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Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies" (so, we have a full functor $Diff(\mathcal{A})\to Diff'(\mathcal{A})$ for the category $Diff'(A)$ that I want to describe; there also exists a functor $Diff'(\mathcal{A})\to K(\mathcal{A})$). The shift functor for $Diff'(\mathcal{A})$ may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff'(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?

P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff'(\mathcal{A})$ is (Heller) triangulated.

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies" (so, we have a full functor $Diff(\mathcal{A})\to Diff'(\mathcal{A})$ for the category $Diff'(A)$ that I want to describe; there also exists a functor $Diff'(\mathcal{A})\to K(\mathcal{A})$). The shift functor for $Diff'(\mathcal{A})$ may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff'(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?

P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff'(\mathcal{A})$ is (Heller) triangulated.

P.S. The proof appears to be a rather simple application of differential graded arguments. However, I wonder whether this construction and its relation to periodic derived categories was considered somewhere.

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies". The shift functor for differential objects may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?

P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff(\mathcal{A})$ is (Heller) triangulated.

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies" (so, we have a full functor $Diff(\mathcal{A})\to Diff'(\mathcal{A})$ for the category $Diff'(A)$ that I want to describe; there also exists a functor $Diff'(\mathcal{A})\to K(\mathcal{A})$). The shift functor for $Diff'(\mathcal{A})$ may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff'(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?

P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff'(\mathcal{A})$ is (Heller) triangulated.

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies". The shift functor for differential objects may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category obtained this way triangulated? Did anybody study it or apply it (note that it is $2$-periodic and "almost $1$-periodic")?

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\to} A\stackrel{-a}{\to}A\stackrel{a}{\to}A\stackrel{-a}{\to}A\to \dots$. Then one can consider "shift-stable" morphisms between complexes of this sort up to "shift-stable homotopies". The shift functor for differential objects may be defined as $(A,a)[1]=(A,-a)$, and it is very easy to describe cones of morphisms. My question is: is the category $Diff(\mathcal{A})$ obtained this way triangulated? Did anybody study it or apply it somehow (note that it is $2$-periodic and "almost $1$-periodic")?

P.S. I suspect that the same arguments that prove that the homotopy category $K(\mathcal{A})$ is triangulated (and also Heller triangulated) also yield that $Diff(\mathcal{A})$ is (Heller) triangulated.