since is a homotopy, we see that . This implies

(5.1)

and so . Similarly, we also have .

• 1. If $n=1$, then (5.1) shows that is a section in the complex , and hence this complex is a split short exact sequence by [16, Claim 2.15]. In particular, we have that . So we put $Y'_{1} :=Y'_{0}\oplus Y'_{2}$ and define to be this composition of isomorphisms. As , and and are isomorphic to objects in $\widehat{\mathcal {C}}$, by Lemma 2.12 there is an isomorphism for some .

• 2. If $n \geqslant 2$, then the form of the homotopy implies that the identities and hold. Therefore, we see that

  

  which shows that and are mutually inverse isomorphisms. We now define $Y'_{1} :=Y'_{0}$ and . Because there are isomorphisms , and and are isomorphic to objects in $\widehat{\mathcal {C}}$, by Lemma 2.12 there is an isomorphism for some . In a similar way, one can show that and are mutually inverse isomorphisms. We set $Y'_{\textit{n}} :=Y'_{n+1}$ and . In addition, there is an isomorphism for some .

The complex with object in degree $0\leqslant i \leqslant n+1$ and differential in degree $0\leqslant i\leqslant n$ is isomorphic to via . Furthermore, as and are identity morphisms, we have that is $\mathfrak {t}$-distinguished by [16, Cor. 2.26(2)]. The complex $\widetilde{Y}'_{\bullet }$ with object in degree $0\leqslant i \leqslant n+1$ and differential in degree $0\leqslant i \leqslant n$ is isomorphic to via $\tilde{s}'_{\bullet }$. Moreover, this induces an isomorphism of $\mathbb {F}$-attached complexes, and hence of $\mathfrak {t}$-distinguished n-exangles. It is clear that

by the construction of $\widetilde{Y}'_\bullet$. Lastly, is $\mathfrak {s}$-distinguished using [16, Prop. 3.3] and that $\mathfrak {s}$ is an exact realisation of $\mathbb {E}$. $\square$

From Proposition 5.1 we see that $\widehat{\mathcal {C}}$ is n-extension-closed in $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ in the following sense.

Definition 5.2

[17, Def. 4.1] Let $(\mathcal {C}',\mathbb {E}',\mathfrak {s}')$ be an n-exangulated category. A full subcategory $\mathcal {D}\subseteq \mathcal {C}'$ is said to be n-extension-closed if, for all $A,C\in \mathcal {D}$ and each $\mathbb {E}'$-extension $\delta \in \mathbb {E}'(C,A)$, there is an object such that for all $1\leqslant i \leqslant n$ and .

Let us now define the biadditive functor and realisation with which we wish to equip $\widehat{\mathcal {C}}$.

Definition 5.3

• (i) Let be the restriction of .

• (ii) For a $\mathbb {G}$-extension , there is a $\mathfrak {t}$-distinguished n-exangle with by Proposition 5.1. We put , the isomorphism class of in .

• (iii) Recall from Sect. 2.2 that is the inclusion functor defined by on objects $X \in \mathcal {C}$. Let be the restriction of the natural transformation defined in Lemma 4.35. This means for .

Since $\widehat{\mathcal {C}}$ is an n-extension closed subcategory of $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ by Proposition 5.1, one can use [17, Prop. 4.2(1)] to deduce axioms (EA2) and (EA2) hold for the triplet $(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})$. The difficult part is then to show that (EA1) is satisfied; this follows from Lemma 5.4 below. We note here, however, it has been shown in [23, Thm. A] that any n-extension-closed subcategory of an n-exangulated category that is also closed under isomorphisms inherits an n-exangulated structure in the expected way. Although the isomorphism-closure in [23] is assumed only for convenience, we highlight that the weak idempotent completion is not necessarily closed under isomorphisms in the idempotent completion using the constructions in Sect. 2. Indeed, one can show that $\widehat{\mathcal {C}}$ is isomorphism-closed in $\widetilde{\mathcal {C}}$ if and only if $\mathcal {C}$ is already weakly idempotent complete.

Lemma 5.4

Let be a $\mathfrak {t}$-inflation with . Then there is a $\mathfrak {t}$-distinguished n-exangle with and .

Proof

By Lemma 4.25 there is an object $C \in \mathcal {C}$, a morphism and an $\mathfrak {s}$-distinguished n-exangle and . The solid morphisms of the diagram

form a commutative diagram. By Lemma 4.24 there exists an idempotent morphism of n-exangles , and for $3 \leqslant i \leqslant n+1$, which makes the diagram above commute. Let and . Notice that the underlying $\mathbb {E}$-extension of ${\tilde{\rho }}'$ is $\rho$. Set . Then as $\mathfrak {t}$-distinguished n-exangles by Lemma 4.38.

We claim that there is a split short exact sequence

Since realises the trivial $\mathbb {F}$-extension , we have that is a section by [16, Claim 2.15]. If $n=1$, then this is enough to see that (5.2) is split short exact. For $n\geqslant 2$ we notice that there is an isomorphism in $\widetilde{\mathcal {C}}$, so . Thus, since is a weak kernel of , we see that factors through . In particular, this implies is a cokernel of . Again, (5.2) is split short exact.

In particular, we have an isomorphism . We know that the objects and are isomorphic to objects in by Lemma 2.4, as and are split idempotents. This implies that is isomorphic to an object in $\widehat{\mathcal {C}}$ by Lemma 2.12. Again Lemma 2.12 and the isomorphism imply that there is an isomorphism for some .

The morphism with underlying morphism is an isomorphism. Finally, put for $i = 0$ and $3 \leqslant i \leqslant n+1$. Then the complex

is isomorphic to via in . With , we see that is $\mathfrak {t}$-distinguished by [16, Cor. 2.26(2)], as desired. $\square$

We may state and prove our main result of this section.

Theorem 5.5

Suppose that $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ is an n-exangulated category. Then $(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})$ is a weakly idempotent complete n-exangulated category, and is an n-exangulated functor, such that the following 2-universal property is satisfied. Suppose $(\mathscr {F},\Lambda ) :(\mathcal {C}, \mathbb {E}, \mathfrak {s}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ is an n-exangulated functor to a weakly idempotent complete n-exangulated category $(\mathcal {C}', \mathbb {E}', \mathfrak {s}')$. Then the following statements hold.

• (i) There is an n-exangulated functor $(\mathscr {E}, \Psi ) :(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ and an n-exangulated natural isomorphism .

• (ii) In addition, for any n-exangulated functor $(\mathscr {G}, \Theta ) :(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r}) \rightarrow (\mathcal {C}', \mathbb {E}', \mathfrak {s}')$ and any n-exangulated natural transformation , there is a unique n-exangulated natural transformation with .

Proof

Since $\widehat{\mathcal {C}}$ is a full subcategory of $\widetilde{\mathcal {C}}$ and because $(\widetilde{\mathcal {C}}, \mathbb {F}, \mathfrak {t})$ is n-exangulated, we can apply [17, Prop. 4.2] and Definition 5.2. We showed above that $\widehat{\mathcal {C}}$ is n-extension-closed in $\widetilde{\mathcal {C}}$; see Proposition 5.1. Moreover, it follows immediately from Lemma 5.4 and its dual that $(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})$ satisfies (EA1). Therefore, we deduce that $(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})$ is an n-exangulated category.

One may argue that is an n-exangulated functor as in Proposition 4.36, by using the definition of $\mathfrak {r}$ and noting that lies in $\widehat{\mathcal {C}}$ for all $X\in \mathcal {C}$.

(i)  One argues like in the proof of Theorem 4.39(i), but using Proposition 2.13 instead of Proposition 2.8, and Proposition 5.1 instead of Lemma 4.38. In particular, we note that the isomorphism of $\mathfrak {t}$-distinguished n-exangles from the statement of Proposition 5.1 induces an isomorphism

of $\mathfrak {r}$-distinguished n-exangles.

(ii)  Similarly, one adapts the proof of Theorem 4.39(ii), using Proposition 2.13 instead of Proposition 2.8. $\square$

Finally, we have an analogue of Corollary 4.34 as a consequence.

Corollary 5.6

Suppose $(\mathcal {C}, \mathbb {E}, \mathfrak {s})$ is n-exact. Then $(\widehat{\mathcal {C}}, \mathbb {G}, \mathfrak {r})$ is n-exact.

Proof

The n-exangulated category $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$ is n-exact by Corollary 4.34. As $(\widehat{\mathcal {C}},\mathbb {G},\mathfrak {r})$ inherits its structure as an n-extension closed subcategory of $(\widetilde{\mathcal {C}},\mathbb {F},\mathfrak {t})$, the result follows from the proof of [23, Cor. 4.15]. $\square$