Idempotent completion of right suspended categories

1.   Introduction

Let ${\mathcal{A}}$ be an additive category. An idempotent morphism $e^2 = e\colon A\to A$ in ${\mathcal{A}}$ is said to be split if there are two morphisms $p\colon A\to B$ and $q\colon B\to A$ such that $e = qp$ and $pq = 1_B$. The category ${\mathcal{A}}$ is said to be idempotent complete if every idempotent morphism splits. Note that ${\mathcal{A}}$ is idempotent complete if and only if every idempotent morphism has a kernel if and only if every idempotent morphism has a cokernel, see ^[1]. Every additive category ${\mathcal{A}}$ can be embedded fully faithfully into an idempotent complete category $\widetilde{{\mathcal{A}}}$. Balmer and Schlichting ^[2] proved that the idempotent completion of a triangulated category is a triangulated category. Bühler showed that the idempotent completion of an exact category is an exact category. Liu and Sun ^[4] showed that the idempotent completion of a right triangulated category is again right triangulated.

Recently, suspended categories were introduced by Li in ^[3] as a simultaneous generalization of exact categories, triangulated categories and right triangulated categories. In this article, we will unify these conclusions stated above by showing that when ${\mathcal{A}}$ is a suspended category then the idempotent completion of ${\mathcal{A}}$ is also a suspended category.

2.   Preliminaries

We first recall some notions and facts on the idempotent completion of additive categories.

Definition 2.1. [2, Definition 1.2] Let ${\mathcal{A}}$ be an additive category. The idempotent completion of ${\mathcal{A}}$ is denoted by $\widetilde{ \mathcal A}$ which be defined as follows. The objects of $\widetilde{{\mathcal{A}}}$ are pairs $(A, p)$, where $A$ is an object of $\mathcal A$ and $p\colon A\to A$ is an idempotent morphism. A morphism in $\widetilde{ \mathcal A}$ from $(A, p)$ to $(B, q)$ is a morphism $f\colon A\to B$ such that $qf = fp = f$. For any object $(A, p)$ in $\widetilde{{\mathcal{A}}}$, the identity morphism $1_{(A, \; p)} = p$.

Remark 2.2. [1, Remark 6.3] Let ${\mathcal{A}}$ be an additive category and $\widetilde{{\mathcal{A}}}$ be an idempotent complete of ${\mathcal{A}}$. The biproduct in $\widetilde{{\mathcal{A}}}$ is defined as

There exists a fully faithful additive functor $\ell_{{\mathcal{A}}}\colon {\mathcal{A}}\to\widetilde{{\mathcal{A}}}$ defined as follows. For an object $A$ in ${\mathcal{A}}$, we have that $\ell_{{\mathcal{A}}}(A) = (A, 1_A)$ and for a morphism $f$ in ${\mathcal{A}}$, we have that $\ell_{{\mathcal{A}}}(f) = f$. Since the functor $\ell_{{\mathcal{A}}}$ is fully faithful, we can view ${\mathcal{A}}$ as a full subcategory of $\widetilde{{\mathcal{A}}}$.

Proposition 2.3. [1, Proposition 6.10] Let ${\mathcal{A}}$ be an additive category and ${\mathcal{B}}$ be anidempotent complete category. For every additive functor $\mathbb{F}\colon {\mathcal{A}}\to{\mathcal{B}}$, there exists a functor $\widetilde{\mathbb{F}}\colon \widetilde{{\mathcal{A}}}\to \widetilde{{\mathcal{B}}}$ and a natural isomorphism $\phi\colon \mathbb{F}\Rightarrow \widetilde{\mathbb{F}}\ell_{{\mathcal{A}}}$.

Now we recall the notion of suspended categories from ^[3].

Let ${\mathcal{A}}$ be an additive category and ${\mathscr{X}}$ be a full subcategory of ${\mathcal{A}}$. Recall that we say a morphism $f\colon A \to B$ in ${\mathcal{C}}$ is an ${\mathscr{X}}$-monic if

is an epimorphism for all $X\in{\mathscr{X}}$. Similarly, we say that $f$ is a left ${\mathscr{X}}$-approximation of $A$ if $f$ is an ${\mathscr{X}}$-monic and $B\in{\mathscr{X}}$. The subcategory ${\mathscr{X}}$ is said to be covariantly finite in ${\mathcal{A}}$, if every object in ${\mathcal{A}}$ has a left ${\mathscr{X}}$-approximation. The notions of left ${\mathscr{X}}$-approximation and covariantly finite subcategories are also known as ${\mathscr{X}}$-preenvelope and preenveloping subcategories, respectively.

Let ${\mathcal{A}}$ be an additive category with an additive endofunctor $\Sigma\colon {\mathcal{A}}\to{\mathcal{A}}$ and ${\mathscr{X}}\subseteq{\mathcal{C}}$ be two full subcategories of ${\mathcal{A}}$. A right $\Sigma$-sequence $A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{h}\Sigma A$ in ${\mathcal{A}}$ is called a right ${\mathcal{C}}$-sequence if $C\in{\mathcal{C}}$, $g$ is a weak cokernel of $f$ (i.e. the induced sequence ${\rm Hom}_{{\mathcal{A}}}(C, {\mathcal{A}})\to {\rm Hom}_{{\mathcal{A}}}(B, {\mathcal{A}})\to{\rm Hom}_{{\mathcal{A}}}(A, {\mathcal{A}})$ is exact) and $h$ is a weak cokernel of $g$.

Dually, a left $\Sigma$-sequence $\Sigma A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{h} A$ is called a left ${\mathcal{C}}$-sequence if $B\in{\mathcal{C}}$, $f$ is a weak kernel of $g$ and $g$ is a weak kernel of $h$.

Definition 2.4. [3, Definition 3.1] Let ${\mathcal{A}}$ be an additive category with an additive endofunctor $\Sigma\colon {\mathcal{A}}\to{\mathcal{A}}$ and ${\mathscr{X}}\subseteq{\mathcal{C}}$ be two full subcategories of ${\mathcal{A}}$. A triple $({\mathcal{A}}, R({\mathcal{C}}, \Sigma), {\mathscr{X}})$ is a right suspended category where $R({\mathcal{C}}, \rm\Sigma)$ is a class of right ${\mathcal{C}}$-sequences (whose elements are also called right ${\mathcal{C}}$-triangles) if $R(\mathcal C, \rm\Sigma)$ is closed under isomorphisms and finite direct sums and the following conditions are satisfied:

(RS1) (a) For any $A\in{\mathcal{C}}$, there exists a sequence $A \stackrel{i}{\longrightarrow} X \longrightarrow U \longrightarrow \Sigma(A)$ in $R({\mathcal{C}}, \rm\Sigma)$ where $i$ is an ${\mathscr{X}}$-preenvelope such that for any morphism $f\colon A\longrightarrow B$ in $\mathcal C$, there exists a sequence

in $R(\mathcal{C}, \Sigma)$.

$(b)$ For any morphism $f\colon A\longrightarrow B$ in ${\mathcal{C}}$, there exists a sequence

in $R(\mathcal{C}, \Sigma).$

(RS2) For any commutative diagram of sequences in $R(\mathcal{C}, \Sigma)$

with $X\in {\mathscr{X}}$, if $\alpha$ factors through $f$, then $\gamma$ factors through $v$.

(RS3) For each solid commutative diagram

with rows in $R(\mathcal{C}, \Sigma)$, the dotted morphism exists which makes the whole diagram commutative.

(RS4) If any three sequences

are in $R(\mathcal{C}, \rm\Sigma)$ and $f, g$ are ${\mathscr{X}}$-monic, then there exists two morphisms $\alpha:D\longrightarrow F$ and $\beta:F\longrightarrow E$ of $\mathcal C$, such that the diagram below is commutative:

where the third column from the left is in $R(\mathcal{C}, \rm\Sigma)$, with $\alpha$ is an ${\mathscr{X}}$-monic.

Dually, we can define the notion of a left suspended category.

Now we give some examples of right suspended categories from ^[3].

Example 2.5. (1) If $(\mathcal A, \Sigma, \Delta)$ is a right triangulated category, we take ${\mathscr{X}} = 0, \; \mathcal{C = A}$ and $R(\mathcal A, \Sigma) = \Delta$. Then the triple $(\mathcal A, R(\mathcal A, \rm\Sigma) = \Delta, 0)$ is a right suspended category. We know that any triangulated category can be viewed as a right triangulated category. Hence any triangulated category can be viewed as a right suspended category.

(2) Let $({\mathcal{A}}, \mathcal E)$ be an exact category and

Then $({\mathcal{A}}, R(\mathcal A, \Sigma = 0), {\mathcal{A}})$ is a right suspended category.

(3) Let $({\mathcal{A}}, \mathcal E)$ be an exact category with enough injectives. We denote by $\mathcal I$ the full subcategory of all injectives objects in ${\mathcal{A}}$. Then $({\mathcal{A}}, R(\mathcal A, \Sigma = 0), \mathcal I)$ is a right suspended category, where

We collect some useful lemmas which can be used in the sequel.

Lemma 2.6. Assume $(\mathcal A, R(\mathcal C, \Sigma), {\mathscr{X}})$ be satisfies (RS1), (RS2), (RS3). If

are in $R(\mathcal C, \rm\Sigma)$, then there exists an isomorphism $\gamma\colon C\longrightarrow C ^{\prime}$ which makes the following diagram commutative:

Proof. It can be proved in a similar way as in [3, Lemma 3.2]

Lemma 2.7. Let $(\mathcal A, R(\mathcal C, \Sigma), \mathcal X)$ be a right suspended category.Given a commutative diagram

with rows in $R(\mathcal C, \Sigma)$.If $p\colon A\to A$ and $q\colon B\to B$ are idempotent morphisms, then there exists an idempotent morphism $\alpha\colon C\to C$ such that the diagram

commutes.

Proof. The proof is very similar to [2, Lemma 1.13], we omit it.

3.   Idempotent completion of right suspended categories

Let $({\mathcal{A}}, R({\mathcal{C}}, \Sigma), {\mathscr{X}})$ be a right suspended category. Then the additive endofunctor $\Sigma$ of ${\mathcal{A}}$ induces the endofunctor $\widetilde{\Sigma}$ of idempotent completion $\widetilde{{\mathcal{A}}}$ given by $\widetilde{\Sigma}(A, e) = (\Sigma A, \Sigma e)$. Moreover, it is easy to see that there is a commutative diagram

Clearly, $\ell_{{\mathcal{A}}}({\mathcal{C}})\subseteq \widetilde{{\mathcal{C}}}$, and $\ell_{{\mathcal{A}}}({\mathscr{X}})\subseteq \widetilde{{\mathscr{X}}}$.

We define a right $\widetilde{\Sigma}$-sequence in $\widetilde{{\mathcal{A}}}$,

to be a right $\widetilde{{\mathcal{C}}}$-sequence in $R(\widetilde{{\mathcal{C}}}, \widetilde{\Sigma})$ if there is a right $\widetilde{{\mathcal{C}}}$-sequence in $R(\widetilde{{\mathcal{C}}}, \widetilde{\Sigma})$

such that $\Delta\oplus \Delta'$ is isomorphic to a right ${\mathcal{C}}$-sequence in $R({\mathcal{C}}, \Sigma)$ or equivalently, it is a direct summand of a right ${\mathcal{C}}$-sequence in $R({\mathcal{C}}, \Sigma)$. It is easy to see that $R(\widetilde{{\mathcal{C}}}, \widetilde{\Sigma})$ is closed under isomorphisms and finite direct sums. For convenience, we usually write $\widetilde{\rm\Sigma}$ as $\rm\Sigma$.

Lemma 3.1. Let $(\mathcal A, R(\mathcal C, \rm\Sigma), {\mathscr{X}} = 0)$ be a right suspended category. A sequence

is a right ${\mathcal{C}}$-sequence in $R(\mathcal C, \rm{\Sigma})$ if and only if both two sequences

are right ${\mathcal{C}}$-sequences in $R(\mathcal C, \rm{\Sigma})$.

Proof. Since $R({\mathcal{C}}, \Sigma)$ is closed under finite direct sums, it is enough to show the necessity. By axiom (RS1), there are two right ${\mathcal{C}}$-sequences in $R({\mathcal{C}}, \Sigma)$

By axiom (RS3), there exists a commutative diagram

Thus, we have $fy = a$ and $bf = z$. Similarly, one can find a morphism $f':C'\rightarrow C'_{1}$ such that $f'y' = a'$ and and $b'f' = z'$. Hence, we have the following commutative diagram

By Lemma 2.6, we know that $\left(\begin{smallmatrix} f & 0\\ 0 & f' \end{smallmatrix} \right)$ is an isomorphism. It follows that $f$ and $f'$ are isomorphisms. It is easy to see that there exists a commutative diagram

where the second row lies in $R({\mathcal{C}}, \Sigma)$. It follows that $A \xrightarrow{x} B\xrightarrow{y} C\xrightarrow{z} \Sigma A$ lies in $R({\mathcal{C}}, \Sigma)$. Similarly, we can show that $A' \xrightarrow{x'} B'\xrightarrow{y'} C'\xrightarrow{z'} \Sigma A'$ lies in $R({\mathcal{C}}, \Sigma)$.

Now we state and prove our main result in this article.

Theorem 3.2. Let $\rm\Sigma$ be an endofunctor when restricted to $\mathcal C, \; (\mathcal A, R(\mathcal C, \rm\Sigma), {\mathscr{X}} = 0)$ be a right suspended category. Then the triple $(\widetilde{ \mathcal A}, R({\widetilde{ \mathcal C}, \widetilde{\rm\Sigma}}), \widetilde{{\mathscr{X}}} = 0)$ is a right suspended category.

Proof. We will check the axioms of suspended categories.

(RS1) (a) Let $A$ be an arbitrary object in $\widetilde{ \mathcal C}$. Then there is $A ^{\prime}$ in $\widetilde{ \mathcal C}$ such that $A\oplus {A ^{\prime}}\in \mathcal C$ actually, if $A = (N, e)$ take $A' = (N, id_N-e)$ we have $A\oplus A'\cong\ell_{{\mathcal{A}}}(N)$). Note that $A \oplus A^{\prime} \stackrel{0}{\longrightarrow} 0 \longrightarrow 0 \longrightarrow \Sigma\left(A \oplus A^{\prime}\right)$ is a right ${ \mathcal C}$-sequence in $R({ \mathcal C}, \rm\Sigma)$. It is clear that $0$ is an ${\mathscr{X}}$-preenvelope. By the definition of right $\widetilde{ \mathcal C}$-sequences in $\widetilde{ \mathcal A}$, we obtain $A \stackrel{0}{\longrightarrow} 0 \longrightarrow 0 \longrightarrow \Sigma(A)$ with $0$ is an $\widetilde{{\mathscr{X}}}$-preenvelope.

For any morphism $f\colon A\to B$ in $\widetilde{ \mathcal C}$, there exists two objects $A ^{\prime}, B ^{\prime}\in\widetilde{ \mathcal C}$ such that $A\oplus A ^{\prime}, B\oplus B ^{\prime}\in \mathcal C$. For the morphism $A\oplus A ^{\prime}\xrightarrow{\left(\begin{smallmatrix} f & 0\\ 0 & 0 \end{smallmatrix}\right)}B\oplus B ^{\prime}$ in $\mathcal C$, by axiom (RS1)(a), there exists a right ${ \mathcal C}$-sequence

in $R(\mathcal C, \rm\Sigma)$. By Lemma 2.7, there exists an idempotent morphism $p = p^2\colon N\to N$ which makes the following diagram commutative:

Therefore, the sequence $A\xrightarrow{f}B\xrightarrow{pa_1}(N, p)\xrightarrow{a_2p}\Sigma (A)$ is in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$.

(b) For each morphism $f\colon A\to B$ in $\widetilde{ \mathcal C}$, there are two objects $A ^{\prime}, B ^{\prime}\in\widetilde{ \mathcal C}$ such that $A\oplus A ^{\prime}, B\oplus B ^{\prime}\in \mathcal C$. For the morphism $A\oplus A ^{\prime}\xrightarrow{\left(\begin{smallmatrix} f & 0\\ 0 & 0 \end{smallmatrix}\right)}B\oplus B ^{\prime}$ in $\mathcal C$, by axiom (RS1)(b), there is a right ${ \mathcal C}$-sequence in $R({ \mathcal C}, \rm\Sigma)$

which guarantees

is a right $\widetilde{ \mathcal C}$-sequence in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$.

(RS2) For any two right $\widetilde{ \mathcal C}$-sequences

lies in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$. For any commutative diagram

with $\alpha$ factors through $f$. Next we will prove $\gamma = 0$, thus we are done.

By the definition of right $\widetilde{ \mathcal C}$-sequences, there are two right $\widetilde{ \mathcal C}$-sequences

lie in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$. Taking the direct sum of right $\widetilde{ \mathcal C}$-sequences (3.2) and (3.4), we get a right $\widetilde{ \mathcal C}$-sequence

in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$ such that (3.6) is isomorphic to a right ${\mathcal{C}}$-sequence in $R({ \mathcal C}, {\rm\Sigma})$.

Similarly, taking the direct sum of right $\widetilde{ \mathcal C}$-sequences (3.3) and (3.5), we get a right $\widetilde{ \mathcal C}$-sequence

in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$ such that (3.7) is isomorphic to a right ${ \mathcal C}$-sequence in $R({ \mathcal C}, {\rm\Sigma})$. Thus we have a commutative diagram in $R({ \mathcal C}, {\rm\Sigma})$

Note that ${\left(\begin{smallmatrix} \alpha & 0\\ 0 & 0 \end{smallmatrix}\right)}$ factors through ${\left(\begin{smallmatrix} f & 0\\ 0&f ^{\prime} \end{smallmatrix}\right)}$ since $\alpha$ factors through $f$, hence ${\left(\begin{smallmatrix} \gamma & 0\\ 0 & 0 \end{smallmatrix}\right)}$ factors through ${\left(\begin{smallmatrix} 0 & 0\\ 0&m ^{\prime} \end{smallmatrix}\right)}$. In particular, we have

which implies $\gamma = 0$.

(RS3) For any two right $\widetilde{ \mathcal C}$-sequences

in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$, the diagram below with the leftmost square is commutative

Next we will prove that there exists a morphism $\gamma\colon C\longrightarrow Z$ which makes the whole diagram commutative in $\widetilde{ \mathcal A}$. By the definition of right $\widetilde{ \mathcal C}$-sequences, there exists right ${ \mathcal C}$-sequence $\Delta ^{\prime}, \Gamma ^{\prime}$ and morphisms $i:\Delta\longrightarrow \Delta ^{\prime}, p:\Delta ^{\prime}\longrightarrow \Delta, j:\Gamma\longrightarrow \Gamma ^{\prime}, q:\Gamma ^{\prime}\longrightarrow \Gamma$, such that $pi = 1_\Delta, qj = 1_\Gamma$, which induce a morphism $j\circ(\alpha, \beta)\circ p:\Delta ^{\prime}\longrightarrow \Gamma ^{\prime}$ in ${ \mathcal A}$, since $\Delta ^{\prime}$ and $\Gamma ^{\prime}$ are right ${ \mathcal C}$-sequence in $R({ \mathcal C}, \rm\Sigma)$. According to axiom (RS3), we have a right ${ \mathcal C}$-sequence map $u:\Delta ^{\prime}\longrightarrow\Gamma ^{\prime}$, which induces a right ${ \mathcal C}$-sequence morphism $q\circ u\circ i:\Delta\longrightarrow\Gamma$ extending $(\alpha, \beta)$ in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$.

(RS4) For any three right $\widetilde{ \mathcal C}$-sequence $A \stackrel{f}{\longrightarrow} B \stackrel{l}{\longrightarrow} D \stackrel{i}{\longrightarrow} \Sigma(A), B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} E \stackrel{j}{\longrightarrow} \Sigma(B)$ and $A \stackrel{g f}{\longrightarrow} C \stackrel{k}{\longrightarrow} E \stackrel{m}{\longrightarrow} \Sigma(A)$ are in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$, with $f, g$ are ${\mathscr{X}}$-monics in $\widetilde{ \mathcal C}$. For the morphism $f:A\longrightarrow B$ in $\widetilde{ \mathcal C}$, there exists $A ^{\prime}, B ^{\prime}$ in $\widetilde{ \mathcal C}$, such that $A\oplus A ^{\prime}, B\oplus B ^{\prime}$ in $\mathcal C$, Clearly

and

are right $\widetilde{ \mathcal C}$-sequences in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$. Take the direct sum of right triangles (3.8)–(3.10), we get the following right $\widetilde{ \mathcal C}$-sequence:

By the proof of (RS1), we know that any morphism in $\mathcal C$ can be embedded into a right ${ \mathcal C}$-sequence, since the morphism $A\oplus A ^{\prime}\xrightarrow{\left(\begin{smallmatrix} f & 0\\ 0 & 0 \end{smallmatrix}\right)}B\oplus B ^{\prime}$ in $\mathcal C$, therefore, it can be extended to a right ${ \mathcal C}$-sequence (3.1). By Lemma 2.6, (3.11) is isomorphic to (3.1) in $R({ \mathcal C}, {\rm\Sigma})$.

Similarly, the following right $\widetilde{ \mathcal C}$-sequence

is isomorphic to a right $\mathcal C$-sequence in $R(\mathcal C, \rm\Sigma)$. Since the morphism $gf\colon A\to C$ in $\widetilde{ \mathcal C}$, similar to above, the following right $\widetilde{ \mathcal C}$-sequence

is isomorphic to a right $\mathcal C$-sequence in $R(\mathcal C, \rm\Sigma)$.

By axiom (RS4), we can get the following commutative diagram in ${ \mathcal A}$:

where the third column is a right ${ \mathcal C}$-sequence in $R({ \mathcal C}, \rm\Sigma)$ and $h_1$ is an ${{\mathscr{X}}}$-monic.

We write

According to the above commutative diagram, we have

Hence

According to $h_2\circ h_1 = 0$, we have

Thus we obtain

For the object $F\oplus C ^{\prime}\oplus\Sigma (A ^{\prime})$, there are morphisms $u, v\colon F\oplus C ^{\prime}\oplus\Sigma (A ^{\prime})\longrightarrow F\oplus C ^{\prime}\oplus\Sigma (A ^{\prime})$ where

such that $u$ and $v$ are inverse of each other. Therefore we can get a commutative diagram as follows:

Note that

we obtain the right $\widetilde{ \mathcal C}$-sequence $D\xrightarrow{a_{11}}F\xrightarrow{b_{11}}E\xrightarrow{\Sigma(l)\circ j}\Sigma (D)$ in $R(\widetilde{ \mathcal C}, \widetilde{\rm\Sigma})$.

Therefore, we can get the following commutative diagram in $\widetilde{ \mathcal A}$:

where $a_{11}$ is an $\widetilde{{\mathscr{X}}}$-monic.

This completes the proof.

Remark 3.3. In Theorem 3.2, when ${\mathcal{A}} = {\mathcal{C}}$ is a triangulated category, it is just Theorem 1.5 in ^[2]; when ${\mathcal{A}} = {\mathcal{C}}$ is an exact category, it is just Proposition 6.13 in ^[1]; when ${\mathcal{A}} = {\mathcal{C}}$ is a right triangulated category, it is just Theorem 2.14 in ^[4].

4.   Conclusions

In this article, we show that the idempotent completion of a right suspended category has a natural structure of right suspended category and dually this is true for a left suspended category.

Acknowledgments

The author would like to thank the anonymous reviewers for their comments and suggestions.

Conflict of interest

The author declares no conflict of interests.